%!PS-Adobe-2.0 %%Creator: dvips 5.482 Copyright 1986-92 Radical Eye Software %%Title: 3surj.dvi %%Pages: 18 -1 %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips 3surj.dvi %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N} B /TR{translate}N /isls false N /vsize 11 72 mul N /@rigin{isls{[0 -1 1 0 0 0] concat}if 72 Resolution div 72 VResolution div neg scale isls{Resolution hsize -72 div mul 0 TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{ CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N} B /I{cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval (Display)eq}{pop false}ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{ /SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 40258431 52099146 1000 300 300 @启动/FA 3 104 df0 D<<33000、38、00、38、00、38、00、38、38、0、38、38、0 0 0 0 0 0 0 0 0 0 0 0 38 38 0 0 0 0 38 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0I E /Fb 28 118 df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dII<03F0180603607390076018180E0001C0E1C0E01C0E1C0E01C0E1001C0600 1807803300 1806000 E1C000 03F000 012127F91 15 I II114 D<1F9030F07F07F807F03FEF0FF00F880,38 80180C018008D0608FC0D127F9110> I< 0400400C01C01C03C00 FFE01C C 01C000 1C000 1C000 1C000 1C000 1C000 1C000 1C101C101C101C101C1001C100C100E1003C00 C1A7F99 10> I iI E /Fc 4 55 df<0C001C00EC000C000C 000C000C000C000C000C000C000C000C000C000C000C000C000C00FFC00A137D9211>49 D<1F0060C06060F070F030603000700070006000C001C00180020004000810101020207FE0FFE0 0C137E9211>I<0FC030707038703870380038003000E00FC0007000380018001C601CF01CF018 E03860701FC00E137F9211>I<07C00C201070207060006000C000CF00D0C0E060C020C030C030 C03040306020206010C00F000C137E9211>54 D E /Fd 6 111 df<60F0F07010101010202040 80040C7C830C>59 D<00000C0000000C0000001C0000001C0000003C0000007C0000005C000000 9C0000008E0000010E0000010E0000020E0000040E0000040E0000080E0000080E0000100E0000 200E00003FFE000040070000400700008007000100070001000700020007000200070006000700 1E000700FF807FF01C1D7F9C1F>65 D<01FFC3FF80003C00780000380070000038007000003800 70000038007000007000E000007000E000007000E000007000E00000E001C00000E001C00000E0 01C00000FFFFC00001C003800001C003800001C003800001C00380000380070000038007000003 80070000038007000007000E000007000E000007000E000007000E00000F001E0000FFE1FFC000 211C7E9B23>72 D<000FC100303300400F00800601800603000603000606000406000407000007 000007800003F00001FF0000FFC0003FE00003E00000F000007000003000003020003020003060 00606000606000C0600080F00300CC060083F800181E7E9C19>83 D<007180018B800307800607 800E07000C07001C07001C0700380E00380E00380E00380E00381C00381C00381C00183C0008F8 00073800003800003800007000607000F06000F0E000E180007E0000111A7F9114>103 D<381F004E61804681C04701C08F01C08E01C00E01C00E01C01C03801C03801C03801C07003807 10380710380E10380E2070064030038014127E9119>110 D E /Fe 2 55 df<1F00318060C04040C060C060C060C060C060C060C060C060404060C031801F000B107F8F0F> 48 D<0780184030C060C06000C000CF00F080E040C060C060C060406060C030801F000B107F8F 0F>54 D E /Ff 8 113 df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g 9 89 df<00200040008001000300060006000C000C0018001800380030003000700070 0070006000E000E000E000E000E000E000E000E000E000E000E00004004006000 06000 03009000180900180100C1001C01C000 0C000 0E000 0E000 0E000 0E000 0E000 0E000 0E000 0E000 0E000 0C000 1C000 1C000 1800 00 1800 38 00 3 000 000 3000 0600 0000 C000 0C000 18000 1000 000 400 0800 00 B317F8113> I E3000 0000 07000 000 7000 0700020μdI24 dI<00001800003000 00600000E00000C0000180000380000700000600000E00000C00001C0000380000380000700000 700000E00000E00001E00001C00001C0000380000380000380000780000700000700000F00000E 00000E00001E00001E00001E00001C00001C00003C00003C00003C00003C000038000078000078 0000780000780000780000780000780000780000700000F00000F00000F00000F00000F00000F0 0000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F0 0000F00000F00000F00000F00000F00000F00000F0000070000078000078000078000078000078 00007800007800007800003800003C00003C00003C00003C00001C00001C00001E00001E00001E 00000E00000E00000F000007000007000007800003800003800003800001C00001C00001E00000 E00000E000007000007000003800003800001C00000C00000E0000060000070000038000018000 00C00000E0000060000030000018157C768121>32 DI88μE/FH 20 113 DF0 D<70F8F8F87005057C8D0D>I<400004 C0000C6000183000301800600C00C006018003030001860000CC00007800003000003000007800 00CC000186000303000601800C00C0180060300030600018C0000C40000416187A9623>I<0002 000000060000000600000006000000060000000600000006000000060000000600000006000000 06000000060000000600000006000000060000FFFFFFF0FFFFFFF0000600000006000000060000 000600000006000000060000000600000006000000060000000600000006000000060000000600 00FFFFFFF0FFFFFFF01C207D9E23>6 D<003FFFC000FFFFC003C00000070000000C0000001800 000030000000300000006000000060000000C0000000C0000000C0000000C0000000C0000000C0 000000C000000060000000600000003000000030000000180000000C0000000700000003C00000 00FFFFC0003FFFC0000000000000000000000000000000000000000000000000000000007FFFFF C07FFFFFC01A247C9C23>18 D<000000C0000003C000000F0000003C000000F0000003C0000007 0000001C00000078000001E00000078000001E00000078000000E0000000780000001E00000007 80000001E0000000780000001C0000000700000003C0000000F00000003C0000000F00000003C0 000000C000000000000000000000000000000000000000亿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i 14 121 df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×D<1FC08603038038083F81E38 30838 E039 E039 E079FF1F1E100E7F8D12>97×D<0F9E183037070707070601818C02F802000 603FE03FF83FFC6EC06C00 6C000 6600 C·38 380FE010157F8D12>103 D108 dI<07C01830186C600 CEE900EE900EE900EE00 E701C301818>300 7C000 F0E7F8D12>111 D120 D E /Fj 1 50 df<00C003C0FFC0FFC003C003C003C003C003C003C003C0 03C003C003C003C003C003C003C003C003C003C003C07FFE7FFE0F187D9716>49 D E /Fk 17 118 df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l 37 124 df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μd77μd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d(110)D<07E01C3300 C700 E66E00 7E00 700 E00 7E00 700 E700 7600 67 00 E38 1C1C3807E010107F8F13>I114×D<1F200C020F03F03F03F03F080708080C030C020F0408F10C107F8F0F>I<0400400C01C01C03C0FC01C01C01C01C01C01C01C01C01C01C201C201C201C201C201C200 E400 38 00 B177F960F>I123 D E /Fm 2 49 df<0C000C008C40EDC07F800C007F80EDC08C400C000C000A0B7D8B10>3 D<081C1C3838383070706060C0C0060D7E8D09>48 D E /Fn 17 119 df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o 44 122 df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μd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p 39 118 df<0018007000E001C00380038007000E000E001E001C003C003C0078007800 78007800F800F000F000F000F000F000F000F000F000F000F80078007800780078003C003C001C 001E000E000E0007000380038001C000E0007000180D2D7DA114>40 DI<387CFEFEFE7C3807077C860F>46 D<01FC0007FF001F07C01E03C03E03E0 7C01F07C01F07C01F0FC01F8FC01F8FC01F8FC01F8FC01F8FC01F8FC01F8FC01F8FC01F8FC01F8 FC01F8FC01F8FC01F87C01F07C01F07C01F03E03E01E03C01F8FC007FF0001FC00151D7E9C1A> 48 D<00E00001E0000FE000FFE000F3E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E000FFFF80FFFF80111D7C9C1A>I<07F0001FFE00383F007C1F80FE0FC0FE0FC0FE0F E0FE07E07C07E03807E0000FE0000FC0000FC0001F80001F00003E0000780000F00000E00001C0 000380600700600E00601C00E01FFFC03FFFC07FFFC0FFFFC0FFFFC0131D7D9C1A>I<01FC0007 FF000E0F801E0FC03F07E03F07E03F07E03F07E01E0FC0000FC0000F80001F0001FC0001FC0000 0F800007C00003E00003F00003F83803F87C03F8FE03F8FE03F8FE03F0FC03F07807E03C0FC01F FF8003FC00151D7E9C1A>I<0001C00003C00007C00007C0000FC0001FC0003BC00073C00063C0 00C3C00183C00383C00703C00E03C00C03C01803C03803C07003C0E003C0FFFFFEFFFFFE0007C0 0007C00007C00007C00007C00007C000FFFE00FFFE171D7F9C1A>I<3803803FFF803FFF003FFE 003FFC003FF0003F800030000030000030000030000033F80037FE003C1F00380F801007C00007 C00007E00007E07807E0FC07E0FC07E0FC07E0FC07C0780FC0600F80381F001FFC0007F000131D 7D9C1A>I<003F0001FFC007E0E00F81E01F03F01E03F03E03F07C03F07C01E07C0000FC1000FC FF00FDFFC0FD03E0FE01F0FE01F0FC01F8FC01F8FC01F8FC01F87C01F87C01F87C01F83C01F03E 01F01E03E00F07C007FF8001FE00151D7E9C1A>I<6000007FFFF87FFFF87FFFF07FFFE07FFFC0 E00180C00300C00300C00600000C0000180000380000380000780000700000F00000F00001F000 01F00001F00001F00003F00003F00003F00003F00003F00003F00001E00000C000151E7D9D1A> I<01FC0007FF000F07801E03C01C01E03C01E03C01E03E01E03F01E03FC3C01FE3801FFF000FFE 0007FF8007FFC01FFFE03C3FF0780FF07803F8F001F8F000F8F00078F00078F000707800707C00 E03E03C00FFF8003FC00151D7E9C1A>I<01FC000FFF001F07803E03C07C03E07C01E0FC01F0FC 01F0FC01F0FC01F8FC01F8FC01F8FC01F87C03F87C03F83E05F81FFDF807F9F80041F80001F03C 01F07E01F07E03E07E03E07E07C03C0780381F001FFC0007F000151D7E9C1A>I<0000E0000000 00E000000001F000000001F000000001F000000003F800000003F800000006FC00000006FC0000 000EFE0000000C7E0009000C7000 F30000 0130F1000 0701FC0000 FC000 000 FC000 000 FC000 7F0 0 FFFE00 FFFF00 00 1800 3F000 00 1800 3F000 00 3000 01 F800 03000 01F800 0600 0FC000 0F000 0FE00 FFEF0 0 FFF0FFEF0023 1F7E9E28>65μd000 01E F1000 000 E000 F0 000 000 FE000 000 FE000 000 FE000 000 FE000 00 FE000 00 FE000 00 FE000 000 E000 000 E000 000 06000 F6000 0663F000 000 C1F800 00 C0FC0 0 00 180 7E000 03803F000 07000 FE01C000 03FFF800 00 7FC11F1F7D9E26> I I<007FC003FF00EFE03DE03F00FE07E000 03E0FCF80 d82 D<03FC080FFF381E03F83800F8700078700038F00038F00018F00018 F80000FC00007FC0007FFE003FFF801FFFE00FFFF007FFF000FFF80007F80000FC00007C00003C C0003CC0003CC0003CE00038E00078F80070FE01E0E7FFC081FF00161F7D9E1D>I<7FFFFFFC7F FFFFFC7C07E07C7007E01C6007E00C6007E00CE007E00EC007E006C007E006C007E006C007E006 0007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0 000007E0000007E0000007E0000007E0000007E0000007E0000007E00003FFFFC003FFFFC01F1E 7E9D24>I91 d93×D<07FCF3F03F03C03F03E03F03E03E03C03E003E000 07EF07FE00F03E03C03E07C003 E0F803E0F803E0F803E0F05E07E0DE03FF8FE0FE07E17147F9319>97 DI<01FE0007FF801F0FC03E0FC03E0FC07C0FC07C0300FC00 00FC0000FC0000FC0000FC0000FC00007C00007E00003E00603F00C01F81C007FF0001FC001314 7E9317>I<0007F80007F80000F80000F80000F80000F80000F80000F80000F80000F80000F800 00F801F8F80FFEF81F83F83E01F87E00F87C00F87C00F8FC00F8FC00F8FC00F8FC00F8FC00F8FC 00F87C00F87C00F87E00F83E01F81F07F80FFEFF03F8FF18207E9F1D>I<01FE0007FF800F83C0 1E01E03E00F07C00F07C00F8FC00F8FFFFF8FFFFF8FC0000FC0000FC00007C00007C00003E0018 1E00180F807007FFE000FF8015147F9318>I<001F8000FFC001F3E003E7E003C7E007C7E007C3 C007C00007C00007C00007C00007C000FFFC00FFFC0007C00007C00007C00007C00007C00007C0 0007C00007C00007C00007C00007C00007C00007C00007C00007C00007C0003FFC003FFC001320 7F9F10>I104×D<1C03F07F03F03E01C000 000 000 000 000 000 000 000 FF00 FF1FF1 1F00 1F00 1 F00 1F00 1F000 1F00 1F000 1F00 1F00 1F000 1F00 1F000 1F00 F00 FFEF0FFE00 B217EA00 E>107 d109 dI<01FF03F03E0F83E0F8C07C7C07CFC77EFC07EFC07EFC07EFC00×7EFC07E7C000 7C77C3E00 F83E00 F81F83F03FF77FFC01FF00 17147F931 A> II3FFE701EF6FFC03FFC03FEF03FE07F07F07C07E07F00 6F81FEFFC C7F010147E9315> I< 01800 1800、1800、38、00、38、00、808、F803F80FCFF、F800、F800、F800、F080、F800、F800、F800、F800、F860F860F860F860F8607C03F801F00 F1D7F9C14> 114×D<0FE6I E /Fq 85 128 df<001F83E000F06E3001C078780380F8780300F0300700700007007000070070000700700007 0070000700700007007000FFFFFF80070070000700700007007000070070000700700007007000 070070000700700007007000070070000700700007007000070070000700700007007000070070 0007007000070070007FE3FF001D20809F1B>11 D<003F0000E0C001C0C00381E00701E00701E0 070000070000070000070000070000070000FFFFE00700E00700E00700E00700E00700E00700E0 0700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E07FC3FE 1720809F19>I<003FE000E0E001C1E00381E00700E00700E00700E00700E00700E00700E00700 E00700E0FFFFE00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700 E00700E00700E00700E00700E00700E00700E00700E07FE7FE1720809F19>I<001F81F80000F0 4F040001C07C06000380F80F000300F00F000700F00F0007007000000700700000070070000007 0070000007007000000700700000FFFFFFFF000700700700070070070007007007000700700700 070070070007007007000700700700070070070007007007000700700700070070070007007007 000700700700070070070007007007000700700700070070070007007007007FE3FE3FF0242080 9F26>I<07070F1C383060C00808779F17>19 D<3E004100808080808080808041003E00090874 A022>23 D<7038F87CFC7EFC7E743A0402040204020804080410081008201040200F0E7E9F17> 34 D<007800000084000001840000030200000702000007020000070200000702000007040000 0704000007080000070800000310000003A00FFC03C003E0038001C001C0008001C0010003E001 0004E0020008F00200187004003078080070380800701C1000F01E1000F00E2000F0074000F003 C0087003C0087801C010380670301C18386007E00F801E227EA023>38 D<70F8FCFC7404040408 0810102040060E7C9F0D>I<0020004000800100020006000C000C001800180030003000300070 00600060006000E000E000E000E000E000E000E000E000E000E000E000E0006000600060007000 300030003000180018000C000C000600020001000080004000200B2E7DA112>I<800040002000 100008000C00060006000300030001800180018001C000C000C000C000E000E000E000E000E000 E000E000E000E000E000E000E000C000C000C001C001800180018003000300060006000C000800 10002000400080000B2E7DA112>I<000600000006000000060000000600000006000000060000 000600000006000000060000000600000006000000060000000600000006000009000FFFFFF90006000 000 0600 000 0000 000 0600 000 000 0600 000 0600 000 000 0600 000 000 06 0000000 600 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 1 1 C207 D9A23>43 D<70F8CF40740404080101020400 60E7C840D> II<70F8F8F8700505 7C840D>I<03F0000E1C001C0E00180600380700700380700380700380700380F003C0F003C0F0 03C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C070038070038070 03807807803807001806001C0E000E1C0003F000121F7E9D17>48 D<018003800F80F380038003 800380038003800380038003800380038003800380038003800380038003800380038003800380 03800380038007C0FFFE0F1E7C9D17>I<03F0000C1C00100E00200700400780800780F007C0F8 03C0F803C0F803C02007C00007C0000780000780000F00000E00001C0000380000700000600000 C0000180000300000600400C00401800401000803FFF807FFF80FFFF80121E7E9D17>I<03F000 0C1C00100E00200F00780F80780780780780380F80000F80000F00000F00000E00001C00003800 