%!PS-Adobe-2.0 %%Copyright: Copyright (c) 1993 AT&T, All Rights Reserved %%Version: 3.4 %%DocumentFonts: (atend) %%Pages: (atend) %%BoundingBox: (atend) %%EndComments /DpostDict 200 dict def DpostDict begin % % Copyright (c) 1993 AT&T, All Rights Reserved % % Version 3.4 prologue for troff files. % /#copies 1 store /Prologue (dpost.ps) def /aspectratio 1 def /formsperpage 1 def /landscape false def /linewidth .3 def /magnification 1 def /margin 0 def /orientation 0 def /resolution 720 def /rotation 1 def /xoffset 0 def /yoffset 0 def /roundpage true def /useclippath true def /pagebbox [0 0 612 792] def /R /Times-Roman def /I /Times-Italic def /B /Times-Bold def /BI /Times-BoldItalic def /H /Helvetica def /HI /Helvetica-Oblique def /HB /Helvetica-Bold def /HX /Helvetica-BoldOblique def /CW /Courier def /CO /Courier def /CI /Courier-Oblique def /CB /Courier-Bold def /CX /Courier-BoldOblique def /PA /Palatino-Roman def /PI /Palatino-Italic def /PB /Palatino-Bold def /PX /Palatino-BoldItalic def /Hr /Helvetica-Narrow