03F000003C00000E00000F000007800007800007C02007C0F807C0F807C0F807C0F00780400780 400F00200E001C3C0003F000121F7E9D17>I<000600000600000E00000E00001E00002E00002E 00004E00008E00008E00010E00020E00020E00040E00080E00080E00100E00200E00200E00400E 00C00E00FFFFF0000E00000E00000E00000E00000E00000E00000E0000FFE0141E7F9D17>I<18 03001FFE001FFC001FF8001FE00010000010000010000010000010000010000011F000161C0018 0E001007001007800003800003800003C00003C00003C07003C0F003C0F003C0E0038040038040 0700200600100E000C380003E000121F7E9D17>I<007C000182000701000E03800C07801C0780 380300380000780000700000700000F1F000F21C00F40600F80700F80380F80380F003C0F003C0 F003C0F003C0F003C07003C07003C07003803803803807001807000C0E00061C0001F000121F7E 9D17>I<4000007FFFC07FFF807FFF804001008002008002008004000008000008000010000020 0000200000400000400000C00000C00001C0000180000380000380000380000380000780000780 00078000078000078000078000078000030000121F7D9D17>I<03F0000C0C0010060030030020 01806001806001806001807001807803003E03003F06001FC8000FF00003F80007FC000C7E0010 3F00300F806003804001C0C001C0C000C0C000C0C000C0C000806001802001001002000C0C0003 F000121F7E9D17>I<03F0000E18001C0C00380600380700700700700380F00380F00380F003C0 F003C0F003C0F003C0F003C07007C07007C03807C0180BC00E13C003E3C0000380000380000380 000700300700780600780E00700C002018001070000FC000121F7E9D17>I<70F8F8F870000000 0000000000000070F8F8F87005147C930D>I<70F8F8F8700000000000000000000070F0F8F878 080808101010202040051D7C930D>I<7FFFFFE0FFFFFFF0000000000000000000000000000000 0000000000000000000000000000000000FFFFFFF07FFFFFE01C0C7D9023>61 D<0FC0307040384038E03CF03CF03C603C0038007000E000C00180018001000300020002000200 0200020002000000000000000000000007000F800F800F8007000E207D9F15>63 D<000100000003800000038000000380000007C0000007C0000007C0000009E0000009E0000009 E0000010F0000010F0000010F00000207800002078000020780000403C0000403C0000403C0000 801E0000801E0000FFFE0001000F0001000F0001000F00020007800200078002000780040003C0 0E0003C01F0007E0FFC03FFE1F207F9F22>65 D000 03C01C03C0000 40407C000 000 40 7800 000 F800 000 F800 00 F800 00 F800 00 F800 00 F800 00 F800 00 00 F800 00 0 000 7000 000 C000 00 404 C000 000 401 C000 000 401 E000 00 800 000 7000 000 1000 38 000 000 1C0 0 0400 000 7038 000 000 0FC000 01A217D9F21> I I<00FC04000·7030C01C09C030355C07II1E000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0年1月1日10月1日1月1日24日1月1日1月1日I<00FE02000 781860E04E03082E000 7000 01 E00 F000 E01E00 00 60II0FFFC000、C000、3000、C000、3000、C000、3000、C000、3000、C000、3000、C000、3000、C000、3000、C000、3000、C000、3000、C000、3000、C000、3000、C000、3000、C000、C000、C000、C000、C000、C3000、C000、C000、C3000、C000、C000、C000、C3000、C00、F83C000、F03000、407000、407000、3000、E000、F800、02012、209、E9E1717IIII<001F800000F0F00001C0380007801E000F000F000E0007001E000780 3C0003C03C0003C07C0003E0780001E0780001E0F80001F0F80001F0F80001F0F80001F0F80001 F0F80001F0F80001F0F80001F0F80001F0780001E07C0003E07C0003E03C0003C03C0003C01E00 07800E0007000F000F0007801E0001C0380000F0F000001F80001C217D9F23>II82 D<07E0800C1980100780300380 600180600180E00180E00080E00080E00080F00000F000007800007F00003FF0001FFC000FFE00 03FF00001F800007800003C00003C00001C08001C08001C08001C08001C0C00180C00380E00300 F00600CE0C0081F80012217D9F19>I<7FFFFFE0780F01E0600F0060400F0020400F0020C00F00 30800F0010800F0010800F0010800F0010000F0000000F0000000F0000000F0000000F0000000F 0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F000000 0F0000000F0000000F0000000F0000001F800007FFFE001C1F7E9E21>III3C000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 E40 0 00000 F800 0 00 F800 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 77 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 77 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I<7FF83FF80FE0FC007C000 7000I.7FFFF09000E0C03C03C0C0 03C0800、F8000、F8000、E000 000、E000 000、E000、3000、C000、700、F800、F800、F00、00、E000、1E000、3、C000 403C000 407F4040F10C1E000 0C1E000 083C000 187800、C80187800、F801F8FFFF8161F7D9E、1C9II0804100201020104020408080408040B8CFC7EFC7E7C3E131C0F0E7B9F17> II<1FE000303000781800781C00300E 00000E00000E00000E0000FE00078E001E0E00380E00780E00F00E10F00E10F00E10F01E10781E 103867200F83C014147E9317>97 D<0E0000FE00000E00000E00000E00000E00000E00000E0000 0E00000E00000E00000E00000E3E000EC3800F01C00F00E00E00E00E00700E00700E00780E0078 0E00780E00780E00780E00780E00700E00700E00E00F00E00D01C00CC300083E0015207F9F19> I<03F80E0C1C1E381E380C70007000F000F000F000F000F000F00070007000380138011C020E0C 03F010147E9314>I<000380003F80000380000380000380000380000380000380000380000380 00038000038003E380061B801C0780380380380380700380700380F00380F00380F00380F00380 F00380F003807003807003803803803807801C07800E1B8003E3F815207E9F19>I<03F0000E1C 001C0E00380700380700700700700380F00380F00380FFFF80F00000F00000F000007000007000 003800801800800C010007060001F80011147F9314>I<007C00C6018F038F0706070007000700 0700070007000700FFF00700070007000700070007000700070007000700070007000700070007 000700070007007FF01020809F0E>I<0000E003E3300E3C301C1C30380E00780F00780F00780F 00780F00780F00380E001C1C001E380033E0002000002000003000003000003FFE001FFF800FFF C03001E0600070C00030C00030C00030C000306000603000C01C038003FC00141F7F9417>I<0E 0000FE00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E3E000E 43000E81800F01C00F01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E 01C00E01C00E01C00E01C00E01C0FFE7FC16207F9F19>I<1C003E003E003E001C000000000000 000000000000000E007E000E000E000E000E000E000E000E000E000E000E000E000E000E000E00 0E000E000E00FFC00A1F809E0C>I<00E001F001F001F000E00000000000000000000000000070 07F000F00070007000700070007000700070007000700070007000700070007000700070007000 700070007000706070F060F0C061803F000C28829E0E>I<0E0000FE00000E00000E00000E0000 0E00000E00000E00000E00000E00000E00000E00000E0FF00E03C00E03000E02000E04000E0800 0E10000E30000E70000EF8000F38000E1C000E1E000E0E000E07000E07800E03800E03C00E03E0 FFCFF815207F9F18>I<0E00FE000E000E000E000E000E000E000E000E000E000E000E000E000E 000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E00FFE00B20809F 0C>I<0E1F01F000FE618618000E81C81C000F00F00E000F00F00E000E00E00E000E00E00E000E 00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E00 0E00E00E000E00E00E000E00E00E000E00E00E00FFE7FE7FE023147F9326>I<0E3E00FE43000E 81800F01C00F01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E 01C00E01C00E01C00E01C0FFE7FC16147F9319>I<01F800070E001C03803801C03801C07000E0 7000E0F000F0F000F0F000F0F000F0F000F0F000F07000E07000E03801C03801C01C0380070E00 01F80014147F9317>I<0E3E00FEC3800F01C00F00E00E00E00E00F00E00700E00780E00780E00 780E00780E00780E00780E00700E00F00E00E00F01E00F01C00EC3000E3E000E00000E00000E00 000E00000E00000E00000E00000E0000FFE000151D7F9319>I<03E0800619801C05803C078038 0380780380700380F00380F00380F00380F00380F00380F003807003807803803803803807801C 0B800E138003E380000380000380000380000380000380000380000380000380003FF8151D7E93 18>I<0E78FE8C0F1E0F1E0F0C0E000E000E000E000E000E000E000E000E000E000E000E000E00 0E00FFE00F147F9312>I<1F9030704030C010C010C010E00078007F803FE00FF0007080388018 8018C018C018E030D0608F800D147E9312>I<0200020002000600E00E080E080E080E1008031001100E1D1C7F 9B12> I< 0E01C0E01C0E01C0E01C0E01C0E01C0E01C0E01C0E01C0E01C0E01 01 C0E01C0E01C0E01C0E01C0E03C0603C0603C030DC000 1FC1FC16147F9319FI I 060E00E03EFFF8080E00II<7FC3F01E0701C00 0701801001C000 E400 00 EC000 07000 38 000 000 3C000 00 4000 E000 00 8000 0107000 030380 0 0201C0601E01E01E0FF007FE171480931818II<3FFF380E>E201201C4038 7407000 E00 1E00 1C00 38 700 800 7010 E011E011C033 8027 767 0 00 EFFE10147F 9314I<30307878F87C787830300E057C9E17>127 D E /Fr 64 123 df<00003F03E00000C386700001878CF00003879CF000030318600007003800 00070038000007003800000E003800000E007000000E007000000E00700000FFFFFF80001C0070 00001C00E000001C00E000001C00E000001C00E000003800E000003801C000003801C000003801 C000003801C000007001C0000070038000007003800000700380000070038000006003800000E0 07000000E007000000E007000000E007000000C006000001C00E000001C00E000031860C000079 8F180000F31E100000620C6000003C07C000002429829F1C>11 D<00003FE00000E01000018038 000380780003007800070030000700000007000000070000000E0000000E0000000E000000FFFF E0000E00E0001C01C0001C01C0001C01C0001C01C0001C03800038038000380380003803800038 070000380700007007000070071000700E2000700E2000700E2000E00E2000E0064000E0038000 E0000000C0000001C0000001C000003180000079800000F3000000620000003C0000001D29829F 1A>I<000100020004000800100020006000C0018001800300070006000E000C001C0018003800 380030007000700060006000E000E000C000C000C000C000C000C000C000C000C000C000C000C0 00C0004000600060002000100010000800102E79A113>40 D<0010000008000004000006000002 000003000003000003000001000001800001800001800001800001800001800001800003800003 80000380000300000300000300000700000700000600000600000E00000C00000C00001C000018 0000380000300000700000600000E00000C0000180000100000300000600000C00001800003000 00600000800000112E80A113>I<1C3C3C3C3C040408081020204080060E7D840E>44 D<7FF0FFE07FE00C037D8A10>I<70F8F8F0E005057B840E>I<000F800030E000E07001C0700380 300380380700380F00780F00780E00781E00781E00703C00F03C00F03C00F03C00F07801E07801 E07801E07801C07003C0F003C0F00380F00780F00700700700700E00701C003038001870000FC0 00151F7C9D17>48 D<000200020006000E003C00DC031C001C0038003800380038007000700070 007000E000E000E000E001C001C001C001C003800380038003800780FFF80F1E7B9D17>I<001F 000061800080E00100E00200700220700420700410700820F00820F00820F00840E00881E00703 C0000380000700000C000018000060000080000300000400000800401000401000802001807E03 0047FF0041FE0080FC00807800141F7C9D17>I<001F800060E000807001003002003804203804 20380410380420700460700380600000E00001C000030000FE00001C0000060000070000078000 0780000780300780780780780780F00F00800F00401E00401C0040380020E0001F8000151F7C9D 17>I<0000600000E00000E00000E00001C00001C00001C0000380000380000300000700000700 000600000E00000C0000180000180000300000300000630000C700008700010700030700060E00 040E00080E003F8E00607C00801FC0001C00001C00003800003800003800003800007000007000 00600013277E9D17>I<00C06000FFC001FF8001FE000100000100000200000200000200000200 00040000047800058C000606000C0700080700000780000780000780000780000F00700F00F00F 00F00E00E01E00801C0080380080300040600061C0001F0000131F7B9D17>I<0007C0001C2000 30200060E000C1E00181E00380C00700000F00000E00001E00001E78001D84003E06003E07003C 07007C0780780780780780780780700F00700F00F00F00F00E00F01E00701C00601C0070380030 700010C0000F8000131F7B9D17>I<08E0100BF01017F8201FF8603E19C0380E80200080600100 400300800300000600000E00000C00001C00001C0000380000380000700000700000F00000F000 01E00001E00001E00003C00003C00003C00007C000078000078000030000141F799D17>I<001F 000061800080C00100600300600600600600600600600E00C00F00800F818007C30007E40003F8 0001F80003FC00047E00183F00300F00200700600700C00300C00300C00300800600800600C00C 00C008004030003060001F8000131F7B9D17>I<00000200000006000000060000000E0000001E 0000001E0000003F0000002F0000004F0000004F0000008F0000010F0000010F0000020F000002 0F0000040F00000C0F0000080F0000100F0000100F0000200F80003FFF800040078000C0078000 80078001000780010007800200078002000780060007801E000F80FF807FF81D207E9F22>65 D<01FFFFC0001E00F0001E0078001E0038001E003C003C003C003C003C003C003C003C003C0078 007800780078007800F0007801E000F0078000FFFE0000F00F8000F003C001E001C001E001E001 E001E001E001E003C001E003C001E003C001E003C001C0078003C00780078007800F0007801E00 0F007800FFFFE0001E1F7D9E20>I<0000FE0200078186001C004C0038003C0060003C00C0001C 01C0001803800018070000180F0000181E0000101E0000103C0000003C00000078000000780000 007800000078000000F0000000F0000000F0000000F0000000F000008070000080700000807000 01003800010038000200180004000C001800060020000381C00000FE00001F217A9F21>I<01FF FF80001E00E0001E0070001E0038001E001C003C001C003C000E003C000E003C000E0078000E00 78000E0078000E0078000E00F0001E00F0001E00F0001E00F0001E01E0003C01E0003C01E0003C 01E0007803C0007003C0007003C000E003C001C0078001C00780038007800E0007801C000F0070 00FFFFC0001F1F7D9E22>I<01FFFFFE001E001C001E000C001E0004001E0004003C0004003C00 04003C0004003C00040078080800780800007808000078180000F0300000FFF00000F0300000F0 300001E0200001E0200001E0200001E0001003C0002003C0002003C0004003C000400780008007 80018007800100078007000F001F00FFFFFE001F1F7D9E1F>I<01FFFFFC001E0038001E001800 1E0008001E0008003C0008003C0008003C0008003C000800780010007808000078080000780800 00F0100000F0300000FFF00000F0300001E0200001E0200001E0200001E0200003C0000003C000 0003C0000003C00000078000000780000007800000078000000F800000FFF800001E1F7D9E1E> I<0000FC040007030C001C00980030007800E0007801C000380380003003800030070000300E00 00301E0000201E0000203C0000003C00000078000000780000007800000078000000F0000000F0 00FFF0F0000780F0000780F0000F0070000F0070000F0070000F0070001E0038001E0018003E00 1C002E000E00CC000383040000FC00001E217A9F23>I<01FFF3FFE0001F003E00001E003C0000 1E003C00001E003C00003C007800003C007800003C007800003C007800007800F000007800F000 007800F000007800F00000F001E00000FFFFE00000F001E00000F001E00001E003C00001E003C0 0001E003C00001E003C00003C007800003C007800003C007800003C007800007800F000007800F 000007800F000007800F00000F801F0000FFF1FFE000231F7D9E22>I<01FFF0001F00001E0000 1E00001E00003C00003C00003C00003C0000780000780000780000780000F00000F00000F00000 F00001E00001E00001E00001E00003C00003C00003C00003C0000780000780000780000780000F 8000FFF800141F7D9E12>I<001FFF0000F80000F00000F00000F00001E00001E00001E00001E0 0003C00003C00003C00003C0000780000780000780000780000F00000F00000F00000F00001E00 001E00301E00781E00F83C00F83C00F0780080700040E00021C0001F000018207D9E18>I<01FF F800001F0000001E0000001E0000001E0000003C0000003C0000003C0000003C00000078000000 780000007800000078000000F0000000F0000000F0000000F0000001E0000001E0000001E00000 01E0008003C0010003C0010003C0030003C00200078006000780060007800C0007801C000F0078 00FFFFF800191F7D9E1D>76 D<01FE00007FC0001E0000FC00001E0000F8000017000178000017 0001780000270002F00000270004F00000270004F00000270008F00000470009E00000470011E0 0000470021E00000470021E00000870043C00000838043C00000838083C00000838083C0000103 810780000103820780000103820780000103840780000203840F00000203880F00000203900F00 000203900F00000401E01E00000401E01E00000401C01E00000C01801E00001C01803E0000FF81 03FFC0002A1F7D9E29>I<01FF007FE0001F000F00001F00040000178004000017800400002780 08000023C008000023C008000023C008000041E010000041E010000041F010000040F010000080 F0200000807820000080782000008078200001003C400001003C400001003C400001001E400002 001E800002001E800002000F800002000F800004000F0000040007000004000700000C00070000 1C00020000FF80020000231F7D9E22>I<0001FC0000070700001C01C0003000E000E0006001C0 00700380007007800038070000380E0000381E0000381C0000383C0000383C0000387800007878 0000787800007878000078F00000F0F00000F0F00000E0F00001E0F00001C0F00003C070000380 7000070078000F0038001E0038003C001C0070000E00E0000783800001FC00001D217A9F23>I< 01FFFF80001E00E0001E0070001E0038001E003C003C003C003C003C003C003C003C003C007800 7800780078007800F0007800E000F003C000F00F0000FFFC0000F0000001E0000001E0000001E0 000001E0000003C0000003C0000003C0000003C00000078000000780000007800000078000000F 800000FFF000001E1F7D9E1F>I<0001FC0000070700001C01C0003000E000E000E001C0007003 80007007800078070000380F0000381E0000381E0000383C0000383C0000787800007878000078 7800007878000078F00000F0F00000F0F00000E0F00001E0F00001C0F00003C070000380701C07 0070200F0038411E0038413C001C4170000E41E0000743808001FD008000010180000101000003 8300000386000003FE000003FC000001F8000000F0001D297A9F23>I<01FFFF00001E03C0001E 00E0001E0070001E0078003C0078003C0078003C0078003C0078007800F0007800F0007801E000 7801C000F0070000F01E0000FFF00000F0380001E01C0001E01E0001E00E0001E00F0003C01E00 03C01E0003C01E0003C01E0007803C0007803C0807803C0807803C100F801C10FFF00C20000007 C01D207D9E21>I<0007E040001C18C0003005800060038000C0038001C0018001800100038001 0003800100038001000380000003C0000003C0000003F8000001FF800001FFE000007FF000001F F0000001F800000078000000780000003800000038002000380020003800200030006000700060 0060006000E0007000C000E8038000C606000081F800001A217D9F1A>I<0FFFFFF01E0780E018 0780201007802020078020200F0020600F0020400F0020400F0020801E0040001E0000001E0000 001E0000003C0000003C0000003C0000003C00000078000000780000007800000078000000F000 0000F0000000F0000000F0000001E0000001E0000001E0000001E0000003E00000FFFF00001C1F 789E21>I<007FFFF800FC00F000E001E000C003C0008007800180078001000F0001001E000100 3C00020078000000F8000000F0000001E0000003C00000078000000F0000000F0000001E000000 3C00000078010000F0010001F0020001E0020003C00600078004000F000C001E0008003E001800 3C0038007801F000FFFFF0001D1F7D9E1C>90 D<001FC0001FC000180000180000300000300000 300000300000600000600000600000600000C00000C00000C00000C00001800001800001800001 80000300000300000300000300000600000600000600000600000C00000C00000C00000C000018 00001800001800001800003000003000003000003000006000006000006000007F0000FE000012 2D7EA10E>I<001FC0001FC00000C00000C0000180000180000180000180000300000300000300 000300000600000600000600000600000C00000C00000C00000C00001800001800001800001800 00300000300000300000300000600000600000600000600000C00000C00000C00000C000018000 0180000180000180000300000300000300007F0000FE0000122D82A10E>93 D<00F1800389C00707800E03801C03803C0380380700780700780700780700F00E00F00E00F00E 00F00E20F01C40F01C40703C40705C40308C800F070013147C9317>97 D<07803F800700070007 0007000E000E000E000E001C001C001CF01D0C3A0E3C0E380F380F700F700F700F700FE01EE01E E01EE01CE03CE038607060E031C01F0010207B9F15>I<007E0001C1000300800E07801E07801C 07003C0200780000780000780000F00000F00000F00000F00000F0000070010070020030040018 380007C00011147C9315>I<0000780003F80000700000700000700000700000E00000E00000E0 0000E00001C00001C000F1C00389C00707800E03801C03803C0380380700780700780700780700 F00E00F00E00F00E00F00E20F01C40F01C40703C40705C40308C800F070015207C9F17>I<007C 01C207010E011C013C013802780C7BF07C00F000F000F000F0007000700170023804183807C010 147C9315>I<00007800019C00033C00033C000718000700000700000E00000E00000E00000E00 000E0001FFE0001C00001C00001C00001C00003800003800003800003800003800007000007000 00700000700000700000700000E00000E00000E00000E00000C00001C00001C000018000318000 7B0000F300006600003C00001629829F0E>I<003C6000E27001C1E00380E00700E00F00E00E01 C01E01C01E01C01E01C03C03803C03803C03803C03803C07003C07001C0F001C17000C2E0003CE 00000E00000E00001C00001C00301C00783800F0700060E0003F8000141D7E9315>I<01E0000F E00001C00001C00001C00001C000038000038000038000038000070000070000071E000763000E 81800F01C00E01C00E01C01C03801C03801C03801C0380380700380700380700380E10700E2070 0C20701C20700C40E00CC060070014207D9F17>I<00C001E001E001C000000000000000000000 000000000E003300230043804300470087000E000E000E001C001C001C00384038803080708031 0033001C000B1F7C9E0E>I<0001800003C00003C0000380000000000000000000000000000000 000000000000003C00004600008700008700010700010700020E00000E00000E00000E00001C00 001C00001C00001C0000380000380000380000380000700000700000700000700000E00000E000 30E00079C000F180006300003C00001228829E0E>I<01E0000FE00001C00001C00001C00001C0 000380000380000380000380000700000700000703C00704200E08E00E11E00E21E00E40C01C80 001D00001E00001FC00038E000387000387000383840707080707080707080703100E03100601E 0013207D9F15>I<03C01FC0038003800380038007000700070007000E000E000E000E001C001C 001C001C0038003800380038007000700070007100E200E200E200E200640038000A207C9F0C> I<1C0F80F0002630C318004740640C004780680E004700700E004700700E008E00E01C000E00E0 1C000E00E01C000E00E01C001C01C038001C01C038001C01C038001C01C0708038038071003803 806100380380E10038038062007007006600300300380021147C9325>I<1C0F802630C0474060 4780604700704700708E00E00E00E00E00E00E00E01C01C01C01C01C01C01C0384380388380308 3807083803107003303001C016147C931A>I<007C0001C3000301800E01C01E01C01C01E03C01 E07801E07801E07801E0F003C0F003C0F003C0F00780F00700700F00700E0030180018700007C0 0013147C9317>I<01C1E002621804741C04781C04701E04701E08E01E00E01E00E01E00E01E01 C03C01C03C01C03C01C0380380780380700380E003C1C0072380071E000700000700000E00000E 00000E00000E00001C00001C0000FFC000171D809317>I<00F0400388C00705800E03801C0380 3C0380380700780700780700780700F00E00F00E00F00E00F00E00F01C00F01C00703C00705C00 30B8000F380000380000380000700000700000700000700000E00000E0000FFE00121D7C9315> I<1C1E002661004783804787804707804703008E00000E00000E00000E00001C00001C00001C00 001C000038000038000038000038000070000030000011147C9313>I<00FC030206010C030C07 0C060C000F800FF007F803FC003E000E700EF00CF00CE008401020601F8010147D9313>I<0180 01C0038003800380038007000700FFF007000E000E000E000E001C001C001C001C003800380038 003820704070407080708031001E000C1C7C9B0F>I<0E00C03300E02301C04381C04301C04701 C08703800E03800E03800E03801C07001C07001C07001C07101C0E20180E20180E201C1E200C26 4007C38014147C9318>I<0E03803307802307C04383C04301C04700C08700800E00800E00800E 00801C01001C01001C01001C02001C02001C04001C04001C08000E300003C00012147C9315>I< 0E00C1C03300E3C02301C3E04381C1E04301C0E04701C060870380400E0380400E0380400E0380 401C0700801C0700801C0700801C0701001C0701001C0602001C0F02000C0F04000E13080003E1 F0001B147C931E>I<0383800CC4401068E01071E02071E02070C040E00000E00000E00000E000 01C00001C00001C00001C040638080F38080F38100E5810084C60078780013147D9315>I<0E00 C03300E02301C04381C04301C04701C08703800E03800E03800E03801C07001C07001C07001C07 001C0E00180E00180E001C1E000C3C0007DC00001C00001C00003800F03800F07000E06000C0C0 004380003E0000131D7C9316>I<01C04003E08007F1800C1F0008020000040000080000100000 20000040000080000100000200000401000802001002003E0C0063FC0041F80080E00012147D93 13>I E /Fs 30 122 df<001C0000003E0000003E0000003E0000003E0000203E0200F81C0F80 FC1C1F80FE1C3F80FF9CFF803FDDFE0007FFF00001FFC000003E000001FFC00007FFF0003FDDFE 00FF9CFF80FE1C3F80FC1C1F80F81C0F80203E0200003E0000003E0000003E0000003E0000001C 0000191B7CAC22>42 D45 D<1C003E007F00FF80FF80FF807F003E001C0009097B8813>I<000E00001E00007E0007FE00FF FE00FFFE00F8FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000 FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000 FE0000FE0000FE0000FE0000FE0000FE007FFFFE7FFFFE7FFFFE17277BA622>49 D<00FF800003FFF0000FFFFC001F03FE003800FF007C007F80FE003FC0FF003FC0FF003FE0FF00 1FE0FF001FE07E001FE03C003FE000003FE000003FC000003FC000007F8000007F000000FE0000 00FC000001F8000003F0000003E00000078000000F0000001E0000003C00E0007000E000E000E0 01C001C0038001C0070001C00FFFFFC01FFFFFC03FFFFFC07FFFFFC0FFFFFF80FFFFFF80FFFFFF 801B277DA622>I<007F800003FFF00007FFFC000F81FE001F00FF003F80FF003F807F803F807F 803F807F801F807F800F007F800000FF000000FF000000FE000001FC000001F8000007F00000FF C00000FFF0000001FC0000007E0000007F0000007F8000003FC000003FC000003FE000003FE03C 003FE07E003FE0FF003FE0FF003FE0FF003FC0FF007FC07E007F807C007F003F01FE001FFFFC00 07FFF00000FF80001B277DA622>I<000003800000000007C00000000007C0000000000FE00000 00000FE0000000000FE0000000001FF0000000001FF0000000003FF8000000003FF8000000003F F80000000073FC0000000073FC00000000F3FE00000000E1FE00000000E1FE00000001C0FF0000 0001C0FF00000003C0FF80000003807F80000007807FC0000007003FC0000007003FC000000E00 3FE000000E001FE000001E001FF000001C000FF000001FFFFFF000003FFFFFF800003FFFFFF800 00780007FC0000700003FC0000700003FC0000E00001FE0000E00001FE0001E00001FF0001C000 00FF0001C00000FF00FFFE001FFFFEFFFE001FFFFEFFFE001FFFFE2F297EA834>65 DI<00003FF001800003FFFE0380000FFFFF878000 3FF007DF8000FF8001FF8001FE00007F8003FC00003F8007F000001F800FF000000F801FE00000 07801FE0000007803FC0000007803FC0000003807FC0000003807F80000003807F8000000000FF 8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF800000 0000FF8000000000FF80000000007F80000000007F80000000007FC0000003803FC0000003803F C0000003801FE0000003801FE0000007000FF00000070007F000000E0003FC00001E0001FE0000 3C0000FF8000F800003FF007E000000FFFFFC0000003FFFF000000003FF8000029297CA832>I< FFFFFCFFFFFCFFFFFC01FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE00 01FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE00 01FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE00FFFFFC FFFFFCFFFFFC16297EA81A>73 D82 D<01FF800007FFF0000F81F8001FC07E001FC07E001FC03F000F803F8007003F8000003F800000 3F8000003F80000FFF8000FFFF8007FC3F800FE03F803F803F803F003F807F003F80FE003F80FE 003F80FE003F80FE003F807E007F807F00DF803F839FFC0FFF0FFC01FC03FC1E1B7E9A21>97 D<001FF80000FFFE0003F01F0007E03F800FC03F801F803F803F801F007F800E007F0000007F00 0000FF000000FF000000FF000000FF000000FF000000FF000000FF0000007F0000007F0000007F 8000003F8001C01F8001C00FC0038007E0070003F01E0000FFFC00001FE0001A1B7E9A1F>99 D<00003FF80000003FF80000003FF800000003F800000003F800000003F800000003F800000003 F800000003F800000003F800000003F800000003F800000003F800000003F800000003F800001F E3F80000FFFBF80003F03FF80007E00FF8000FC007F8001F8003F8003F8003F8007F0003F8007F 0003F8007F0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800 FF0003F8007F0003F8007F0003F8007F0003F8003F8003F8001F8003F8000F8007F80007C00FF8 0003F03BFF8000FFF3FF80003FC3FF80212A7EA926>I<003FE00001FFF80003F07E0007C01F00 0F801F801F800F803F800FC07F000FC07F0007C07F0007E0FF0007E0FF0007E0FFFFFFE0FFFFFF E0FF000000FF000000FF0000007F0000007F0000007F0000003F8000E01F8000E00FC001C007E0 038003F81F0000FFFE00001FF0001B1B7E9A20>I<0007F0003FFC00FE3E01F87F03F87F03F07F 07F07F07F03E07F00007F00007F00007F00007F00007F00007F000FFFFC0FFFFC0FFFFC007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F0007FFF807FFF807FFF80182A7EA915>I<00FF 81F003FFE7F80FC1FE7C1F80FC7C1F007C383F007E107F007F007F007F007F007F007F007F007F 007F007F007F003F007E001F007C001F80FC000FC1F8001FFFE00018FF80003800000038000000 3C0000003E0000003FFFF8001FFFFF001FFFFF800FFFFFC007FFFFE01FFFFFF03E0007F07C0001 F8F80000F8F80000F8F80000F8F80000F87C0001F03C0001E01F0007C00FC01F8003FFFE00007F F0001E287E9A22>I<07000F801FC03FE03FE03FE01FC00F800700000000000000000000000000 0000FFE0FFE0FFE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00F E00FE00FE00FE00FE00FE0FFFEFFFEFFFE0F2B7DAA14>105 D107 dII03F0F007F007F0FF007F8F0FF00F7F0FF00F7F0FF00F7F0FF007F8F00 0 07F87F007F03FF03F800 F0F01F800 FC01FC01F07F07F001FFFC000 3FE001D 1B7E9A22> I I<33FE09001FFFC003F07E000 0FC01F801F800 FCF114 D<03FE300FFFF01E03F03800F0700070F00070F00070 F80070FC0000FFE0007FFE007FFF803FFFE01FFFF007FFF800FFF80003FC0000FC60007CE0003C F0003CF00038F80038FC0070FF01E0F7FFC0C1FF00161B7E9A1B>I<0070000070000070000070 0000F00000F00000F00001F00003F00003F00007F0001FFFF0FFFFF0FFFFF007F00007F00007F0 0007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F03807F03807F0 3807F03807F03807F03803F03803F87001F86000FFC0001F8015267FA51B>IIII121 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300 TeXDict begin %%EndSetup %%Page: 17 1 bop 60 74 a Fq([57])21 b(M.)15 b(A.)f(Tsfasman)g(and)h(S.)g(G.)f(Vladuts,)i Fr(A)o(lgebr)n(aic-Ge)n(ometric)f(Co)n(des)p Fq(,)f(Klu)o(w)o(er,)h(Dordrec)o (h)o(t,)e(Hol-)153 159 y(land,)j(1991.)60 281 y([58])21 b(T.)f(V)l(erho)q (e\013,)h(\\An)f(up)q(dated)g(table)h(of)e(minim)o(um-distance)j(b)q(ounds)f (for)e(binary)i(linear)g(co)q(des,")153 366 y Fr(IEEE)16 b(T)m(r)n(ans.)e (Inform.)i(The)n(ory)f Fp(33)g Fq(\(1987\),)e(665{680.)967 2853 y(17)p eop %%Page: 16 2 bop 60 74 a Fq([44])21 b(J.)e(Naor)f(and)g(M.)g(Naor,)g(\\Small-bias)i (probabilit)o(y)f(spaces:)27 b(e\016cien)o(t)19 b(constructions)g(and)f (appli-)153 159 y(cations,")23 b(in)f Fr(Pr)n(o)n(c.)f(20th)h(A)o(nnual)f(A)o (CM)g(Symp.)g(The)n(ory)h(of)g(Computing)p Fq(,)g(A)o(CM)e(Press,)i(1990,)153 244 y(pp.)16 b(213{223.)60 366 y([45])21 b(P)l(.)15 b(R.)g(J.)327 354 y(\177)320 366 y(Osterg)-6 b(\027)-28 b(ard,)15 b(\\Constructions)f(of)h (mixed)h(co)o(v)o(ering)f(co)q(des,")g(Rep)q(ort)g(18,)f(Digital)i(Systems) 153 451 y(Lab)q(oratory)l(,)f(Helsinki)i(Univ.)f(T)l(ec)o(hnology)l(,)f(Dec.) g(1991.)60 573 y([46])21 b(S.)f(P)o(oljak,)g(A.)f(Pultr)h(and)g(V.)f(R\177) -23 b(odl,)22 b(\\On)e(qualitativ)o(ely)h(indep)q(enden)o(t)h(partitions)e (and)g(related)153 657 y(problems,")c Fr(Discr.)g(Appl.)g(Math.)f Fp(6)h Fq(\(1983\),)d(193{205.)60 779 y([47])21 b(S.)11 b(P)o(oljak)g(and)g (Z.)f(T)l(uza,)i(\\On)f(the)g(maxim)o(um)g(n)o(um)o(b)q(er)g(of)g(qualitativ) o(ely)h(indep)q(enden)o(t)h(partitions,")153 864 y Fr(J.)j(Combin.)g(The)n (ory,)g(Ser.)f Fp(A)i(51)e Fq(\(1989\),)e(111{116.)60 986 y([48])21 b(A.)15 b(R)o(\023)-21 b(en)o(yi,)15 b Fr(F)m(oundations)h(of)g(Pr)n(ob)n (ability)p Fq(,)e(Wiley)l(,)j(New)e(Y)l(ork,)f(1971.)60 1108 y([49])21 b(C.)16 b(T.)g(Retter,)g(\\In)o(tersecting)g(Goppa)g(co)q(des,")g Fr(IEEE)g(T)m(r)n(ans.)g(Inform.)g(The)n(ory)g Fp(35)h Fq(\(1989\),)d(822{) 153 1193 y(828.)60 1315 y([50])21 b(G.)12 b(Roux,)g(\\)p Fo(k)q Fq(-propri)o(\023)-21 b(et)o(\023)g(es)11 b(dans)h(des)g(tableaux)h(de)f Fr(n)g Fq(colonnes;)h(cas)f(particulier)i(de)e(la)h Fo(k)q Fq(-surjectivit)o(\023)-21 b(e)153 1399 y(et)15 b(de)h(la)g Fo(k)q Fq(-p)q(erm)o(utivit)o(\023)-21 b(e,")15 b(Ph.D.)f(Dissertation,)h (Univ)o(ersit)o(y)g(of)g(P)o(aris)g(6,)f(Marc)o(h)h(1987.)60 1522 y([51])21 b(G.)e(Seroussi)h(and)g(N.)f(H.)g(Bshout)o(y)l(,)i(\\V)l (ector)e(sets)g(for)g(exhaustiv)o(e)h(testing)f(of)h(logical)g(circuits,")153 1606 y Fr(IEEE)c(T)m(r)n(ans.)e(Information)i(The)n(ory)f Fp(34)g Fq(\(1988\),)f(513{522.)60 1728 y([52])21 b(G.)f(B.)f(Sherw)o(o)q(o)q(d,)i (\\Constrained)f(Arra)o(y)f(T)l(est)h(System)g(\(CA)l(TS\))f(User's)h (Guide,")h(A)l(T&T)f(Bell)153 1813 y(Lab)q(oratories,)15 b(Middleto)o(wn,)h (NJ,)f(1992.)60 1935 y([53])21 b(E.)e(Sp)q(erner,)j(\\Ein)e(Satz)g(\177)-24 b(ub)q(er)20 b(Un)o(termengen)g(einer)g(endlic)o(her)i(Menge,")e Fr(Math.)g(Z.)f Fp(27)h Fq(\(1928\),)153 2020 y(544{548.)60 2142 y([54])h(D.)e(T.)h(T)l(ang)f(and)h(C.-L.)f(Chen,)i(\\Logic)f(test)g (pattern)f(generation)h(using)g(linear)h(co)q(des,")g Fr(IEEE)153 2226 y(T)m(r)n(ans.)15 b(Computers)g Fp(33)h Fq(\(1984\),)d(845{850.)60 2348 y([55])21 b(D.)d(T.)f(T)l(ang)h(and)g(C.-L.)f(Chen,)i(\\Iterativ)o(e)f (exhaustiv)o(e)g(pattern)f(generation)h(for)g(logic)h(testing,")153 2433 y Fr(IBM)d(J.)g(R)n(es.)g(Dev.)f Fp(28)g Fq(\(1984\),)e(212{219.)60 2555 y([56])21 b(D.)f(T.)g(T)l(ang)g(and)g(L.)h(S.)f(W)l(o)q(o,)h (\\Exhaustiv)o(e)f(test)g(pattern)g(generation)g(with)h(constan)o(t)e(w)o (eigh)o(t)153 2640 y(v)o(ectors,")14 b Fr(IEEE)h(T)m(r)n(ans.)g(Computers)g Fp(32)h Fq(\(1983\),)d(1145{1150.)967 2853 y(16)p eop %%Page: 15 3 bop 60 74 a Fq([29])21 b(I.)d(S.)f(Honk)m(ala,)h(\\A)g(Graham-Sloane)f(t)o (yp)q(e)g(construction)h(for)f Fo(s)p Fq(-surjectiv)o(e)h(matrices,")g (preprin)o(t,)153 159 y(1991.)60 281 y([30])j(G.)c(O.)h(H.)f(Katona,)h(\\Tw)o (o)e(applications)j(\(for)e(searc)o(h)g(theory)h(and)f(truth)h(functions\))g (of)f(Sp)q(erner)153 366 y(t)o(yp)q(e)f(theorems,")e Fr(Perio)n(dic)n(a)i (Math.)g(Hung.)f Fp(3)g Fq(\(1973\),)e(19{26.)60 488 y([31])21 b(G.)11 b(O.)g(H.)f(Katona,)h(\\Extremal)g(problems)h(for)e(h)o(yp)q (ergraphs,")h(in)h Fr(Combinatorics)p Fq(,)f(ed.)g(M.)g(Hall,)g(Jr.)153 573 y(and)16 b(J.)f(H.)g(v)m(an)g(Lin)o(t,)h(Reidel,)h(Dordrec)o(h)o(t,)d (Holland,)i(1975,)e(pp.)h(215{244.)60 695 y([32])21 b(G.)14 b(O.)g(H.)g(Katona)g(and)h(J.)f(Sriv)m(asta)o(v)m(a,)g(\\Minimal)i(2-co)o(v)o (erings)e(of)g(a)g(\014nite)h(a\016ne)g(space)f(based)h(on)153 779 y Fo(GF)6 b Fq(\(2\),")14 b Fr(J.)i(Stat.)h(Planning)d(Infer)n(enc)n(e)f Fp(8)i Fq(\(1983\),)e(375{388.)60 901 y([33])21 b(G.)c(L.)g(Katsman,)g(M.)f (A.)h(Tsfasman)f(and)h(S.)g(G.)g(Vladuts,)g(\\Mo)q(dular)h(curv)o(es)f(and)g (co)q(des)h(with)f(a)153 986 y(p)q(olynomial)g(construction,")e Fr(IEEE)g(T)m(r)n(ans.)g(Information)h(The)n(ory)f Fp(30)g Fq(\(1984\),)e(353{355.)60 1108 y([34])21 b(D.)16 b(J.)f(Kleitman)i(and)f(J.) g(Sp)q(encer,)h(\\F)l(amilies)g(of)e Fo(k)q Fq(-indep)q(enden)o(t)k(sets,")c Fr(Discr.)h(Math.)g Fp(6)g Fq(\(1973\),)153 1193 y(255{262.)60 1315 y([35])21 b(J.)16 b(K\177)-23 b(orner)15 b(and)g(M.)g(Lucertini,)i (\\Compressing)e(inconsisten)o(t)h(data,")e(preprin)o(t,)h(1992.)60 1437 y([36])21 b(D.)15 b(L.)g(Kreher,)h(p)q(ersonal)f(comm)o(unication.)60 1559 y([37])21 b(D.)15 b(L.)g(Kreher)h(and)f(V.)g(D.)g(T)l(onc)o(hev,)g(p)q (ersonal)h(comm)o(unication.)60 1681 y([38])21 b(A.)c(Lemp)q(el)j(and)d(S.)g (Winograd,)h(\\A)f(new)g(approac)o(h)g(to)g(error-correcting)g(co)q(des,")h Fr(IEEE)f(T)m(r)n(ans.)153 1766 y(Inform.)f(The)n(ory)f Fp(23)h Fq(\(1977\))d(503{508.)60 1888 y([39])21 b(S.)15 b(N.)g(Litsyn,)h(p)q (ersonal)g(comm)o(unication,)f(1992.)60 2010 y([40])21 b(F.)d(J.)f (MacWilliams)j(and)e(N.)f(J.)h(A.)g(Sloane,)h Fr(The)f(The)n(ory)h(of)g(Err)n (or-Corr)n(e)n(cting)f(Co)n(des)p Fq(,)f(North-)153 2094 y(Holland,)g (Amsterdam,)d(1977.)60 2216 y([41])21 b(C.)15 b(L.)g(Mallo)o(ws,)g(p)q (ersonal)h(comm)o(unication.)60 2338 y([42])21 b(R.)13 b(J.)g(McEliece,)i(E.) d(R.)h(Ro)q(demic)o(h,)h(H.)f(C.)f(Rumsey)l(,)h(Jr.,)g(and)g(L.)g(R.)g(W)l (elc)o(h,)h(\\New)e(upp)q(er)i(b)q(ounds)153 2423 y(on)k(the)h(rate)e(of)h(a) g(co)q(de)g(via)h(the)f(Delsarte-MacWilliams)h(inequalities,")i Fr(IEEE)d(T)m(r)n(ans.)f(Inform.)153 2508 y(The)n(ory)e Fp(23)h Fq(\(1977\))d(157{166.)60 2630 y([43])21 b(D.)10 b(Mikl\023)-23 b(os,)12 b(\\Linear)f(binary)g(co)q(des)g(with)g(in)o(tersection)h(prop)q (erties,")f Fr(Discr.)h(Appl.)g(Math.)e Fp(9)h Fq(\(1984\),)153 2715 y(187{196.)967 2853 y(15)p eop %%Page: 14 4 bop 60 74 a Fq([15])21 b(G.)14 b(D.)f(Cohen)i(and)f(G.)f(Simon)o(yi,)i(\\Co)q (ding)g(for)e(write-directional)k(memories)d(and)g(con\015ict)h(resolu-)153 159 y(tion,")g Fr(Discr.)h(Appl.)g(Math.)g Fp(24)f Fq(\(1989\),)e(103{114.)60 281 y([16])21 b(J.)16 b(H.)e(Con)o(w)o(a)o(y)l(,)g(p)q(ersonal)i(comm)o (unication.)60 403 y([17])21 b(J.)14 b(H.)f(Con)o(w)o(a)o(y)g(and)h(N.)f(J.)h (A.)f(Sloane,)h Fr(Spher)n(e)h(Packing,)f(L)n(attic)n(es)g(and)h(Gr)n(oups)p Fq(,)f(Springer-V)l(erlag,)153 488 y(NY,)h(2nd)h(ed.,)f(1992,)e(to)i(app)q (ear.)60 610 y([18])21 b(W.)15 b(Co)q(ok,)f(p)q(ersonal)i(comm)o(unication.) 60 732 y([19])21 b(I.)g(I.)g(Dumer,)h(\\Asymptotically)g(optimal)f(co)q(des)h (correcting)f(memory)g(defects)g(of)f(\014xed)i(m)o(ulti-)153 817 y(plicit)o(y)l(,")c Fr(Pr)n(oblemy)e(Per)n(e)n(dachi)h(Informatsii)e Fp(25)h Fq(\(No.)f(4,)h(1989\),)e(pp.)i(3{10.)e(English)j(translation,)153 901 y Fr(Pr)n(oblems)f(of)g(Information)g(T)m(r)n(ansmission)d Fp(25)i Fq(\(1989\),)e(259{265.)60 1023 y([20])21 b(P)l(.)c(Erd\177)-23 b(os,)18 b(P)l(.)f(F)l(rankl)g(and)h(Z.)e(F)q(\177)-24 b(uredi,)19 b(\\F)l(amilies)f(of)f(\014nite)h(sets)f(in)i(whic)o(h)f(no)f(set)g(is)h(co)o (v)o(ered)f(b)o(y)153 1108 y(the)f(union)g(of)f(t)o(w)o(o)f(others,")g Fr(J.)i(Combin.)f(The)n(ory)p Fq(,)g(Ser.)g(A)g Fp(33)h Fq(\(1982\),)d (158{166.)60 1230 y([21])21 b(P)l(.)c(Erd\177)-23 b(os,)17 b(C.)f(Ko)h(and)h(R.)f(Rado,)g(\\In)o(tersection)g(theorems)g(for)g(systems)f (of)h(\014nite)h(sets,")e Fr(Quart.)153 1315 y(J.)g(Math.)h(Oxfor)n(d)e Fp(12)h Fq(\(1961\),)d(313{318.)60 1437 y([22])21 b(R.)h(A.)f(Fisher)h(and)g (F.)e(Y)l(ates,)j Fr(Statistic)n(al)e(T)m(ables)f(for)j(Biolo)n(gic)n(al,)f (A)n(gricultur)n(al,)h(and)f(Me)n(dic)n(al)153 1522 y(R)n(ese)n(ar)n(ch)p Fq(,)14 b(Hafner,)h(NY,)g(6th)g(ed.,)g(1963.)60 1644 y([23])21 b(L.)16 b(Gargano,)d(J.)j(K\177)-23 b(orner)15 b(and)h(U.)f(V)l(accaro,)f (\\Sp)q(erner)i(theorems)f(on)h(directed)g(graphs)f(and)h(quali-)153 1728 y(tativ)o(e)f(indep)q(endence)q(,")i Fr(J.)f(Combinatorial)g(The)n(ory)f Fq(\(submitted\).)60 1850 y([24])21 b(L.)16 b(Gargano,)e(J.)i(K\177)-23 b(orner)16 b(and)g(U.)f(V)l(accaro,)h(\\Sp)q(erner)g(capacities,")h Fr(Gr)n(aphs)g(and)g(Combinatorics)153 1935 y Fq(\(submitted\).)60 2057 y([25])k(L.)14 b(Gargano,)f(J.)h(K\177)-23 b(orner)14 b(and)g(U.)g(V)l(accaro,)g(\\Capacities:)19 b(from)14 b(information)g(theory) f(to)h(extremal)153 2142 y(set)h(theory)l(,")g Fr(J.)h(A)o(mer.)g(Math.)g(So) n(c.)f Fq(\(submitted\).)60 2264 y([26])21 b(C.)14 b(Greene,)h(\\Sp)q(erner)g (families)g(and)g(partitions)f(of)g(a)g(partially)i(ordered)e(set,")g(in)h Fr(Combinatorics)p Fq(,)153 2348 y(ed.)h(M.)e(Hall,)i(Jr.)f(and)g(J.)g(H.)g (v)m(an)h(Lin)o(t,)f(Reidel,)j(Dordrec)o(h)o(t,)13 b(Holland,)j(1975,)e(pp.)h (277{290.)60 2470 y([27])21 b(M.)15 b(Hall,)h(Jr.,)e Fr(Combinatorial)i(The)n (ory)p Fq(,)f(Wiley)l(,)h(NY,)f(2nd)g(ed.,)g(1986.)60 2592 y([28])21 b(I.)13 b(S.)g(Honk)m(ala,)h(\\Mo)q(di\014ed)f(b)q(ounds)h(for)f (co)o(v)o(ering)f(co)q(des,")i Fr(IEEE)f(T)m(r)n(ans.)f(Information)i(The)n (ory)f Fp(37)153 2677 y Fq(\(1991\),)g(351{365.)967 2853 y(14)p eop %%Page: 13 5 bop 60 74 a Fs(References)83 204 y Fq([1])21 b(N.)11 b(Alon,)h(\\Explicit)h (construction)f(of)e(exp)q(onen)o(tial-sized)k(families)f(of)d Fo(k)q Fq(-indep)q(endent)j(sets,")e Fr(Discr.)153 289 y(Math.)16 b Fp(58)f Fq(\(1986\),)e(191{193.)83 411 y([2])21 b(D.)15 b(Applegate,)g(p)q (ersonal)h(comm)o(unication.)83 533 y([3])21 b(E.)11 b(F.)f(Assm)o(us,)h(Jr.) f(and)i(J.)f(D.)f(Key)l(,)i(\\Hadamard)e(matrices)h(and)h(their)f(designs:)19 b(a)11 b(co)q(ding)h(theoretic)153 618 y(approaC)O(H),j(r)n(ANS)f(a),(h)(n)(c)d fq(to)h(APP)q(耳)83(740)y((4))21μb(b)16 b(BEC)O(K)O(ER)g((Hg))f(西蒙,)i(\HO)O(W)D(鲁棒)I(IS)G(G)g Fo(n)p Fq(-Cub)q(e),")i Fr(Info.)e(Computation)g Fp(77)g Fq(\(1988\),)153 824 y(162{178.)83 946 y([5])21 b(A.)15 b(Blokh)o(uis)i(and)e (K.)g(Metsc)o(h,)g(p)q(ersonal)h(comm)o(unication.)83 1068 y([6])21 b(A.)15 b(E.)g(Brou)o(w)o(er,)f(p)q(ersonal)i(comm)o(unication.)83 1190 y([7])21 b(A.)14 b(E.)f(Brou)o(w)o(er,)g(J.)h(B.)f(Shearer,)h(N.)f(J.)h (A.)f(Sloane)i(and)f(W.)f(D.)g(Smith,)h(\\A)f(new)h(table)h(of)e(constan)o(t) 153 1275 y(w)o(eigh)o(t)i(co)q(des,")g Fr(IEEE)h(T)m(r)n(ans.)e(Inform.)i (The)n(ory)f Fp(36)g Fq(\(1990\),)e(1334{1380.)83 1397 y([8])21 b(P)l(.)11 b(Bussc)o(h)o(bac)o(h,)h(\\Constructiv)o(e)e(metho)q(ds)h(to)f (solv)o(e)h(the)g(problems)h(of)e Fo(s)p Fq(-surjectivit)o(y)l(,)i (con\015ict)g(reso-)153 1482 y(lution,)i(co)q(ding)f(in)g(defectiv)o(e)g (memories,")f(Rep)q(ort)g(84D005,)1235 1470 y(\023)1231 1482 y(Ecole)g(Nationale)h(Sup)o(\023)-21 b(er.)11 b(T)o(\023)-21 b(el)o(\023)g(ecomm.,)153 1567 y(P)o(aris,)15 b(1984.)83 1689 y([9])21 b(A.)14 b(R.)g(Calderbank)h(and)f(W.)g(M.)f(Kan)o(tor,)g(\\The)h (geometry)f(of)h(t)o(w)o(o-w)o(eigh)o(t)f(co)q(des,")h Fr(Bul)r(l.)h(L)n (ondon)153 1773 y(Math.)i(So)n(c.)d Fp(18)i Fq(\(1986\),)d(97{122.)60 1895 y([10])21 b(P)l(.)15 b(J.)g(Cameron,)g(\\Biplanes,")h Fr(Math.)g(Z.)f Fp(131)h Fq(\(1973\),)d(85{101.)60 2017 y([11])21 b(A.)e(K.)f(Chandra,)h(L.)g(T.)f(Kou,)h(G.)f(Mark)o(o)o(wsky)e(and)j(S.)f (Zaks,)h(\\On)g(sets)f(of)g(Bo)q(olean)h Fo(n)p Fq(-v)o(ectors)153 2102 y(with)d(all)g Fo(k)q Fq(-pro)s(jections)f(surjectiv)o(e,")g Fr(A)n(cta.)h(Inform.)f Fp(20)g Fq(\(1983\),)e(103{111.)60 2224 y([12])21 b(G.)13 b(D.)g(Cohen,)h(\\Applications)i(of)d(co)q(ding)i (theory)e(to)g(comm)o(unication)i(com)o(binatorial)f(problems,")153 2309 y Fr(Discr.)j(Math.)e Fp(83)g Fq(\(1990\),)e(237{248.)60 2431 y([13])21 b(G.)12 b(D.)g(Cohen,)h(P)l(.)f(Go)q(dlewski)i(and)e(M.)g (Karp)q(o)o(vsky)l(,)h(\\T)l(est)f(exhaustif)h(de)g(circuits)g(com)o (binatoires,")153 2515 y Fr(T)m(r)n(aitment)j(de)g(Signal)e Fp(1)h Fq(\(1984\),)e(223{226.)60 2638 y([14])21 b(G.)14 b(D.)g(Cohen)h(and)g (A.)f(Lemp)q(el,)i(\\Linear)f(in)o(tersecting)h(co)q(des,")e Fr(Discr.)i(Math.)e Fp(56)h Fq(\(1985\),)e(35{43.)967 2853 y(13)p eop %%Page: 12 6 bop 60 354 a Fq(T)l(able)17 b(3:)k(Upp)q(er)c(b)q(ounds)g(to)f Fo(g)621 361 y Fi(3)640 354 y Fq(\()p Fo(n)p Fq(\),)g(the)g(minimal)h(n)o(um) o(b)q(er)g(of)f(v)o(ectors)f(in)i(a)f(binary)g(3-co)o(v)o(ering)g(arra)o(y)60 410 y(of)f(length)h Fo(n)p Fq(.)k(If)15 b Fo(n)h Fq(is)f(missing,)h(use)g (the)f(follo)o(wing)h(en)o(try)l(.)655 549 y Fd(n)110 b(g)810 555 y Fc(3)828 549 y Fb(\()p Fd(n)p Fb(\))50 b(Construction)595 621 y(3)174 b(8)124 b Fa(f)p Fb(0)p Fd(;)7 b Fb(1)p Fa(g)1038 606 y Fc(3)595 671 y Fb(4)174 b(8)124 b(\(1000\))p Fd(;)7 b Fb(\(0111\))595 721 y(5)174 b(10)103 b(\(10000\))p Fd(;)7 b Fb(\(01111\))595 771 y(11)153 b(12)103 b Fd(H)970 777 y Fc(12)595 821 y Fb(14)153 b(16)103 b Fd(H)970 827 y Fc(16)1014 821 y Fa(\000)10 b Fb(2)j(columns)595 870 y(16)153 b(17)103 b(2-\(16)p Fd(;)7 b Fb(6)p Fd(;)g Fb(2\))12 b(biplane)h(or)h(\(1\))595 920 y(20)153 b(18)103 b(\(13\))595 970 y(22)153 b(19)103 b(\(13\))595 1020 y(28)153 b(23)103 b(\(13\))595 1070 y(30)153 b(24)103 b(\(13\))595 1120 y(32)153 b(25)103 b(\(13\))595 1169 y(40)153 b(26)103 b(\(13\))595 1219 y(44)153 b(27)103 b(\(13\))595 1269 y(56)153 b(31)103 b(\(13\))595 1319 y(64)153 b(32)103 b(\(14\))595 1369 y(70)153 b(34)103 b(\(13\))595 1418 y(80)153 b(35)103 b(\(13\))595 1468 y(121)132 b(36)103 b(\(15\))595 1518 y(128)132 b(42)103 b(\(13\))595 1568 y(176)132 b(44)103 b Fd(A)p Fb(\(22)p Fd(;)7 b Fb(8)p Fd(;)g Fb(7\))595 1618 y(253)132 b(46)103 b Fd(S)r Fb(\(4)p Fd(;)7 b Fb(7)p Fd(;)g Fb(23\))595 1668 y(254)132 b(50)103 b Fd(A)p Fb(\(25)p Fd(;)7 b Fb(8)p Fd(;)g Fb(7\))595 1717 y(256)132 b(51)103 b(\(15\))595 1767 y(260)132 b(52)103 b Fd(S)r Fb(\(3)p Fd(;)7 b Fb(5)p Fd(;)g Fb(26\))595 1817 y(361)132 b(54)103 b(\(15\))595 1867 y(420)132 b(57)103 b(\(13\))595 1917 y(506)132 b(58)103 b(\(13\))595 1966 y(1331)111 b(60)103 b(\(15\))595 2016 y(1584)111 b(73)103 b(\(13\))595 2066 y(2662)111 b(74)103 b(\(13\))595 2116 y(14641)90 b(84)103 b(\(15\))595 2166 y(22880)90 b(101)82 b(\(13\))595 2215 y(29282)90 b(102)82 b(\(13\))595 2265 y(161051)69 b(108)82 b(\(15\))595 2315 y(184756)69 b(128)82 b(\(13\))595 2365 y(322102)69 b(129)82 b(\(13\))595 2415 y(1771561)48 b(132)82 b(\(15\))967 2853 y Fq(12)p eop %%Page: 11 7 bop 60 74 a Fr(\(ii\))18 b(If)g(a)h(Steiner)f(system)g Fo(S)s Fq(\()p Fo(T)t(;)8 b(K)q(;)f(V)f Fq(\))19 b Fr(c)n(ontaining)e Fo(M)22 b Fq(=)1120 40 y Fg(\000)1139 54 y Fn(V)1141 90 y(T)1168 40 y Fg(\001)1187 74 y Fo(=)1210 40 y Fg(\000)1229 54 y Fn(K)1232 90 y(T)1261 40 y Fg(\001)1298 74 y Fr(blo)n(cks)c(exists)g(with)h Fo(V)26 b(>)18 b Fq(3)p Fo(K)j Fr(and)60 159 y Fo(K)15 b(>)e Fq(2\()p Fo(T)j Fh(\000)10 b Fq(1\))p Fr(,)16 b(then)g Fo(g)485 166 y Fi(3)504 159 y Fq(\()p Fo(M)5 b Fq(\))12 b Fh(\024)h Fq(2)p Fo(V)d Fr(.)60 294 y Fp(Pro)q(of.)20 b Fq(\(i\))e(Let)g Fo(A)g Fq(b)q(e)h(the)f Fo(N)e Fh(\002)d Fo(M)23 b Fq(arra)o(y)17 b(whose)g(columns)i(are)f(the)g(co)q(dew)o(ords.)28 b(An)o(y)18 b(t)o(w)o(o)f(distinct)60 378 y(co)q(dew)o(ords)e(in)o(tersect)g(in)h(at)e (most)g Fo(W)h Fh(\000)10 b Fo(D)q(=)p Fq(2)k(co)q(ordinates,)h(and)g(since)h Fo(W)j(>)13 b Fq(2\()p Fo(W)i Fh(\000)10 b Fo(D)q(=)p Fq(2\))k(this)h(means) 60 463 y(that)c(in)i(an)o(y)f(three)g(columns)h(all)g(of)e(100,)g(010,)h(001) f(o)q(ccur.)19 b(Since)14 b Fo(N)j(>)c Fq(3)p Fo(W)6 b Fq(,)12 b(000)f(also)h(o)q(ccurs.)19 b(Therefore)60 548 y(the)c(ro)o(ws)f(of)h Fo(A)g Fq(and)p 433 512 35 2 v 16 w Fo(A)g Fq(form)g(a)f(3-co)o(v)o(ering)h (arra)o(y)l(.)k(\(ii\))d(is)g(a)f(sp)q(ecial)i(case)e(of)g(\(i\).)131 632 y(The)h(maximal)g(n)o(um)o(b)q(er)g(of)f(co)q(dew)o(ords)h(in)g(a)g (constan)o(t)f(w)o(eigh)o(t)g(co)q(de)i(of)e(length)h Fo(N)5 b Fq(,)15 b(w)o(eigh)o(t)h Fo(W)22 b Fq(and)60 717 y(minimal)17 b(distance)g Fo(D)f Fq(is)g(usually)h(denoted)f(b)o(y)g Fo(A)p Fq(\()p Fo(N)r(;)8 b(D)q(;)g(W)e Fq(\),)13 b(and)j([7)o(])f(giv)o(es)h (extensiv)o(e)g(tables)g(of)f(lo)o(w)o(er)60 802 y(b)q(ounds)i(on)f(this)g (quan)o(tit)o(y)l(.)21 b(An)c(en)o(try)e(mark)o(ed)g Fo(A)p Fq(\()p Fo(N)r(;)8 b(D)q(;)g(W)e Fq(\))14 b(or)i Fo(S)s Fq(\()p Fo(T)t(;)8 b(K)q(;)f(V)f Fq(\))16 b(in)h(T)l(able)f(3)g(indicates)h(an)60 886 y(arra)o(y)d(obtained)i(b)o(y)f(applying)i(Theorem)e(9.)131 971 y(T)l(able)k(3)f(summarizes)g(all)i(these)e(results,)h(giving)h(upp)q(er) f(b)q(ounds)g(on)f Fo(g)1416 978 y Fi(3)1436 971 y Fq(\()p Fo(n)p Fq(\))g(for)f Fo(n)i Fh(\024)f Fq(11)1734 955 y Fi(6)1753 971 y Fq(.)29 b(If)19 b Fo(n)f Fq(is)60 1056 y(missing,)c(the)f(follo)o(wing) h(en)o(try)e(should)i(b)q(e)g(used.)20 b(Explicit)15 b(arra)o(ys)c(are)i(kno) o(wn)g(in)h(ev)o(ery)f(case.)19 b(It)13 b(follo)o(ws)60 1140 y(from)i(\(12\))g(that)h(the)g(en)o(tries)h(in)g(T)l(able)g(3)e(for)h Fo(n)e Fh(\024)h Fq(11)h(are)f(exact.)23 b(Ho)o(w)o(ev)o(er,)15 b(the)h(author)g(exp)q(ects)h(that)60 1225 y(most)c(of)f(the)i(other)f(en)o (tries)g(\(as)g(w)o(ell)h(as)f(those)g(in)i(T)l(able)f(1\))e(can)i(b)q(e)g (considerably)h(impro)o(v)o(ed,)e(and)h(o\013ers)60 1310 y(these)h(tables)h (as)f(a)g(c)o(hallenge)h(to)f(the)g(reader.)60 1453 y Fs(Ac)n(kno)n (wledgemen)n(ts)146 1583 y Fq(I)h(should)h(lik)o(e)f(to)f(thank)h(Da)o(vid)f (Applegate,)h(Aart)f(Blokh)o(uis,)i(Andries)g(Brou)o(w)o(er,)d(G)o(\023)-21 b(erard)14 b(Cohen,)60 1667 y(John)j(Con)o(w)o(a)o(y)l(,)e(Bill)k(Co)q(ok,)c (J\023)-23 b(anos)17 b(K\177)-23 b(orner,)16 b(Donald)h(Kreher,)g(Simon)g (Litsyn,)g(Colin)h(Mallo)o(ws,)e(Klaus)60 1752 y(Metsc)o(h)f(and)g(Vladimir)i (T)l(onc)o(hev,)e(all)h(of)f(whom)g(ha)o(v)o(e)g(con)o(tributed)h(to)e(this)i (pap)q(er.)967 2853 y(11)p eop %%Page: 10 8 bop 60 74 a Fp(Remark.)31 b Fq(This)20 b(is)g(a)f(strengthening)g(of)g(some)g (constructions)g(in)o(tro)q(duced)i(b)o(y)e(Bussc)o(h)o(bac)o(h)h([8)o(])f (and)60 159 y(also)14 b(used)h(b)o(y)g(Roux)f([50)o(].)20 b([8)o(])14 b(has)g(2)704 143 y Fn(K)747 159 y Fh(\000)8 b Fq(1)14 b(rather)g(than)h(2) 1092 143 y Fn(K)1140 159 y Fq(in)g(\(14\),)e(and)h(establishes)i(\(15\))d (only)i(under)60 244 y(the)g(stricter)g(h)o(yp)q(othesis)h(that)824 328 y Fo(K)f Fh(\024)926 269 y Fg(\024)953 298 y Fq(4)p Fo(n)1003 305 y Fi(0)p 953 318 70 2 v 977 360 a Fq(9)1028 269 y Fg(\025)1060 328 y Fq(+)c(1)j Fo(:)683 b Fq(\(16\))60 466 y Fp(Pro)q(of.)18 b Fq(F)l(or)d(part)g(\(ii\))i(w)o(e)e(let)i Fh(C)h Fq(b)q(e)f(an)f([)p Fo(N)i Fq(=)c(2)p Fo(K)f Fh(\000)e Fq(1)p Fo(;)d(K)q(;)g(K)s Fq(])13 b(MDS)i(in)o(tersecting)i(co)q(de)g(o)o(v)o(er)e(the)h(\014eld)60 551 y(of)g(order)f Fo(n)259 558 y Fi(0)279 551 y Fq(.)22 b(Suc)o(h)17 b(a)e(co)q(de)i(exists)f(b)o(y)g(Theorem)g(1\(v\).)21 b(F)l(or)15 b(part)h(\(i\))f(w)o(e)h(set)g Fo(n)1481 558 y Fi(0)1515 551 y Fq(=)e Fo(k)1588 558 y Fi(0)1621 551 y Fq(=)g(2,)i Fo(A)e Fq(=)1819 516 y Fg(\000)1838 531 y Fi(0)h(1)1838 567 y(1)g(0)1888 516 y Fg(\001)1907 551 y Fq(.)60 636 y(Then)g(for)f(b)q(oth)g(parts)g(w)o(e)g (arbitrarily)h(lab)q(el)h(the)e(columns)h(of)f Fo(A)h Fq(b)o(y)f(the)g (elemen)o(ts)h(of)f(the)h(\014eld)g(of)f(order)60 720 y Fo(n)87 727 y Fi(0)107 720 y Fq(,)h(form)f(the)h Fo(N)f Fh(\002)d Fo(n)444 704 y Fn(K)444 732 y Fi(0)493 720 y Fq(arra)o(y)j(whose)h(columns)h(are)f (the)g(co)q(dew)o(ords)g(of)f Fh(C)s Fq(,)g(and)i(replace)g(ev)o(ery)f(en)o (try)f(of)60 805 y(this)j(arra)o(y)f(b)o(y)g(the)h(corresp)q(onding)h(column) f(of)f Fo(A)p Fq(.)25 b(This)17 b(pro)q(duces)h(a)e Fo(k)1355 812 y Fi(0)1374 805 y Fo(N)g Fh(\002)11 b Fo(n)1500 788 y Fn(K)1500 816 y Fi(0)1552 805 y Fq(arra)o(y)l(.)23 b(F)l(or)16 b(part)g(\(i\))60 890 y(w)o(e)f(further)g(adjoin)h(the)g(all-0's)f(and)h(all-1's)g(ro)o(ws.)j (Theorem)c(3)g(no)o(w)g(implies)j(that)c(the)i(resulting)g(arra)o(y)60 974 y(is)g(a)f(3-co)o(v)o(ering.)131 1059 y(F)l(or)h(example,)j(let)f(us)f (apply)h(Theorem)f(7\(ii\))h(with)g Fo(n)1091 1066 y Fi(0)1127 1059 y Fq(=)f(11,)g Fo(k)1279 1066 y Fi(0)1314 1059 y Fq(=)g(12.)26 b(W)l(e)17 b(ma)o(y)g(tak)o(e)f Fo(K)j Fq(=)e(6)g(in)60 1144 y(\(13\),)c(and)i(deduce)g(that)f Fo(g)526 1151 y Fi(3)546 1144 y Fq(\(11)610 1127 y Fi(6)628 1144 y Fq(\))f Fh(\024)g Fq(132.)18 b(This)d(arra)o(y)f(w)o(as)f(also)i(giv)o(en)g(b)o(y)f(Roux,)h (but)f(his)h(claim)h(that)e(it)60 1228 y(is)i(a)f(3-co)o(v)o(ering)g(is)g (unjusti\014ed)i(since)f Fo(K)g Fq(=)d(6)i(violates)g(\(16\).)k(A)c(second)h (iteration)f(giv)o(es)740 1351 y Fo(g)762 1358 y Fi(3)782 1351 y Fq(\(11)846 1332 y Fi(3)p Ff(\002)p Fi(11)927 1320 y Fe(6)945 1351 y Fq(\))d Fh(\024)h Fq(132)c Fh(\002)i Fq(11)1193 1332 y Fi(6)1227 1351 y Fo(;)60 1473 y Fq(and)20 b(so)f(on.)33 b(This)20 b(sequence)g(of)f(3-co)o(v)o(erings)h(has)f Fo(k)q(=)8 b Fq(log)g Fo(n)20 b Fh(\031)g Fq(6)p Fo(:)p Fq(4)e(for)h Fo(n)h Fh(\024)h Fq(10)1511 1456 y Fi(6)1530 1473 y Fq(,)f Fh(\031)g Fq(12)p Fo(:)p Fq(719)e(for)h Fo(n)h Fh(\024)60 1558 y Fq(10)106 1541 y Fi(5000000)250 1558 y Fq(\(compare)e(with)g(\(10\)\),)g(while,)i(for)e (large)h Fo(n)p Fq(,)g Fo(k)q(=)8 b Fq(log)g Fo(n)19 b Fq(gro)o(ws)e(as)h (constan)o(t)12 b Fh(\002)g Fq(log)1717 1539 y Ff(\003)1744 1558 y Fo(n)p Fq(,)20 b(where)60 1642 y(log)119 1624 y Ff(\003)146 1642 y Fo(n)15 b Fq(is)h(the)f(n)o(um)o(b)q(er)h(of)f(logarithms)g(needed)h (to)f(reduce)h Fo(n)f Fq(to)g(a)g(n)o(um)o(b)q(er)g(less)h(than)f(1.)131 1727 y(If)g(instead)g(of)g(MDS)g(co)q(des)g(w)o(e)g(use)g(the)g (algebraic-geometric)h(co)q(des)g(men)o(tioned)g(in)f(\(8\),)f(w)o(e)h (obtain)60 1812 y(the)g(follo)o(wing)h(result.)60 1937 y Fp(Theorem)h(8)h (\([39].\))k Fr(If)14 b(ther)n(e)h(exists)f(a)i(3-c)n(overing)e(arr)n(ay)i (of)f(size)f Fo(k)1305 1944 y Fi(0)1339 1937 y Fr(and)h(length)f Fo(n)1585 1944 y Fi(0)1606 1937 y Fr(,)h(wher)n(e)g Fo(n)1789 1944 y Fi(0)1824 1937 y Fr(is)f(an)60 2021 y(even)k(p)n(ower)i(of)f(a)h (prime,)g(then)f(ther)n(e)g(is)g(an)g(in\014nite)f(se)n(quenc)n(e)f(of)j(3-c) n(overing)f(arr)n(ays)g(of)g(size)g Fo(k)h Fr(and)60 2106 y(length)c Fo(n)g Fr(with)641 2191 y Fq(lim)8 b(inf)658 2216 y Fn(n)p Ff(!1)830 2160 y Fo(k)p 795 2180 94 2 v 795 2222 a Fq(log)h Fo(n)907 2191 y Fh(\024)1051 2160 y Fq(1)p 960 2180 207 2 v 965 2207 a Fi(1)p 965 2214 18 2 v 965 2240 a(2)997 2225 y Fh(\000)1096 2207 y Fi(1)p 1048 2214 114 2 v 1048 2219 a Ff(p)p 1077 2219 39 2 v 21 x Fn(n)1098 2245 y Fe(0)1116 2240 y Ff(\000)p Fi(1)1227 2160 y Fo(k)1251 2167 y Fi(0)p 1192 2180 114 2 v 1192 2222 a Fq(log)f Fo(n)1285 2229 y Fi(0)1327 2191 y Fo(:)60 2309 y Fr(The)16 b(c)n(omplexity)g(of)h(c)n(onstructing)e(these)h(arr)n(ays)g(gr)n (ows)g(as)g(a)h(p)n(olynomial)f(in)f Fo(n)p Fr(.)131 2434 y Fq(F)l(or)f(example,)i(using)g Fo(n)549 2441 y Fi(0)581 2434 y Fq(=)d(14641,)h Fo(k)795 2441 y Fi(0)827 2434 y Fq(=)f(84)h(w)o(e)h(obtain) h Fo(k)d Fh(\024)g Fq(12)p Fo(:)p Fq(347)8 b(log)f Fo(n)30 b Fq(\(1)10 b(+)g Fo(o)p Fq(\(1\)\).)131 2519 y(The)15 b(\014nal)h(theorem)f (is)h(due)f(to)g(Kreher)h([36)o(])f(and)g(the)g(author.)60 2644 y Fp(Theorem)i(9.)22 b Fr(\(i\))15 b(If)g(a)g(binary)h(c)n(onstant)e (weight)i(c)n(o)n(de)f(with)h(length)f Fo(N)5 b Fr(,)15 b(c)n(onstant)f (weight)i Fo(W)6 b Fr(,)15 b(minimal)60 2729 y(distanc)n(e)g Fo(D)i Fr(and)f(c)n(ontaining)e Fo(M)21 b Fr(c)n(o)n(dewor)n(ds)16 b(exists)f(with)h Fo(N)h(>)c Fq(3)p Fo(W)22 b Fr(and)16 b Fo(D)e(>)f(W)6 b Fr(,)16 b(then)f Fo(g)1677 2736 y Fi(3)1697 2729 y Fq(\()p Fo(M)5 b Fq(\))12 b Fh(\024)h Fq(2)p Fo(N)5 b Fr(.)967 2853 y Fq(10)p eop %%Page: 9 9 bop 131 74 a Fq(Before)14 b(lea)o(ving)h(the)f(asymptotic)g(theory)g(w)o(e)g (men)o(tion)h(another)f(result)g(from)g(Roux's)g(thesis)h(that)e(is)60 159 y(at)g(\014rst)g(glance)h(quite)g(surprising.)21 b(F)l(or)12 b(an)o(y)i Fo(t)f Fq(and)h(an)o(y)f Fh(E)k Fo(>)c Fq(0)g(there)h(is)f(a)h (constan)o(t)e Fo(k)1573 166 y Fi(0)1606 159 y Fq(suc)o(h)i(that)f Fh(E)t Fq(-bad)60 244 y Fo(t)p Fq(-co)o(v)o(ering)18 b(arra)o(ys)e(of)h(size) i Fo(k)578 251 y Fi(0)615 244 y Fq(exist)f(for)f(all)h(lengths)g Fo(n)p Fq(.)27 b(The)18 b(size)h(is)f(indep)q(enden)o(t)i(of)d(the)g(n)o(um)o (b)q(er)h(of)60 328 y(columns.)24 b(F)l(or)16 b(example,)i(at)d(most)h(68)g (v)o(ectors)g(are)g(needed)i(for)d(an)i(0.001-bad)e(3-co)o(v)o(ering)h(arra)o (y)g(with)60 413 y(an)o(y)i(n)o(um)o(b)q(er)i(of)e(columns.)32 b(The)19 b(result)g(is)g(easily)h(established)h(b)o(y)d(c)o(ho)q(osing)i (arra)o(ys)d(at)h(random)h(and)60 498 y(using)d(a)f(probabilistic)i(argumen)o (t.)131 583 y(Roux)e(also)g(giv)o(es)h(t)o(w)o(o)e(useful)i(b)q(ounds.)60 703 y Fp(Theorem)h(6.)22 b Fq(\(Roux)16 b([50)o(]\))773 788 y Fo(g)795 795 y Fn(t)p Fi(+1)855 788 y Fq(\()p Fo(n)10 b Fq(+)g(1\))i Fh(\025)h Fq(2)p Fo(g)1101 795 y Fn(t)1115 788 y Fq(\()p Fo(n)p Fq(\))j Fo(;)632 b Fq(\(12\))750 893 y Fo(g)772 900 y Fi(3)791 893 y Fq(\(2)p Fo(n)p Fq(\))12 b Fh(\024)h Fo(g)959 900 y Fi(3)979 893 y Fq(\()p Fo(n)p Fq(\))d(+)g Fo(g)1119 900 y Fi(2)1138 893 y Fq(\()p Fo(n)p Fq(\))16 b Fo(:)609 b Fq(\(13\))60 1014 y Fp(Pro)q(of.)17 b Fq(\(12\))d(is)i(clear.)21 b(\(13\))14 b(Let)h Fo(A)p Fq(,)g Fo(B)j Fq(b)q(e)e(arra)o(ys)e(ac)o(hieving)j Fo(g)1200 1021 y Fi(3)1219 1014 y Fq(\()p Fo(n)p Fq(\),)e Fo(g)1332 1021 y Fi(2)1351 1014 y Fq(\()p Fo(n)p Fq(\).)20 b(Then)c(it)f(is)h(easy)f (to)g(c)o(hec)o(k)60 1098 y(that)934 1155 y Fo(A)44 b(A)932 1211 y(B)p 1011 1175 37 2 v 45 w(B)60 1289 y Fq(is)16 b(a)f(3-co)o(v)o(ering) g(arra)o(y)f(\(where)h(the)g(bar)g(indicates)i(the)e(complemen)o(tary)g(arra) o(y\).)131 1373 y(W)l(e)i(no)o(w)g(giv)o(e)g(some)g(constructions)h(for)e (3-co)o(v)o(ering)h(arra)o(ys.)25 b(The)18 b(ro)o(ws)e(of)h Fo(I)1527 1380 y Fn(n)1567 1373 y Fq(and)p 1658 1337 24 2 v 18 w Fo(I)1681 1380 y Fn(n)1722 1373 y Fq(sho)o(w)g(that)60 1458 y Fo(g)82 1465 y Fi(3)101 1458 y Fq(\()p Fo(n)p Fq(\))e Fh(\024)h Fq(2)p Fo(n)h Fq(for)f Fo(n)f Fh(\025)h Fq(4.)24 b(It)17 b(is)g(also)g(trivial)h(to)e(v)o(erify)h(\(b)o(y)g(computer\))f(that) g(a)h(normalized)h(Hadamard)60 1543 y(matrix)f(\()p Fo(H)266 1550 y Fi(12)303 1543 y Fq(\))h(of)f(order)g(12)g(yields)j(a)d(3-co)o(v)o (ering)h(arra)o(y)e(with)i Fo(n)f Fq(=)h(11)f(and)h Fo(k)f Fq(=)h(12)f(when)h(the)g(initial)60 1627 y(column)f(of)e(1's)g(is)h(deleted)h (and)f Fh(\000)p Fq(1's)f(are)g(replaced)i(b)o(y)f(0's.)k(None)c(of)f(the)h (\014v)o(e)g(Hadamard)f(matrices)g(of)60 1712 y(order)h(16)g(pro)q(duces)i(a) e(3-co)o(v)o(ering)g(of)h(length)g(15)f(in)h(this)g(w)o(a)o(y)l(.)23 b(Ho)o(w)o(ev)o(er,)16 b(Mallo)o(ws)g([41])g(found)h(that)e(if)60 1797 y(the)h(\014rst)f(and)h(nin)o(th)g(columns)g(are)g(omitted)f(from)g(the) h(fourth)f(Hadamard)g(matrix)g(of)g(order)h(16)f(\()p Fo(B)1847 1804 y Fi(3)1882 1797 y Fq(in)60 1881 y(the)e(notation)g(of)g(Assm)o(us)g (and)g(Key)h([3)o(]\),)e(a)h(3-co)o(v)o(ering)g(arra)o(y)f(is)i(obtained)g (with)f(length)h(14)f(and)g(size)h(16.)60 1966 y(Kreher)j(and)h(T)l(onc)o (hev)f([37)o(])f(observ)o(ed)h(that)f(the)h(incidence)j(matrix)d(of)f(the)h (nicest)h(2-\(16)p Fo(;)8 b Fq(6)p Fo(;)g Fq(2\))13 b(biplane)60 2051 y(\([10)o(];)i([22)o(],)g(T)l(able)h(XVI)q(I,)h(No.)e(5;)g([27)o(],)g(T) l(able)i(I.1,)e(No.)g(10\),)f(supplemen)o(ted)k(b)o(y)d Fp(0)p Fq(,)h(forms)f(a)g(3-co)o(v)o(ering)60 2135 y(with)h Fo(n)c Fq(=)h(16,)i Fo(k)e Fq(=)g(17.)20 b(Another)15 b(arra)o(y)f(with)h(the)h (same)e(parameters)h(is)h(giv)o(en)f(in)h(\(1\).)60 2256 y Fp(Theorem)h(7.)22 b Fr(\(i\))16 b(If)f(an)i Fq([)p Fo(N)r(;)8 b(K)q(;)g(D)p Fq(])14 b Fr(binary)i(interse)n(cting)e(c)n(o)n(de)i Fh(C)j Fr(exists)c(then)807 2374 y Fo(g)829 2381 y Fi(3)849 2374 y Fq(\(2)890 2355 y Fn(K)923 2374 y Fq(\))d Fh(\024)h Fq(2)p Fo(N)i Fq(+)10 b(2)16 b Fo(:)666 b Fq(\(14\))60 2492 y Fr(\(ii\))15 b(If)h(a)h(3-surje)n(ctive)f(arr)n(ay)h Fo(A)f Fr(exists)f(of)i(size)e Fo(k)932 2499 y Fi(0)968 2492 y Fr(and)h(length)g Fo(n)1217 2499 y Fi(0)1237 2492 y Fr(,)g(wher)n(e)g Fo(n)1422 2499 y Fi(0)1458 2492 y Fr(is)g(a)h(prime)f(p)n(ower,)h(then)780 2611 y Fo(g)802 2618 y Fi(3)821 2611 y Fq(\()p Fo(n)866 2592 y Fn(K)866 2622 y Fi(0)900 2611 y Fq(\))c Fh(\024)g Fo(k)1003 2618 y Fi(0)1022 2611 y Fq(\(2)p Fo(K)f Fh(\000)f Fq(1\))638 b(\(15\))60 2729 y Fr(holds)16 b(for)h(al)r(l)f Fo(K)j Fr(with)e Fq(1)12 b Fh(\024)h Fo(K)i Fh(\024)656 2719 y Fl(1)676 2729 y(/)695 2738 y(2)714 2729 y Fo(n)741 2736 y Fi(0)771 2729 y Fq(+)c(1)p Fr(.)979 2853 y Fq(9)p eop %%Page: 8 10 bop 60 74 a Fp(Pro)q(of)17 b(of)f(upp)q(er)h(b)q(ound)e Fq(\(after)e(Roux)i ([50)o(]\).)k(Let)c Fo(n)1044 58 y Ff(0)1068 74 y Fq(=)e(3)p Fo(n=)p Fq(2,)h Fo(k)g Fq(=)f(2)p Fo(r)q Fq(.)19 b(W)l(e)14 b(consider)i(the)e(collection)60 159 y(of)19 b Fo(k)14 b Fh(\002)f Fo(n)229 143 y Ff(0)260 159 y Fq(arra)o(ys)18 b Fo(A)h Fq(formed)g(b)o(y)g(c) o(ho)q(osing)h(the)f(columns)h(to)e(b)q(e)i(random)f(v)o(ectors)f(of)h (length)h(2)p Fo(r)g Fq(and)60 244 y(w)o(eigh)o(t)13 b Fo(r)q Fq(.)19 b(F)l(or)12 b(an)o(y)h(triple)i Fo(s)e Fq(=)g Fo(s)642 251 y Fi(1)662 244 y Fo(s)683 251 y Fi(2)703 244 y Fo(s)724 251 y Fi(3)757 244 y Fq(of)g(co)q(ordinates)g(and)h(an)o(y)e(binary)i(v)o (ector)f Fo(v)h Fq(=)f Fo(v)1598 251 y Fi(1)1618 244 y Fo(v)1640 251 y Fi(2)1660 244 y Fo(v)1682 251 y Fi(3)1701 244 y Fq(,)h(de\014ne)g(the) 60 328 y(random)e(v)m(ariable)j Fo(Q)429 335 y Fn(s;v)475 328 y Fq(\()p Fo(A)p Fq(\))d(to)g(b)q(e)i(0)e(if)i(the)f(pro)s(jection)f(of)h Fo(A)f Fq(on)o(to)g Fo(s)h Fq(includes)i Fo(v)r Fq(,)e(and)g(to)f(b)q(e)i(1)e (otherwise.)60 413 y(Also)k(let)783 498 y Fo(Q)p Fq(\()p Fo(A)p Fq(\))c(=)949 457 y Fg(X)957 545 y Fn(s;v)1017 498 y Fo(Q)1053 505 y Fn(s;v)1099 498 y Fq(\()p Fo(A)p Fq(\))j Fo(:)60 620 y Fq(If)g Fo(Q)p Fq(\()p Fo(A)p Fq(\))d(=)h(0)i(then)g Fo(A)g Fq(is)g(3-co)o(v)o(ering)g(arra)o(y)l(,)e(while)k(if)e Fo(Q)p Fq(\()p Fo(A)p Fq(\))d Fo(<)h(b)h Fq(then)h Fo(A)g Fq(is)h(an)e Fh(E)t Fq(-bad)i(3-co)o(v)o(ering)e(arra)o(y)60 705 y(with)i Fh(E)g Fq(=)d Fo(b=)295 671 y Fg(\000)313 685 y Fn(n)334 673 y Fm(0)321 721 y Fi(3)346 671 y Fg(\001)365 705 y Fq(.)131 790 y(The)i(probabilit)o(y)h(that)f Fo(Q)591 797 y Fn(s;v)637 790 y Fq(\()p Fo(A)p Fq(\))e(=)f(1)j(is)h(equal)g(to)705 863 y Fg( )738 904 y Fq(2)p Fo(r)750 966 y(r)783 863 y Fg(!)(816)874 Y FF(000)P FI(2)895 882 Y FN(R)873 894 Y FG(x)870 985 985 Y FN(U)p Fi(=0)944τy fg()y yy y(r)y y(u)αy fg(!)1035 874 Y FI(2)1055 863 Y FG()1088 1088 904 y Fq(2)p fo(r)10 b Fh(000)h FO(u)1140 966 y(r)966 y fg(!)1262 935 y Fo(:)60 1070 y Fq(T)l(o)16 b(see)g(this,)g(tak)o(e)f(for)g(example)i Fo(s)c Fq(=)h(123,)h Fo(v)h Fq(=)d(111,)i(and)h(supp)q(ose)h(the)f(\014rst)f (column)i(consists)f(of)f Fo(r)i Fq(1's)60 1155 y(follo)o(w)o(ed)g(b)o(y)g(r) g(0's.)25 b(The)17 b(n)o(um)o(b)q(er)h(of)f(c)o(hoices)h(for)e(columns)i(2)f (and)g(3)g(is)1368 1121 y Fg(\000)1387 1135 y Fi(2)p Fn(r)1396 1171 y(r)1422 1121 y Fg(\001)1441 1127 y Fi(2)1461 1155 y Fq(.)25 b(Let)18 b Fo(u)f Fq(b)q(e)g(the)h(n)o(um)o(b)q(er)60 1240 y(of)d(common)g(1's)g(in)h(columns)g(1)f(and)h(2.)k Fo(Q)812 1247 y Fn(s;v)858 1240 y Fq(\()p Fo(A)p Fq(\))13 b(=)g(1)i(if)h(and)g(only)g (if)f(the)h(1's)f(in)h(column)g(3)f(are)g(disjoin)o(t)60 1325 y(from)e(these)i(common)e(1's,)h(an)g(ev)o(en)o(t)g(whic)o(h)h(can)f(happ)q (en)h(in)1146 1290 y Fg(\000)1167 1304 y Fn(r)1165 1340 y(u)1186 1290 y Fg(\001)1205 1301 y Fi(2)1224 1290 y Fg(\000)1243 1304 y Fi(2)p Fn(r)q Ff(\000)p Fn(u)1276 1340 y(r)1326 1290 y Fg(\001)1359 1325 y Fq(w)o(a)o(ys.)j(The)d(exp)q(ected)g(v)m(al(60)1409 y y(q)p fq(\)(p fO(a)p fq(\))g(is)g(is)642,1507 y(8)665,1435 y y fg()697,1476 y y(n)724 1459 y f(0)0 y fq(γ)y fg(!)802 1476 Y FQ(2)p FO(r)813 813 y(r)847 1435 y y fg(!)879 1446 Y FF(000)p FI(2)958 1454 y fn(r)937 1466 y fg(x)934 1557 1557 y fn(u)pFi(=0)1007τy fg()y yy y(r)y y(u)y y fg(!)1099 1446 Y FI(2)1119 1435 Y FG()1152,1476 y y Fq(2)p FO(r)10 b Fh(000)H FO(u)1202 1538 y(u)1538 y y fg(!)1326 1507 y Fo(:)500 b Fq(\(11\))60 1626 y(W)l(e)18 b(c)o(ho)q(ose)g Fo(r)h Fq(to)f(b)q(e)h(the)f(largest)g(in)o(teger)g(so)g (that)g(this)g(quan)o(tit)o(y)g(is)h(less)g(than)f Fo(n=)p Fq(2.)28 b(It)18 b(follo)o(ws)h(that)60 1711 y(there)f(exists)g(a)g(2)p Fo(r)13 b Fh(\002)f Fo(n)479 1694 y Ff(0)509 1711 y Fq(arra)o(y)17 b Fo(A)h Fq(whic)o(h)h(con)o(tains)f(at)f(most)g Fo(n=)p Fq(2)h(bad)g (triples.)30 b(By)18 b(deleting)h(at)f(most)60 1796 y Fo(n=)p Fq(2)13 b(co)q(ordinates)g(w)o(e)g(can)g(eliminate)i(these)e(bad)h(triples,)g (pro)q(ducing)g(a)f(3-co)o(v)o(ering)g(arra)o(y)f(of)g(size)i(2)p Fo(r)7 b Fh(\002)f Fo(n)p Fq(.)60 1880 y(The)17 b(sum)g(in)h(\(11\))d(is)i (dominated)h(b)o(y)e(the)h(terms)g(near)f Fo(u)g Fq(=)f Fo(\013r)q Fq(,)i(where)g Fo(\013)f Fq(=)f(\(3)c Fh(\000)1554 1843 y(p)p 1591 1843 23 2 v 1591 1880 a Fq(5\))p Fo(=)p Fq(2)k(=)g Fo(:)p Fq(3819)8 b Fo(:)g(:)g(:)l Fq(,)60 1965 y(and)15 b(w)o(e)g(\014nd)h Fo(k)e Fq(=)f(2)p Fo(r)g Fq(=)g(7)p Fo(:)p Fq(56444)8 b Fo(:)g(:)f(:)t Fq(log)h Fo(n)16 b Fq(\(1)9 b(+)i Fo(o)p Fq(\(1\)\),)i(as)i(claimed.)22 b(The)15 b(constan)o(t)f(is)965 2064 y(4)p 641 2085 671 2 v 641 2126 a(4)c Fh(\000)g Fq(2)p Fo(H)t Fq(\()p Fo(\013)p Fq(\))f Fh(\000)h Fq(\(2)g Fh(\000)g Fo(\013)p Fq(\))p Fo(H)t Fq(\(1)p Fo(=)p Fq(\(2)e Fh(\000)j Fo(\013)p Fq(\)\))1332 2095 y Fo(:)131 2226 y Fq(The)k(upp)q(er)i(b)q(ound)g(in)f(Theorem)g(5)f(is)h(nonconstructiv) o(e.)22 b(Alon)16 b([1)o(])g(ga)o(v)o(e)e(an)i(explicit)i(construction)60 2310 y(of)f Fo(t)p Fq(-co)o(v)o(ering)i(arra)o(ys)d(with)i Fo(k)g Fh(\024)f Fo(c)8 b Fq(log)g Fo(n)p Fq(,)18 b(but)g(his)g(constan)o(t)f (is)h(extremely)h(large.)27 b(In)19 b(the)e(case)h Fo(t)f Fq(=)h(3,)60 2395 y(Cohen)e([12)o(])f(found)h(that)f(the)h(constan)o(t)f(in)h(Alon's)g (construction)g(is)g(ab)q(out)f(10)1455 2378 y Fi(656)1510 2395 y Fq(.)21 b(Roux)16 b([50)o(])f(w)o(as)g(able)60 2480 y(to)f(reduce)i(the)g(constan)o(t)e(to)g(10)622 2463 y Fi(256)677 2480 y Fq(.)20 b(The)15 b(construction)g(using)h(non)o(binary)g(in)o (tersecting)g(co)q(des)g(giv)o(en)f(in)60 2564 y(Theorem)i(7\(ii\))h(pro)q (duces)g(arra)o(ys)f(with)h(a)f(p)q(olynomial-time)i(construction)f(and)g Fo(k)f Fh(\024)g Fq(12)p Fo(:)p Fq(347)8 b(log)e Fo(n)p Fq(,)18 b(as)60 2649 y(w)o(ell)e(as)f(explicit)i(arra)o(ys)d(with)i(a)f(small)h (constan)o(t)e(for)g(a)h(wide)h(range)f(of)g(v)m(alues)h(of)f Fo(n)p Fq(.)979 2853 y(8)p eop %%Page: 7 11 bop 356 104 a Fq(T)l(able)16 b(2:)k(Generator)14 b(matrices)h([)p Fo(A)g(I)t Fq(])f(for)h(binary)h(in)o(tersecting)g(co)q(des.)p 119 216 1742 2 v 118 273 2 57 v 146 256 a Fo(N)p 214 273 V 58 w(K)p 309 273 V 58 w(D)p 404 273 V 498 w Fq(Ro)o(ws)e(of)h Fo(A)g Fq(in)h(hexadecimal)p 1860 273 V 119 275 1742 2 v 118 331 2 57 v 144 314 a(21)p 214 331 V 68 w(7)p 309 331 V 73 w(7)p 404 331 V 53 w(35ED,)6 b(216E,)h(38B8,)f(1389,)g(2E5D,)h(339E,)f(2FF7)p 1860 331 V 118 387 V 214 387 V 309 387 V 404 387 V 1860 387 V 118 417 V 144 400 a(25)p 214 417 V 68 w(8)p 309 417 V 73 w(9)p 404 417 V 53 w(001FF,)g(01FF0,)f(0333A,)i(07C3C,)f(0B996,)g(0D6B7,)g (13A77,)g(14A9D)p 1860 417 V 118 473 V 214 473 V 309 473 V 404 473 V 1860 473 V 118 503 V 144 486 a(41)p 214 503 V 49 w(11)p 309 503 V 50 w(11)p 404 503 V 49 w(3C8DF41E,)f(33EC23BF,)h(22543C8B,)g (112FD46A,)f(14FF5B3D,)g(2ED72BF9,)p 1860 503 V 118 559 V 214 559 V 309 559 V 404 559 V 430 542 a(3EE1EF78,)h(0C6EFF07,)f(357CC3AD,)g (3D75F)-5 b(ADE,)6 b(2FFFD2BB)p 1860 559 V 119 561 1742 2 v 60 739 a Fs(3.)52 b(Binary)23 b(3-co)n(v)n(ering)g(arra)n(ys)147 869 y Fq(In)16 b(this)h(section)f(w)o(e)g(study)g(binary)g(3-co)o(v)o(ering)g (arra)o(ys,)f(b)q(eginning)j(with)e(the)g(asymptotic)g(results.)60 954 y(Let)i Fo(g)166 961 y Fn(t)180 954 y Fq(\()p Fo(n)p Fq(\))f(denote)h (the)g(minimal)h(size)f(of)f(a)g(binary)h Fo(t)p Fq(-co)o(v)o(ering)g(arra)o (y)e(of)h(length)i Fo(n)p Fq(.)26 b(Then)18 b Fo(g)1734 961 y Fi(1)1754 954 y Fq(\()p Fo(n)p Fq(\))e(=)h(2,)60 1039 y(and)e Fo(g)170 1046 y Fi(2)190 1039 y Fq(\()p Fo(n)p Fq(\))g(is)g(determined)i(b)o (y)e(\(2\).)60 1173 y Fp(Theorem)i(5.)22 b Fr(As)16 b Fo(n)d Fh(!)g(1)p Fr(,)342 1303 y Fq(3)p Fo(:)p Fq(21256)8 b Fo(:)g(:)f(:)t Fq(log)i Fo(n)16 b Fq(\(1)10 b(+)g Fo(o)p Fq(\(1\)\))i Fo(<)h(g)946 1310 y Fi(3)965 1303 y Fq(\()p Fo(n)p Fq(\))f Fo(<)h Fq(7)p Fo(:)p Fq(56444)8 b Fo(:)g(:)f(:)t Fq(log)i Fo(n)16 b Fq(\(1)10 b(+)g Fo(o)p Fq(\(1\)\))15 b Fo(:)201 b Fq(\(10\))60 1468 y Fp(Remarks.)25 b Fq(The)17 b(lo)o(w)o(er)g(b)q(ound)h(is)g(due)g(to)f (Kleitman)h(and)g(Sp)q(encer)g([34)o(].)26 b(The)17 b(constan)o(t)g(is)h (equal)g(to)60 1553 y(\()p Fo(H)t Fq(\()138 1543 y Fl(1)156 1553 y(/)175 1562 y(4)194 1553 y Fq(\))5 b Fh(\000)257 1543 y Fl(1)276 1553 y(/)295 1562 y(2)315 1553 y Fq(\))333 1536 y Ff(\000)p Fi(1)380 1553 y Fq(,)12 b(where)h Fo(H)t Fq(\()p Fo(x)p Fq(\))f(=)h Fh(\000)p Fo(x)8 b Fq(log)g Fo(x)d Fh(\000)g Fq(\(1)g Fh(\000)g Fo(x)p Fq(\))j(log)o(\(1)d Fh(\000)g Fo(x)p Fq(\).)18 b(There)13 b(has)g(b)q(een)g(some)g(confusion)g(in)60 1637 y(the)h(literature)h(concerning)h(this)e(b)q(ound.)21 b(An)14 b(unfortunate)g(misprin)o(t)h(in)g(Eq.)f(\(11\))f(of)h([34)o(])g (misleadingly)60 1722 y(suggests)h(that)h(a)g(stronger)f(lo)o(w)o(er)g(b)q (ound)i(has)f(b)q(een)i(established,)f(and)f(this)h(error)e(w)o(as)g(rep)q (eated)i(in)g([8)o(])60 1807 y(and)e([12)o(].)131 1891 y(Kleitman)g(and)g(Sp) q(encer)g(also)f(obtain)h(an)f(upp)q(er)h(b)q(ound)h(of)d(15.5726)g(log)8 b Fo(n)14 b Fq(b)o(y)h(considering)g(random)60 1976 y(arra)o(ys.)27 b(By)18 b(using)h(the)f(main)h(result)f(of)g(the)g(Erd\177)-23 b(os-F)l(rankl-F)q(\177)f(uredi)19 b(pap)q(er)g([20)o(])e(this)i(can)f(b)q(e) h(reduced)60 2061 y(to)d(11)p Fo(:)p Fq(02)8 b(log)e Fo(n)p Fq(.)24 b(A)17 b(further)f(reduction)i(to)e(9)p Fo(:)p Fq(6377)8 b(log)e Fo(n)17 b Fq(can)g(b)q(e)g(obtained)g(b)o(y)f(using)i(the)e(binary)i (in)o(ter-)60 2145 y(secting)f(co)q(des)g(whose)g(existence)h(is)f(guaran)o (teed)f(b)o(y)h(the)g(righ)o(t-hand)g(side)g(of)g(\(9\))e(in)j(the)f (construction)60 2230 y(of)e(Theorem)g(7\(i\))g(b)q(elo)o(w.)20 b(The)15 b(b)q(ound)h(of)f(7)p Fo(:)p Fq(56444)8 b(log)e Fo(n)15 b Fq(uses)h(an)f(argumen)o(t)f(due)i(to)f(Roux)g([50)o(].)20 b(Roux)60 2315 y(actually)d(claims)h(an)e(upp)q(er)h(b)q(ound)h(of)e(6)p Fo(:)p Fq(294)8 b(log)e Fo(n)p Fq(,)17 b(but)f(this)h(seems)g(to)e(b)q(e)i (an)g(arithmetic)g(error.)23 b(The)60 2399 y(follo)o(wing)16 b(is)g(a)f(corrected)g(v)o(ersion)g(of)g(his)h(argumen)o(t.)131 2484 y(Roux)f(de\014nes)i(a)e Fo(k)c Fh(\002)g Fo(n)k Fq(binary)h(arra)o(y)e Fo(A)i Fq(to)f(b)q(e)h(an)f Fh(E)t Fr(-b)n(ad)h Fo(t)p Fq(-co)o(v)o(ering)g (arra)o(y)e(if)i(the)f(n)o(um)o(b)q(er)h(of)f(bad)60 2569 y Fo(t)p Fq(-tuples,)j(that)e(is,)h Fo(t)p Fq(-tuples)i(of)d(co)q(ordinates)h (suc)o(h)g(that)f(the)h(pro)s(jection)f(of)h Fo(A)f Fq(on)o(to)g(those)h(co)q (ordinates)60 2653 y(do)q(es)f(not)e(include)k(all)e(2)486 2637 y Fn(t)516 2653 y Fq(p)q(ossibilities,)i(is)e(at)e(most)g Fh(E)1013 2619 y Fg(\000)1033 2633 y Fn(n)1037 2669 y(t)1054 2619 y Fg(\001)1073 2653 y Fq(.)979 2853 y(7)p eop %%Page: 6 12 bop 60 74 a Fq(it)16 b(su\016ces)g(to)f(tak)o(e)g Fo(c)436 81 y Fi(2)455 74 y Fo(K)k Fq(p)q(oin)o(ts,)c(where)h Fo(c)814 81 y Fi(2)847 74 y Fq(=)e(log)8 b(4)p Fo(=)g Fq(log\(4)p Fo(=)p Fq(3\).)19 b(Therefore)d(1)10 b(+)g Fo(f)1518 81 y Fi(2)1538 74 y Fq(\()p Fo(K)s Fq(\))j Fo(<)h(c)1698 81 y Fi(2)1717 74 y Fo(K)s Fq(.)21 b(Retter)60 159 y([49)o(])13 b(sho)o(ws)f(that)g Fo(f)5 b Fq(\()p Fo(K)s Fq(\))12 b(=)h Fo(c)553 166 y Fi(2)572 159 y Fo(K)s Fq(\(1)5 b(+)g Fo(o)p Fq(\(1\)\))12 b(can)h(b)q(e)h(ac)o(hiev)o (ed)g(b)o(y)f(Goppa)f(co)q(des)h(\(although)g(his)h(argumen)o(t)60 244 y(is)i(also)f(nonconstructiv)o(e\).)131 328 y(T)l(able)g(1)g(giv)o(es)g (the)h(b)q(est)f(upp)q(er)h(b)q(ounds)g(presen)o(tly)g(kno)o(wn)f(on)g Fo(f)1292 335 y Fi(2)1312 328 y Fq(\()p Fo(K)s Fq(\))f(for)g(small)i(v)m (alues)g(of)f Fo(K)s Fq(.)20 b(F)l(or)60 413 y Fo(K)e Fh(\024)d Fq(6)h(the)h(v)m(alues)h(of)e Fo(f)498 420 y Fi(2)518 413 y Fq(\()p Fo(K)s Fq(\))g(are)g(easily)i(pro)o(v)o(ed)e(to)g(b)q(e)h(optimal,)h (using)f(Theorems)f(1\(i\),)h(2\(ii\),)f(2\(iii\),)60 498 y(and)c(the)g(b)q (ounds)h(on)e(the)h(minimal)i(distance)e(of)g(binary)g(linear)h(co)q(des)f (giv)o(en)h(in)f([58)o(].)19 b(F)l(or)11 b(example,)i Fo(f)1842 505 y Fi(2)1862 498 y Fq(\(5\))60 583 y(cannot)k(b)q(e)h(less)g(than)f(13)f (since)i(no)g([12)p Fo(;)8 b Fq(5)p Fo(;)g Fq(5)o(])14 b(linear)19 b(co)q(de)e(exists)h([6)o(].)25 b(The)18 b(v)m(alues)g(of)f Fo(f)1641 590 y Fi(2)1661 583 y Fq(\(1\))p Fo(;)8 b(:)g(:)g(:)t(;)g(f)1843 590 y Fi(2)1862 583 y Fq(\(5\))60 667 y(w)o(ere)15 b(\014rst)g(determined)h (in)g([32)o(].)k(Con)o(w)o(a)o(y)14 b([16)o(])g(has)h(sho)o(wn)g(that)g Fo(f)1250 674 y Fi(2)1270 667 y Fq(\(7\))c Fh(\025)i Fq(19.)60 804 y(T)l(able)21 b(1:)31 b(Minimal)22 b(length)f Fo(f)609 811 y Fi(2)629 804 y Fq(\()p Fo(K)s Fq(\))f(of)g(an)o(y)h([)p Fo(N)r(;)8 b(K)q(;)g(D)q Fq(])17 b(binary)k(in)o(tersecting)h(co)q(de.)37 b(The)21 b(en)o(tries)g(for)60 860 y Fo(K)c Fh(\024)e Fq(6)g(are)h(exact)g (while)i(the)e(remaining)h(en)o(tries)f(giv)o(e)h(upp)q(er)g(b)q(ounds)g(on)f Fo(f)1431 867 y Fi(2)1451 860 y Fq(\()p Fo(K)s Fq(\).)22 b(Explicit)c(co)q (des)e(are)60 917 y(kno)o(wn)f(in)h(ev)o(ery)f(case.)444 1067 y Fo(K)61 b(f)5 b Fq(\()p Fo(K)s Fq(\))49 b(Co)q(de)471 1153 y(1)131 b(1)50 b Fh(f)p Fq(0)p Fo(;)8 b Fq(1)p Fh(g)471 1209 y Fq(2)131 b(3)50 b([3)p Fo(;)8 b Fq(2)p Fo(;)g Fq(2])k(ev)o(en)j(w)o(eigh)o (t)g(co)q(de)471 1266 y(3)131 b(6)50 b([6)p Fo(;)8 b Fq(3)p Fo(;)g Fq(3])k(shortened)j(Hamming)h(co)q(de)471 1322 y(4)131 b(9)50 b(Use)15 b(Theorem)g(1\(iv\))471 1379 y(5)109 b(13)49 b(Omit)16 b(co)q(ordinates)f(1,)g(6)g(from)f(follo)o(wing)i(co)q(de)471 1435 y(6)109 b(15)49 b([15)p Fo(;)8 b Fq(6)p Fo(;)g Fq(6])k(BCH)j(co)q(de)h ([14)o(])471 1491 y(7)109 b(21)49 b(See)16 b(T)l(able)g(2)471 1548 y(8)109 b(25)49 b(See)16 b(T)l(able)g(2)471 1604 y(9)109 b(29)49 b(Omit)16 b(co)q(ordinates)f(1,)g(2)g(from)f(follo)o(wing)i(co)q(de) 448 1661 y(10)109 b(31)49 b(Dual)15 b(of)g([31)p Fo(;)8 b Fq(21)p Fo(;)g Fq(5])k(BCH)j(co)q(de)h([14)o(])448 1717 y(11)109 b(41)49 b(See)16 b(T)l(able)g(2)448 1774 y(12)109 b(46)448 1830 y(13)g(51)448 1887 y(14)g(56)448 1943 y(17)g(63)49 b(Dual)15 b(of)g([63)p Fo(;)8 b Fq(46)p Fo(;)g Fq(7])k(BCH)j(co)q(de)131 2133 y(Tw)o(o)e(of)h(the)h (b)q(est)g(co)q(des)g(in)g(T)l(able)h(1)e(are)g(duals)i(of)e(BCH)h(co)q(des.) 20 b(It)14 b(seems)h(lik)o(ely)h(that)e(the)h(duals)g(of)60 2218 y(some)g(longer)g(BCH)g(co)q(des)h(will)h(also)e(pro)o(vide)g(go)q(o)q (d)g(in)o(tersecting)h(co)q(des.)21 b(The)15 b(ob)o(vious)g(approac)o(h)g(is) h(to)60 2302 y(use)g(the)f(Carlitz-Uc)o(hiy)o(ama)h(b)q(ound)g(\([40)o(],)e (p.)h(280\))f(to)h(guaran)o(tee)f(that)h(the)g(w)o(eigh)o(ts)g(satisfy)g (condition)60 2387 y(\(iii\),)h(but)f(unfortunately)h(the)f(resulting)h(co)q (des)g(are)f(quite)h(w)o(eak.)131 2472 y(Generator)d(matrices)i(for)f(some)g (of)h(the)g(other)f(co)q(des)h(men)o(tioned)h(in)f(T)l(able)h(1)e(are)g(giv)o (en)i(in)f(T)l(able)h(2.)60 2556 y(If)h(the)h(generator)e(matrix)h(has)g(the) g(form)f([)p Fo(A)h(I)t Fq(])f(then)i(T)l(able)g(2)e(giv)o(es)i(the)f(ro)o (ws)f(of)h Fo(A)g Fq(in)h(hexadecimal.)60 2641 y(The)d(remaining)i(co)q(des)e (in)h(T)l(able)g(1)f(ma)o(y)g(b)q(e)h(obtained)g(from)e(the)h(author.)979 2853 y(6)p eop %%Page: 5 13 bop 60 74 a Fr(is)16 b(di\013er)n(ent,)f(another)i(c)n(o)n(or)n(dinate)e (wher)n(e)h Fo(v)i Fr(and)e Fo(w)h Fr(agr)n(e)n(e)e(and)i Fo(u)f Fr(is)f(di\013er)n(ent,)h(and)g(a)g(thir)n(d)g(c)n(o)n(or)n(dinate)60 159 y(wher)n(e)g Fo(w)h Fr(and)g Fo(u)f Fr(agr)n(e)n(e)g(and)g Fo(v)i Fr(is)e(di\013er)n(ent.)131 290 y Fq(Of)f(course)g(p)q(ossibilit)o(y)j (\(i\))d(cannot)g(o)q(ccur)g(for)g(binary)h(co)q(des.)60 374 y Fp(Pro)q(of.)h Fq(W)l(e)f(will)i(sho)o(w)d(that)g Fp(0)p Fq(,)g Fo(x)e Fq(=)h Fo(v)e Fh(\000)f Fo(u)p Fq(,)k Fo(y)h Fq(=)e Fo(w)d Fh(\000)f Fo(u)16 b Fq(ha)o(v)o(e)f(the)h(stated)f(prop)q(ert)o (y)l(.)21 b(Then)c Fo(u;)8 b(v)r(;)g(w)14 b Fq(do)60 459 y(also.)26 b(Either)18 b(there)g(is)g(a)f(co)q(ordinate)h Fo(i)f Fq(suc)o(h)g(that)g Fo(x)1014 466 y Fn(i)1045 459 y Fh(6)p Fq(=)f(0,)h Fo(y)1171 466 y Fn(i)1202 459 y Fh(6)p Fq(=)g(0,)g Fo(y)1329 466 y Fn(i)1360 459 y Fh(6)p Fq(=)f Fo(x)1437 466 y Fn(i)1469 459 y Fq(\(and)h(\(i\))g (holds\),)i(or,)e(for)60 544 y(all)h Fo(i)p Fq(,)f Fo(x)197 551 y Fn(i)228 544 y Fh(6)p Fq(=)f(0)g Fh(\))h Fo(y)402 551 y Fn(i)433 544 y Fq(=)f(0)i(or)e Fo(y)604 551 y Fn(i)635 544 y Fq(=)h Fo(x)713 551 y Fn(i)727 544 y Fq(.)26 b(In)18 b(the)g(latter)f(case) g(there)h(m)o(ust)e(b)q(e)i(a)f(co)q(ordinate)h Fo(j)i Fq(suc)o(h)e(that)60 628 y Fo(x)86 635 y Fn(j)118 628 y Fq(=)13 b(0,)i Fo(y)239 635 y Fn(j)271 628 y Fh(6)p Fq(=)f(0)i(\(or)e(else)j Fo(x)10 b Fh(\000)h Fo(y)r Fq(,)k Fo(y)j Fq(do)d(not)h(in)o(tersect\),)f(a)g(co)q (ordinate)h Fo(k)h Fq(suc)o(h)f(that)f Fo(x)1583 635 y Fn(k)1618 628 y Fh(6)p Fq(=)f(0,)h Fo(y)1740 635 y Fn(k)1775 628 y Fq(=)e(0)j(\(or)60 713 y(else)g Fo(y)10 b Fh(\000)g Fo(x)p Fq(,)k Fo(x)h Fq(do)f(not)g(in)o (tersect\),)h(and)f(a)h(co)q(ordinate)g Fo(l)g Fq(suc)o(h)g(that)f Fo(x)1276 720 y Fn(l)1301 713 y Fh(6)p Fq(=)f(0,)h Fo(y)1421 720 y Fn(l)1447 713 y Fq(=)f Fo(x)1521 720 y Fn(l)1549 713 y Fq(\(or)h(else)h Fo(x;)8 b(y)16 b Fq(do)e(not)60 798 y(in)o(tersect\);)h (and)g(\(ii\))h(holds.)131 882 y(In)f(the)g(non)o(binary)h(case,)e(MDS)h(co)q (des)g(pro)o(vide)h(go)q(o)q(d)f(examples)g(of)g(in)o(tersecting)g(co)q(des)h (\(see)f(Theo-)60 967 y(rem)f(1\(v\)\).)19 b(Other)c(examples)g(are)f(giv)o (en)i(b)o(y)e(co)q(des)h(with)g(few)g(w)o(eigh)o(ts,)f(cf.)g([9)o(],)g(but)h (these)g(do)g(not)f(seem)60 1052 y(to)19 b(pro)q(duce)h(e\016cien)o(t)g(3-co) o(v)o(erings.)31 b(S.)19 b(Litsyn)h([39)o(])f(has)g(p)q(oin)o(ted)i(out)d (that)h(for)f(large)i Fo(N)5 b Fq(,)19 b(algebraic-)60 1136 y(geometric)g(co)q(des)g(pro)o(vide)g(go)q(o)q(d)f(examples)h(of)g(in)o (tersecting)g(co)q(des.)30 b(Let)19 b Fo(Q)g Fq(b)q(e)g(an)f(ev)o(en)h(p)q(o) o(w)o(er)f(of)g(a)60 1221 y(prime.)j(Then)15 b(there)h(are)e (algebraic-geometric)j(co)q(des)e(o)o(v)o(er)g(the)g(\014eld)i(of)d(order)h Fo(Q)g Fq(with)566 1348 y Fo(D)f Fq(=)666 1288 y Fg(\030)695 1317 y Fo(N)h Fq(+)10 b(1)p 695 1337 120 2 v 744 1379 a(2)820 1288 y Fg(\031)852 1348 y Fo(;)37 b Fq(lim)8 b(inf)914 1377 y Fn(N)t Ff(!1)1056 1317 y Fo(K)p 1056 1337 42 2 v 1056 1379 a(N)1115 1348 y Fh(\025)1168 1317 y Fq(1)p 1168 1337 23 2 v 1168 1379 a(2)1206 1348 y Fh(\000)1321 1317 y Fq(1)p 1257 1337 153 2 v 1257 1346 a Fh(p)p 1294 1346 36 2 v 1294 1380 a Fo(Q)i Fh(\000)h Fq(1)1862 1348 y(\(8\))60 1475 y(and)16 b(ha)o(ving)h(a)e(p)q (olynomial-time)k(construction)d([33)o(],)g([57)o(,)g(Theorem)f(3.4.15].)21 b(These)16 b(co)q(des)h(are)f(in)o(ter-)60 1559 y(secting.)131 1644 y(Rather)f(more)g(is)g(kno)o(wn)g(in)h(the)g(binary)f(case.)60 1763 y Fp(Theorem)i(4.)22 b Fr(The)16 b(minimal)g(length)g Fo(f)773 1770 y Fi(2)793 1763 y Fq(\()p Fo(K)s Fq(\))f Fr(of)i(an)f Fq([)p Fo(N)r(;)8 b(K)q(;)g(D)q Fq(])13 b Fr(binary)j(interse)n(cting)e(c)n (o)n(de)i(satis\014es)638 1890 y Fo(c)658 1897 y Fi(1)677 1890 y Fo(K)s Fq(\(1)10 b(+)g Fo(o)p Fq(\(1\)\))i Fh(\024)h Fo(f)996 1897 y Fi(2)1016 1890 y Fq(\()p Fo(K)s Fq(\))f Fh(\024)g Fo(c)1173 1897 y Fi(2)1193 1890 y Fo(K)h Fh(\000)d Fq(2)16 b Fo(;)520 b Fq(\(9\))60 2017 y Fr(wher)n(e)16 b Fo(c)208 2024 y Fi(1)240 2017 y Fq(=)d(3)p Fo(:)p Fq(53)8 b Fo(:)g(:)g(:)21 b(;)15 b(c)502 2024 y Fi(2)535 2017 y Fq(=)636 1999 y Fi(2)p 587 2006 115 2 v 587 2033 a(2)p Ff(\000)p Fi(log)6 b(3)719 2017 y Fq(=)13 b(4)p Fo(:)p Fq(8188)8 b Fo(:)g(:)f(:)13 b Fr(.)60 2136 y Fp(Sk)o(etc)o(h)19 b(of)g(pro)q(of.)25 b Fq(The)17 b(lo)o(w)o(er)f(b)q(ound,)i(obtained)f(b)o(y) g(Katona)f(and)h(Sriv)m(asta)o(v)m(a)g([32)o(],)f(follo)o(ws)h(imme-)60 2221 y(diately)f(from)e(Theorem)h(1\(i\))g(and)g(the)g(McEliece-Ro)q(demi)q (c)o(h-Ru)q(msey-W)l(el)q(c)o(h)j(b)q(ound)e(for)e(linear)i(co)q(des)60 2305 y(\([42)o(],)k([40)o(])g(Chap.)g(17\).)34 b(This)20 b(v)o(ersion)h(of)e (the)i(upp)q(er)g(b)q(ound)g(is)g(due)f(to)g(Blokh)o(uis)h(and)g(Metsc)o(h)e ([5],)60 2390 y(although)14 b(essen)o(tially)i(equiv)m(alen)o(t)g(b)q(ounds)f (had)g(b)q(een)g(giv)o(en)g(earlier)g(b)o(y)f(Koml\023)-23 b(os)15 b(\(see)f([14)o(]\))g(and)g(Retter)60 2475 y([49)o(].)20 b(Blokh)o(uis)d(and)f(Metsc)o(h)f(observ)o(e)g(that)g(1)10 b(+)h Fo(f)959 2482 y Fi(2)979 2475 y Fq(\()p Fo(K)s Fq(\))j(is)i(the)g (minimal)h(n)o(um)o(b)q(er)f(of)f(p)q(oin)o(ts)h(needed)h(to)60 2559 y(blo)q(c)o(k)g(all)h(a\016ne)e(subspaces)h Fo(AG)p Fq(\()p Fo(K)d Fh(\000)d Fq(2)p Fo(;)d Fq(2\))15 b(in)i(an)g Fo(AG)p Fq(\()p Fo(K)q(;)8 b Fq(2\).)21 b(Since)d(eac)o(h)f(p)q(oin)o(t)g(blo)q(c)o (ks)g(one)g(quarter)60 2644 y(of)g(all)h Fo(AG)p Fq(\()p Fo(K)c Fh(\000)e Fq(2)p Fo(;)c Fq(2\),)15 b(w)o(e)i(can)g(mak)o(e)g(sure)g(that)g (after)f(c)o(ho)q(osing)i Fo(i)f Fq(p)q(oin)o(ts)g(at)g(most)f(a)h(fraction)g (\(3)p Fo(=)p Fq(4\))1907 2628 y Fn(i)60 2729 y Fq(of)e(all)i Fo(AG)p Fq(\()p Fo(K)c Fh(\000)e Fq(2)p Fo(;)d Fq(2\))14 b(are)h(not)h(blo)q (c)o(k)o(ed.)22 b(Since)c(the)e(total)f(n)o(um)o(b)q(er)h(of)f Fo(AG)p Fq(\()p Fo(K)e Fh(\000)e Fq(2)p Fo(;)d Fq(2\))14 b(is)i(less)h(than)e (4)1873 2712 y Fn(K)1907 2729 y Fq(,)979 2853 y(5)p eop %%Page: 4 14 bop 60 74 a Fs(2.)52 b(In)n(tersecting)21 b(co)r(des)150 204 y Fq(A)e(linear)i(co)q(de)f Fh(C)i Fq(of)d(length)h Fo(N)5 b Fq(,)19 b(dimension)i Fo(K)h Fq(and)e(minimal)h(Hamming)e(distance)h Fo(D)q Fq(,)g(i.e.)33 b(an)60 289 y([)p Fo(N)r(;)8 b(K)q(;)g(D)q Fq(])17 b(co)q(de,)22 b(o)o(v)o(er)d(a)i(\014eld)g(of)f(order)g Fo(Q)g Fq(is)h(called)h Fr(interse)n(cting)d Fq(\(or)g Fr(linke)n(d)p Fq(\))g(if)i(an)o(y)f(t)o(w)o(o)f(nonzero)60 373 y(co)q(dew)o(ords)e(ha)o(v)o (e)g(at)g(least)g(one)g(co)q(ordinate)h(where)f(they)h(are)f(b)q(oth)g (nonzero.)26 b(Suc)o(h)18 b(co)q(des)g(ha)o(v)o(e)f(b)q(een)60 458 y(studied)j(b)o(y)f(sev)o(eral)h(authors)e([14)o(],)h([32)o(],)h([38)o (],)f([43)o(],)g([49].)31 b(Giv)o(en)19 b Fo(K)s Fq(,)h(the)f(problem)h(is)f (to)g(determine)60 543 y Fo(f)82 550 y Fn(Q)112 543 y Fq(\()p Fo(K)s Fq(\),)g(the)g(minimal)h(length)g Fo(N)j Fq(of)c(an)o(y)g([)p Fo(N)r(;)8 b(K)q(;)g(D)q Fq(])15 b(in)o(tersecting)20 b(co)q(de.)32 b(In)20 b(view)f(of)g(Theorem)g(2\(i\),)60 628 y(\014nding)e Fo(f)234 635 y Fi(2)253 628 y Fq(\()p Fo(K)s Fq(\))e(ma)o(y)f(b)q(e)i (regarded)f(as)g(a)g(\\linear)h(Sp)q(erner")g(problem.)131 712 y(The)j(main)g(prop)q(erties)h(of)e(in)o(tersecting)i(co)q(des)g(are)e (the)h(follo)o(wing.)32 b Fo(D)1413 719 y Fi(max)1499 712 y Fq(denotes)20 b(the)f(maximal)60 797 y(distance)11 b(b)q(et)o(w)o(een)g(co)q (dew)o(ords.)18 b(Most)10 b(of)g(the)g(results)h(assem)o(bled)g(in)h (Theorems)e(1)g(and)h(2)f(are)g(elemen)o(tary)l(.)60 929 y Fp(Theorem)17 b(1.)22 b Fr(\(i\))13 b(If)g Fh(C)j Fr(is)d(interse)n(cting)e (then)i Fo(D)h Fh(\025)f Fo(K)j Fr([38)q(].)j(\(ii\))13 b(If)g Fo(D)1305 936 y Fi(max)1385 929 y Fo(<)g Fq(2)p Fo(D)h Fr(then)f Fh(C)j Fr(is)d(interse)n(cting)60 1013 y([14].)21 b(\(iii\))15 b(If)h Fh(C)j Fr(is)d(an)g Fq([)p Fo(N)r(;)8 b(K)q(;)g(D)q Fq(])13 b Fr(interse)n(cting)h(c)n(o)n(de)i(with)h(gener)n(ator)f(matrix)h Fq([)p Fo(A)f(I)t Fq(])g Fr(then)675 1082 y Fg(\024)756 1113 y Fo(A)154 b(I)162 b(I)107 b Fq(0)704 1170 y(00)8 b Fo(:)g(:)g(:)d Fq(0)46 b(11)8 b Fo(:)g(:)g(:)d Fq(1)46 b(00)8 b Fo(:)g(:)g(:)d Fq(0)46 b(1)1283 1082 y Fg(\025)60 1270 y Fr(gener)n(ates)11 b(an)g Fq([)p Fo(N)5 b Fq(+)p Fo(K)t Fq(+1)p Fo(;)j(K)t Fq(+1)p Fo(;)g(K)t Fq(+1])g Fr(interse)n(cting)i(c)n(o)n(de)i([14].)19 b(\(iv\))11 b(The)h(dir)n(e)n(ct)f(pr)n(o)n(duct)h(\([40],)h(p.)f(568\))60 1354 y(of)i Fq([)p Fo(N)161 1361 y Fi(1)180 1354 y Fo(;)8 b(K)240 1361 y Fi(1)259 1354 y Fo(;)g(D)318 1361 y Fi(1)336 1354 y Fq(])14 b Fr(and)h Fq([)p Fo(N)500 1361 y Fi(2)518 1354 y Fo(;)8 b(K)578 1361 y Fi(2)597 1354 y Fo(;)g(D)656 1361 y Fi(2)674 1354 y Fq(])14 b Fr(interse)n(cting)f(c)n(o)n(des)g(is)h(an)g Fq([)p Fo(N)1220 1361 y Fi(1)1239 1354 y Fo(N)1276 1361 y Fi(2)1295 1354 y Fo(;)8 b(K)1355 1361 y Fi(1)1374 1354 y Fo(K)1413 1361 y Fi(2)1432 1354 y Fo(;)g(D)1491 1361 y Fi(1)1510 1354 y Fo(D)1548 1361 y Fi(2)1567 1354 y Fq(])14 b Fr(interse)n(cting)e(c)n(o)n(de)60 1439 y([14].)35 b(\(v\))19 b(If)i Fo(N)k Fq(=)c(2)p Fo(K)16 b Fh(\000)d Fq(1)21 b Fr(then)f(an)h Fq([)p Fo(N)r(;)8 b(K)q(;)g(D)q Fq(])17 b Fr(MDS)k(c)n(o)n(de)f(\([40],)i(p.)f(317\))g(is)f(interse)n(cting)f (\(sinc)n(e)60 1524 y Fo(D)14 b Fq(=)e(\()p Fo(N)j Fq(+)10 b(1\))p Fo(=)p Fq(2)p Fr(\).)20 b(Such)c(c)n(o)n(des)f(exist)h(if)g Fo(N)h Fh(\024)c Fo(Q)d Fq(+)h(1)16 b Fr(\([40],)g(p.)g(323\).)60 1656 y Fp(Pro)q(of)g(of)g(\(i\))p Fq(.)21 b(Let)14 b Fo(u)455 1663 y Fi(1)487 1656 y Fq(=)f(11)8 b Fo(:)g(:)g(:)d Fq(10)j Fo(:)g(:)g(:)t Fq(0)13 b Fh(2)g Fo(C)j Fq(ha)o(v)o(e)e(w)o(eigh)o(t)f Fo(D)q Fq(,)h(and)g(c)o(ho)q(ose)g(a)g(basis)g Fo(u)1605 1663 y Fi(1)1625 1656 y Fo(;)8 b(u)1672 1663 y Fi(2)1691 1656 y Fo(;)g(:)g(:)g(:)t(;)g(u)1818 1663 y Fn(K)1866 1656 y Fq(for)60 1740 y Fo(C)18 b Fq(in)e(whic)o(h)g Fo(u)320 1747 y Fi(2)340 1740 y Fo(;)8 b(:)g(:)g(:)d(;)j(u)468 1747 y Fn(K)516 1740 y Fq(b)q(egin)17 b(with)e(a)g(0.)20 b(Since)c Fo(u)980 1747 y Fi(2)1000 1740 y Fo(;)8 b(:)g(:)g(:)d(;)j(u)1128 1747 y Fn(K)1176 1740 y Fq(m)o(ust)15 b(in)o(tersect)g Fo(u)1498 1747 y Fi(1)1518 1740 y Fq(,)g Fo(K)e Fh(\000)d Fq(1)j Fh(\024)g Fo(D)e Fh(\000)f Fq(1.)60 1872 y Fp(Theorem)17 b(2.)22 b Fr(L)n(et)15 b Fh(C)k Fr(b)n(e)c(a)h(binary)g(interse)n(cting)e(c)n(o)n(de.)20 b(\(i\))15 b(F)m(or)g(any)h(two)g(nonzer)n(o)f(c)n(o)n(dewor)n(ds)h Fo(u)g Fr(and)f Fo(v)60 1957 y Fr(ther)n(e)f(is)g(a)g(c)n(o)n(or)n(dinate)g(wher)n (e)g Fo(u)g Fr(is)f(0)i(and)f Fo(v)i Fr(is)d(1)h([8)q(].)20 b(\(ii\))13 b Fo(D)1131 1964 y Fi(max)1211 1957 y Fh(\024)g Fo(N)d Fh(\000)5 b Fo(K)j Fq(+)d(1)15 b Fr([6].)20 b(\(iii\))13 b Fo(N)k Fh(\025)c Fq(3\()p Fo(K)8 b Fh(\000)d Fq(1\))60 2042 y Fr([6].)60 2173 y Fp(Pro)q(of)21 b(of)h(\(iii\))p Fq(.)33 b(Let)19 b Fo(u)g Fq(b)q(e)g(a)f(minimal)j(w)o(eigh)o(t)d(co)q(dew)o(ord,)i (and)e(c)o(ho)q(ose)h(another)g(w)o(ord)f Fo(v)i Fq(that)e(has)60 2258 y(at)g(least)h Fo(K)c Fh(\000)e Fq(1)18 b(0's)g(in)h(the)g(supp)q(ort)f (of)g Fo(u)p Fq(.)30 b(This)20 b(is)f(p)q(ossible)h(since)g Fo(D)f Fh(\025)f Fo(K)s Fq(.)30 b(Then)19 b Fo(N)e Fh(\000)c Fo(K)i Fq(+)e(1)18 b Fh(\025)60 2343 y Fo(w)q(t)p Fq(\()p Fo(u)10 b Fq(+)g Fo(v)r Fq(\))i Fh(\025)h Fq(2)p Fo(D)e Fh(\000)g Fq(2\()p Fo(D)f Fh(\000)g Fo(K)j Fq(+)e(1\).)131 2428 y(The)k(application)i(to)e(the)g (construction)h(of)e(3-co)o(v)o(ering)h(arra)o(ys)f(giv)o(en)i(in)g(the)g (next)f(section)h(is)g(based)60 2512 y(on)f(the)g(follo)o(wing)h(prop)q(ert)o (y)l(.)60 2644 y Fp(Theorem)h(3.)22 b Fr(L)n(et)14 b Fo(u;)8 b(v)r(;)g(w)14 b Fr(b)n(e)h(distinct)f(c)n(o)n(dewor)n(ds)g(of)h(an)g (interse)n(cting)e(c)n(o)n(de.)20 b(Then)14 b(either)i(\(i\))e(ther)n(e)g(is) 60 2729 y(a)i(c)n(o)n(or)n(dinate)e(wher)n(e)i Fo(u;)8 b(v)r(;)g(w)14 b Fr(ar)n(e)h(distinct,)g(or)h(\(ii\))e(ther)n(e)i(is)e(a)i(c)n(o)n(or)n (dinate)f(wher)n(e)g Fo(u)h Fr(and)f Fo(v)i Fr(agr)n(e)n(e)e(and)g Fo(w)979 2853 y Fq(4)p eop %%Page: 3 15 bop 60 74 a Fq(The)16 b(v)m(alues)h(for)e Fo(n)f Fh(\024)g Fq(5)h(are)h(optimal,)g(while)h(it)f(seems)g(lik)o(ely)i(that)d(those)g(for)g Fo(n)f Fh(\025)g Fq(8)i(can)g(b)q(e)g(impro)o(v)o(ed.)60 159 y(It)f(w)o(ould)h(b)q(e)g(nice)g(to)f(ha)o(v)o(e)f(a)h(more)g(extensiv)o(e)h (table.)131 244 y(Applegate's)c(in)o(teger)h(programming)f(form)o(ulation)h ([2)o(])f(is)h(w)o(orth)f(recording,)h(since)h(it)f(can)g(b)q(e)g(applied)60 328 y(to)h(the)g(general)h(problem)g(of)f(\014nding)i(a)e Fo(t)p Fq(-co)o(v)o(ering)h(arra)o(y)e(of)h(minimal)i(size.)21 b(Let)14 b Fo(V)24 b Fq(b)q(e)16 b(the)e(set)g(of)g(all)i Fo(q)1897 312 y Fn(n)60 413 y Fq(p)q(ossible)f(v)o(ectors)d(that)g(can)i(b)q(e)f(used)h (in)g(the)f(arra)o(y)l(,)f(let)i Fo(S)h Fq(b)q(e)f(the)f(set)g(of)f(all)1395 379 y Fg(\000)1414 393 y Fn(n)1419 429 y(t)1436 379 y Fg(\001)1468 413 y Fo(t)p Fq(-tuples)i(of)f(co)q(ordinates,)60 498 y(and)j(let)h Fo(P)23 b Fq(b)q(e)16 b(the)h(set)f(of)f(all)i Fo(q)620 481 y Fn(t)651 498 y Fo(q)r Fq(-ary)f(v)o(ectors)f(of)h(length)h Fo(t)p Fq(.)23 b(A)16 b Fo(t)p Fq(-co)o(v)o(ering)h(arra)o(y)e Fo(D)g Fh(\022)f Fo(V)26 b Fq(is)17 b(sp)q(eci\014ed)60 583 y(b)o(y)e(setting)g Fo(x)298 590 y Fn(v)331 583 y Fq(=)e(1)h(if)h Fo(v)g Fh(2)d Fo(D)q Fq(,)j Fo(x)629 590 y Fn(v)662 583 y Fq(=)e(0)h(if)i Fo(v)e Fh(62)f Fo(D)q Fq(.)19 b(Then)d(\014nding)g(a)e(minimal)i(arra)o(y)e (is)h(equiv)m(alen)o(t)i(to)d(the)60 667 y(in)o(teger)h(program:)k(c)o(ho)q (ose)c Fo(x)580 674 y Fn(v)613 667 y Fh(2)e(f)p Fq(0)p Fo(;)8 b Fq(1)p Fh(g)14 b Fq(for)g Fo(v)g Fh(2)f Fo(V)25 b Fq(so)15 b(as)g(to)765 798 y(minimize)63 b Fo(k)14 b Fq(=)1096 757 y Fg(X)1091 849 y Fn(v)q Ff(2)p Fn(V)1169 798 y Fo(x)1195 805 y Fn(v)1862 798 y Fq(\(6\))60 937 y(sub)s(ject)h(to)g(the)g(constrain)o(ts) 642 1027 y Fg(X)710 1067 y Fo(x)736 1074 y Fn(v)769 1067 y Fh(\025)e Fq(1)45 b(for)15 b(all)46 b Fo(s)13 b Fh(2)g Fo(S;)37 b(p)13 b Fh(2)g Fo(P)22 b(;)516 b Fq(\(7\))60 1197 y(where)15 b(the)h(sum)f(in)h(\(7\))e(is)i(o)o(v)o(er)e(all)i Fo(v)f Fh(2)e Fo(V)24 b Fq(suc)o(h)16 b(that)e(the)i(pro)s(jection)f(of)f Fo(v)j Fq(on)o(to)e Fo(s)g Fq(is)h Fo(p)p Fq(.)131 1282 y(The)e(main)g(purp)q (ose)g(of)g(the)g(presen)o(t)g(pap)q(er)g(is)g(to)g(study)f(the)h(case)g Fo(t)f Fq(=)g(3,)h Fo(q)g Fq(=)f(2.)19 b(Imp)q(ortan)o(t)14 b(results)60 1367 y(w)o(ere)i(obtained)h(b)o(y)f(Bussc)o(h)o(bac)o(h)h(in)g (an)f(unpublished)k(tec)o(hnical)d(rep)q(ort)f(in)h(1984)f([8)o(],)g(and)g(b) o(y)g(Roux)h(in)60 1451 y(an)h(unpublished)j(thesis)d(in)h(1987)d([50)o(].)28 b(Ho)o(w)o(ev)o(er,)17 b(as)g(w)o(e)h(shall)h(see,)f(their)g(results)g(can)g (b)q(e)h(somewhat)60 1536 y(impro)o(v)o(ed.)g(The)13 b(main)h(results)f(of)f (this)h(pap)q(er)h(are)e(con)o(tained)h(in)h(Theorem)f(5)f(and)h(T)l(able)h (3)e(in)i(Section)g(3.)131 1621 y(One)k(of)f(the)g(b)q(est)h(constructions)f (for)g(3-co)o(v)o(ering)g(arra)o(ys)f(\(due)i(to)e(Bussc)o(h)o(bac)o(h)i([8)o (]\))f(mak)o(es)g(use)g(of)60 1705 y(\\in)o(tersecting)f(co)q(des",)f(and)g (Section)h(2)f(is)h(dev)o(oted)f(to)g(these)g(co)q(des.)131 1790 y(The)i(cases)h Fo(t)f Fh(\025)g Fq(4)g(\(and)h Fo(q)h Fh(\025)e Fq(2\))g(will)i(not)e(b)q(e)h(discussed)i(here.)27 b(Some)18 b(results)g(on)f(these)h(problems)60 1875 y(can)i(b)q(e)g(found)f (in)i([1)o(],)f([8)o(],)g([11)o(],)g([13)o(],)f([29)o(],)h([34)o(],)g([47)o (],)g([50)o(],)f([51],)g([54)o(],)h([55)o(],)g([56)o(].)32 b(Applications)21 b(of)60 1959 y(these)14 b(co)o(v)o(ering)g(arra)o(ys)f(and) h(related)h(structures)e(to)h(circuit)h(testing,)f(digital)h(comm)o (unication,)g(net)o(w)o(ork)60 2044 y(design,)g(etc.,)e(are)g(discussed)i(in) g([4)o(],)e([8],)g([12)o(],)g([15)o(],)g([19)o(],)h([44)o(],)f([51)o(],)g ([52)o(],)g([54],)g([55)o(].)19 b(The)14 b(surv)o(ey)f(article)60 2129 y(b)o(y)h(K\177)-23 b(orner)13 b(and)h(Lucertini)i([35)o(])d(giv)o(es)h (an)g(o)o(v)o(erview)f(of)h(these)g(and)f(sev)o(eral)h(related)g(problems.)20 b(Honk)m(ala)60 2214 y([28)o(])c(uses)h Fo(t)p Fq(-co)o(v)o(ering)f(arra)o (ys)f(in)i(the)g(construction)f(of)g(co)q(des)h(with)g(small)g(co)o(v)o (ering)f(radius.)24 b(Sherw)o(o)q(o)q(d)60 2298 y([52)o(])13 b(describ)q(es)i(a)d(computer)h(program)f(CA)l(TS)i(\(\\Constrained)e(Arra)o (y)h(T)l(est)g(System"\),)f(whic)o(h)i(attempts)60 2383 y(to)e(\014nd)i (small)f(co)o(v)o(ering)g(arra)o(ys)f(for)g(a)g(large)h(class)g(of)g (problems,)g(including)j(ones)d(in)g(whic)o(h)h(the)f(alphab)q(et)60 2468 y(size)i Fo(q)h Fq(v)m(aries)f(from)e(co)q(ordinate)h(to)g(co)q (ordinate.)19 b(Seroussi)c(and)f(Bshout)o(y)g([51)o(])g(sho)o(w)f(that)g(a)h (generalized)60 2552 y(v)o(ersion)h(of)g(the)g(problem)h(of)f(\014nding)i(a)e (minimal)h Fo(t)p Fq(-co)o(v)o(ering)g(arra)o(y)e(is)i(NP-complete.)979 2853 y(3)p eop %%Page: 2 16 bop 60 74 a Fq(Suc)o(h)15 b(an)g(arra)o(y)e(ma)o(y)h(b)q(e)h(constructed)g (as)f(follo)o(ws.)20 b(The)15 b(names)g(of)f(the)g(sym)o(b)q(ols)h(in)h(an)o (y)e(column)i(of)e(the)60 159 y(arra)o(y)g(ma)o(y)h(b)q(e)h(p)q(erm)o(uted,)f (so)g(w)o(e)g(ma)o(y)g(assume)g(the)h(\014rst)f(ro)o(w)f(is)i(the)f(zero)h(v) o(ector)e Fp(0)p Fq(.)21 b(The)15 b(columns)h(of)60 244 y(the)g(remaining)i Fo(k)12 b Fh(\000)f Fq(1)16 b(ro)o(ws)g(are)g(then)g(tak)o(en)g(to)g(b)q(e)h (the)g(c)o(haracteristic)f(v)o(ectors)g(of)g(all)h Fh(d)1665 226 y Fn(k)p 1665 233 20 2 v 1666 259 a Fi(2)1690 244 y Fh(e)p Fq(-subsets)f(of)60 328 y(a)e(\()p Fo(k)d Fh(\000)e Fq(1\)-set.)19 b(The)c(pro)q(of)g(that)f(this)h(is)g(optimal)h(uses)f(Sp)q(erner's)g(lemma)g ([53)o(],)f([26)o(],)g([31)o(])h(\(if)g Fo(k)h Fq(is)f(ev)o(en\))60 413 y(and)g(the)h(Erd\177)-23 b(os-Ko-Rado)15 b(theorem)g([21)o(])g(\(if)g Fo(k)h Fq(is)g(o)q(dd\).)131 498 y(F)l(or)e(large)h Fo(n)p Fq(,)g(\(2\))g(implies)i(that)d(the)i(minimal)g Fo(k)h Fq(satis\014es)735 615 y Fo(k)c Fq(=)g(log)c Fo(n)h Fq(+)974 584 y(1)p 974 604 23 2 v 974 646 a(2)1010 615 y(log)e(log)g Fo(n)23 b Fh(\001)8 b(\001)g(\001)615 b Fq(\(3\))60 731 y(\(all)16 b(logs)f(are)g(to)f(base)i (2\).)131 816 y(In)c(the)g(case)f Fo(t)i Fq(=)g(2,)f Fo(q)i(>)f Fq(2,)f(the)g(rate)f(of)g(gro)o(wth)f(of)i Fo(k)g Fq(with)g Fo(n)g Fq(w)o(as)f(recen)o(tly)h(determined)h(b)o(y)f(Gargano,)60 901 y(K\177)-23 b(orner)15 b(and)h(V)l(accaro)f([24)o(],)f([25)o(],)g(who)h (sho)o(w)g(that)g(for)f(large)h Fo(n)h Fq(the)f(minimal)i Fo(k)f Fq(satis\014es)762 1018 y Fo(k)e Fq(=)853 987 y Fo(q)p 853 1007 V 853 1049 a Fq(2)888 1018 y(log)8 b Fo(n)16 b Fq(\(1)9 b(+)i Fo(o)p Fq(\(1\)\))i Fo(:)644 b Fq(\(4\))60 1134 y(Ho)o(w)o(ev)o(er,)16 b(they)g(do)h(not)f(giv)o(e)h(an)f(explicit)j(construction)e(of)f(2-co)o(v)o (ering)h(arra)o(ys)e(that)h(ac)o(hiev)o(e)h(\(4\),)f(and)60 1219 y(not)f(m)o(uc)o(h)g(seems)g(to)g(b)q(e)h(kno)o(wn)f(ab)q(out)g(exact)g (v)m(alues)h(of)f Fo(k)h Fq(for)f(small)h Fo(n)p Fq(.)131 1304 y(F)l(or)d(example,)j(let)f(us)f(brie\015y)i(discuss)g(the)e(case)h Fo(t)e Fq(=)g(2,)h Fo(q)g Fq(=)f(3)i(\(the)f(\\ternary)g(Sp)q(ernery")h (problem\).)60 1388 y(In)h(an)g(earlier)g(pap)q(er)g(Gargano,)e(K\177)-23 b(orner)16 b(and)g(V)l(accaro)f([23)o(])g(ga)o(v)o(e)g(an)g(explicit)j (construction)e(ac)o(hieving)60 1473 y Fo(k)g Fq(=)f(2)p Fo(:)p Fq(07)8 b(log)f Fo(n)16 b Fq(\(1)11 b(+)g Fo(o)p Fq(\(1\)\).)23 b(Other)16 b(constructions)h(w)o(ere)f(giv)o(en)h(b)o(y)g(P)o(oljak,)f(Pultr) g(and)h(R\177)-23 b(odl)18 b([46)o(])e(and)60 1558 y(P)o(oljak)d(and)g(T)l (uza)h([47)o(].)19 b(F)l(or)12 b Fo(n)h Fq(=)g(2)p Fo(;)8 b Fq(3)p Fo(;)g Fq(4)j(the)i(co)q(dew)o(ords)g(of)g(the)h(tetraco)q(de)f(\([17) n(],)g(p.)g(81\))g(sho)o(w)f(that)h(the)60 1642 y(minimal)i Fo(k)g Fq(is)g(9.)k(F)l(or)13 b Fo(n)g Fq(=)g(5,)598 1631 y(\177)591 1642 y(Osterg)-6 b(\027)-28 b(ard)14 b([45)o(])f(sho)o(w)o(ed)h(that)f Fo(k)g Fh(\024)g Fq(11,)h(and)g(Applegate)g([2)o(])g(recen)o(tly)g(used)60 1727 y(in)o(teger)j(programming)f(to)g(sho)o(w)g(that)f Fo(k)h Fq(=)g(11.)23 b(Applegate's)17 b(solution)g(ma)o(y)f(b)q(e)h(transformed)f (to)g(read)60 1812 y Fp(0)p Fq(,)i Fh(\006)p Fq(\(01112\).)25 b(F)l(or)16 b Fo(n)h Fq(=)g(6)g(Co)q(ok)g([18)o(])g(also)g(used)h(in)o(teger) g(programming)f(to)g(sho)o(w)f(that)h Fo(k)h Fh(\024)e Fq(12.)26 b(His)60 1896 y(solution)14 b(ma)o(y)f(b)q(e)g(written)h(as)f(0\(01221\),)e (111000,)g(100110,)h(120201,)f(102022,)h(202202,)f(210011,)h(221120.)60 1981 y(F)l(or)i(sligh)o(tly)j(larger)e(v)m(alues)h(of)e Fo(n)i Fq(reasonably)f(go)q(o)q(d)g(arra)o(ys)f(can)h(b)q(e)h(formed)e(b)o(y)h (taking)g(three)g(copies)h(of)60 2066 y(the)j(arra)o(y)e(that)h(solv)o(es)g (the)h Fo(q)h Fq(=)e Fo(t)h Fq(=)f(2)g(problem)i(\(see)e(\(2\)\),)g(remo)o (ving)g(the)h Fp(0)f Fq(ro)o(w)g(from)g(eac)o(h,)h(then)60 2151 y(writing)g(one)g(cop)o(y)f(in)i(terms)e(of)g(0's)g(and)h(1's,)f(one)h (cop)o(y)f(in)i(terms)e(of)g(1's)g(and)h(2's,)f(and)h(one)g(cop)K(this)d(Pro)G(an)f((a))f((a))f((a))f(k)e fq(=)f(3)pFo(a)i fq(and)g(长度)842 2352,y fO(n)e fq(=)930 2280 y fg()1008 2321 y y(a)963 f y h(d)αy y fn(a)pFi(+α)pαvvα(α)y Fh(e)y y fg(!)o(y)f(in)60×2235 y(项)d(f)f(2)h(and)g(0)1125 2352 y Fo(:)724 b Fq(\(5\))60 2473 y(\(A)18 b(similar)i(but)e(sligh)o(tly)i(less)f(e\016cien)o(t)g (construction)f(w)o(as)g(giv)o(en)h(in)g([47)o(].\))29 b(In)19 b(summary)l(,)f(for)g Fo(t)g Fq(=)h(2,)60 2558 y Fo(q)c Fq(=)e(3)h(the)i (smallest)f(kno)o(wn)g(v)m(alues)i(of)d Fo(k)j Fq(for)d(giv)o(en)i(v)m(alues) g(of)f Fo(n)h Fq(are)455 2643 y Fo(n)c Fq(:)42 b(2)f(3)g(4)64 b(5)g(6)g(7)g(8)g(9)41 b(10)g(11)g(12)g Fh(\001)8 b(\001)g(\001)455 2728 y Fo(k)13 b Fq(:)44 b(9)d(9)g(9)g(11)g(12)g(15)g(15)g(15)g(15)g(18)g(18) g Fh(\001)8 b(\001)g(\001)21 b Fo(:)979 2853 y Fq(2)p eop %%Page: 1 17 bop 384 74 a Fs(Co)n(v)n(ering)23 b(Arra)n(ys)h(and)g(In)n(tersecting)d(Co)r (des)835 187 y Fr(N.)16 b(J.)g(A.)g(Slo)n(ane)595 272 y Fq(Mathematical)f (Sciences)i(Researc)o(h)f(Cen)o(ter)745 328 y(A)l(T&T)g(Bell)h(Lab)q (oratories)681 385 y(Murra)o(y)d(Hill,)j(New)e(Jersey)h(07974)60 613 y Fs(1.)52 b(In)n(tro)r(duction)149 743 y Fq(Before)19 b(shipping)i(those)e(new)g(mac)o(hines)h(o\013)e(to)g(y)o(our)h(customers,)g (y)o(ou)f(w)o(an)o(t)g(to)g(run)i(some)e(\014nal)60 827 y(tests.)23 b(There)16 b(are)g(sixteen)i(switc)o(hes)e(on)h(the)f(bac)o(k)g(of)g(eac)o(h) h(mac)o(hine)g(that)e(ha)o(v)o(e)h(to)g(b)q(e)h(set,)f(eac)o(h)h(with)60 912 y(t)o(w)o(o)h(p)q(ositions.)35 b(Since)21 b(there)f(are)f(2)720 895 y Fi(16)778 912 y Fq(=)h(65536)e(p)q(ossible)k(com)o(binations,)f(y)o(ou) e(can't)h(test)f(them)g(all.)60 997 y(Instead,)f(y)o(ou)g(w)o(ould)g(lik)o(e) h(to)e(\014nd)i(a)e(small)i(n)o(um)o(b)q(er)f(of)f(test)g(settings)h(for)f (the)h(sixteen)h(switc)o(hes)f(suc)o(h)60 1081 y(that)d(ev)o(ery)g(subset)h (of)f(three)h(switc)o(hes)g(gets)f(exercised)i(in)f(all)g(2)1194 1065 y Fi(3)1229 1081 y Fq(p)q(ossible)i(w)o(a)o(ys.)h(In)e(other)e(w)o (ords,)f(y)o(ou)60 1166 y(w)o(ould)i(lik)o(e)i(a)d(minimal)j(set)e(of)f (binary)i(v)o(ectors)e(of)h(length)g(16)g(with)g(the)g(prop)q(ert)o(y)g(that) f(the)h(pro)s(jection)60 1251 y(on)o(to)g(an)o(y)g(three)h(co)q(ordinates)g (includes)i(all)f(2)876 1234 y Fi(3)912 1251 y Fq(p)q(ossibilities.)28 B(HO)O(W)16 B(man)O(Y)G(测试)H(V)O((S))F(DO)G(Y)O(OH)H(需要?)60 1335 y(The)e(answ)o(er)g(is)h(not)f(more)f(than)h(17:)20 b(y)o(ou)15 b(could)h(for)e(example)i(use)g(the)f(v)o(ectors)808 1437 y(00000000000000)o (00)791 1494 y(\()o(00001011011101)o(11)m(\))1862 1466 y(\(1\))60 1596 y(where)g(the)g(paren)o(theses)g(indicate)h(that)e(all)i(sixteen)g (cyclic)g(shifts)f(of)g(this)g(v)o(ector)f(are)g(to)h(b)q(e)g(used.)20 b(This)60 1681 y(is)c(a)f Fr(3-c)n(overing)h(arr)n(ay)p Fq(.)k(It)15 b(is)h(not)f(kno)o(wn)g(if)g(17)g(is)h(minimal)g(\(the)f(b)q(est)h(lo)o(w)o (er)f(b)q(ound)h(is)f(14\).)131 1765 y(More)h(generally)l(,)j(a)e Fo(t)p Fr(-c)n(overing)g(arr)n(ay)h Fq(with)g(alphab)q(et)g(size)g Fo(q)r Fq(,)f(length)h Fo(n)f Fq(and)h(size)g Fo(k)g Fq(consists)f(of)g Fo(k)60 1850 y Fq(v)o(ectors)11 b(of)h(length)g Fo(n)g Fq(with)h(en)o(tries)f (from)f Fh(f)p Fq(0)p Fo(;)d Fq(1)p Fo(;)g(:)g(:)f(:)t(;)h(q)d Fh(\000)t Fq(1)p Fh(g)11 b Fq(with)h(the)g(prop)q(ert)o(y)g(that)f(the)h(pro) s(jection)g(on)o(to)60 1935 y(an)o(y)k Fo(t)i Fq(co)q(ordinates)f(con)o (tains)g(all)g Fo(q)690 1918 y Fn(t)722 1935 y Fq(p)q(ossibilities.)28 b(Other)17 b(names)f(are)h Fo(t)p Fr(-surje)n(ctive)h(arr)n(ay)p Fq(,)f(or)f(\(for)g(the)60 2019 y(transp)q(osed)c(arra)o(y\))e(a)i Fr(qualitatively)h Fo(t)p Fr(-indep)n(endent)g(family)f Fq(of)g(v)o(ectors.) 18 b(The)12 b(problem,)h(apparen)o(tly)f(\014rst)60 2104 y(studied)k(b)o(y)f (R)o(\023)-21 b(en)o(yi)15 b([48],)f(is)h(to)g(minimize)i Fo(k)f Fq(for)f(giv)o(en)g(v)m(alues)h(of)f Fo(q)r Fq(,)g Fo(t)p Fq(,)g Fo(n)p Fq(,)g(or)f(equiv)m(alen)o(tly)j(to)e(maximize)60 2189 y Fo(n)e Fq(for)g(giv)o(en)h(v)m(alues)g(of)f Fo(q)r Fq(,)g Fo(t)p Fq(,)h Fo(k)q Fq(.)19 b(As)13 b(the)g(bibliograph)o(y)i(sho)o(ws,)e (there)g(is)h(an)f(extensiv)o(e)h(b)q(o)q(dy)g(of)e(literature)60 2273 y(related)k(to)e(this)i(problem.)131 2358 y(In)i(the)g(case)f Fo(t)g Fq(=)g Fo(q)i Fq(=)e(2)g(the)h(problem)g(w)o(as)f(completely)i(solv)o (ed)f(b)o(y)g(R)o(\023)-21 b(en)o(yi)17 b([48])g(\(for)f Fo(k)j Fq(ev)o(en\))f(and)60 2443 y(indep)q(enden)o(tly)i(b)o(y)c(Katona)g([30)o(])g (and)h(Kleitman)h(and)f(Sp)q(encer)h([34)o(])e(\(for)g(all)h Fo(k)q Fq(\).)24 b(The)17 b(answ)o(er)f(is)h(that)60 2528 y(for)e(an)o(y)f Fo(k)q Fq(,)h(the)h(maximal)f(length)h(of)f(a)g(binary)h(2-co)o(v)o(ering)f (arra)o(y)f(of)g(size)i Fo(k)h Fq(is)848 2658 y Fo(n)c Fq(=)936 2586 y Fg( )968 2627 y Fo(k)f Fh(\000)e Fq(1)985 2689 y Fh(d)1010 2671 y Fn(k)p 1010 2678 20 2 v 1011 2705 a Fi(2)1035 2689 y Fh(e)1072 2586 y Fg(!)1120 2658 y Fo(:)729 b Fq(\(2\))p eop %%Page: 1 18 bop 367 74 a Fs(Co)n(v)n(ering)23 b(Arra)n(ys)h(and)g(In)n(tersecting)d(Co)r (des*)835 187 y Fr(N.)16 b(J.)g(A.)g(Slo)n(ane)595 272 y Fq(Mathematical)f (Sciences)i(Researc)o(h)f(Cen)o(ter)745 328 y(A)l(T&T)g(Bell)h(Lab)q (oratories)681 385 y(Murra)o(y)d(Hill,)j(New)e(Jersey)h(07974)820 498 y(Jan)o(uary)f(25,)g(1994)836 639 y Fp(ABSTRA)o(CT)149 802 y Fq(A)f Fo(t)p Fq(-co)o(v)o(ering)h(arra)o(y)e(is)i(a)f(set)g(of)g Fo(k)h Fq(binary)g(v)o(ectors)e(of)h(length)h Fo(n)f Fq(with)h(the)f(prop)q (ert)o(y)g(that,)g(in)h(an)o(y)f Fo(t)60 886 y Fq(co)q(ordinate)i(p)q (ositions,)g(all)h(2)575 870 y Fn(t)605 886 y Fq(p)q(ossibilitie)q(s)h(o)q (ccur)e(at)f(least)h(once.)22 b(Suc)o(h)16 b(arra)o(ys)e(are)i(used)g(for)f (example)60 971 y(in)f(circuit)h(testing,)e(and)h(one)g(wishes)g(to)e (minimize)k Fo(k)f Fq(for)d(giv)o(en)i(v)m(alues)h(of)e Fo(n)g Fq(and)h Fo(t)p Fq(.)20 b(The)13 b(case)h Fo(t)f Fq(=)g(2)g(w)o(as)60 1056 y(solv)o(ed)19 b(b)o(y)g(R)o(\023)-21 b(en)o(yi,)19 b(Katona,)g(and)g (Kleitman)g(and)g(Sp)q(encer.)32 b(The)19 b(presen)o(t)g(pap)q(er)g(is)g (concerned)h(with)60 1140 y(the)e(case)g Fo(t)g Fq(=)g(3,)h(where)f(imp)q (ortan)o(t)g(\(but)g(unpublished\))j(con)o(tributions)d(w)o(ere)g(made)g(b)o (y)h(Bussc)o(h)o(bac)o(h)60 1225 y(and)g(Roux)g(in)g(the)g(1980's.)28 b(One)19 b(of)f(the)h(principal)h(constructions)f(mak)o(es)f(use)h(of)f(in)o (tersecting)h(co)q(des)60 1310 y(\(linear)14 b(co)q(des)g(with)f(the)g(prop)q (ert)o(y)g(that)g(an)o(y)g(t)o(w)o(o)e(nonzero)j(co)q(dew)o(ords)f(meet\).)19 b(This)13 b(pap)q(er)h(studies)g(the)60 1394 y(prop)q(erties)h(of)g(3-co)o(v) o(ering)f(arra)o(ys)g(and)h(in)o(tersecting)g(co)q(des,)g(and)g(giv)o(es)g(a) g(table)g(of)f(the)h(b)q(est)g(3-co)o(v)o(ering)60 1479 y(arra)o(ys)f(presen) o(tly)i(kno)o(wn.)j(F)l(or)c(large)g Fo(n)h Fq(the)f(minimal)i Fo(k)f Fq(satis\014es)f(3)p Fo(:)p Fq(21256)c Fo(<)i(k)q(=)8 b Fq(log)g Fo(n)k(<)h Fq(7)p Fo(:)p Fq(56444.)p 60 2640 744 2 v 111 2667 a Fm(\003)129 2683 y Fl(This)i(pap)q(er)g(ap(i)i(组合)d fl((1)i fl(\(1993))h(51)63(22)b(is)14 b(is)h(也)g(dima)o(cs)f(t)m(EC)o(HNICIC)60πy(Req)q(ORT)G(93 {48),f(8)h(1993)p p OP %%TRAILL结束符/结束钩子已知{端钩},如果%%EOF p)q(ErRead)h(in)f fk(期刊)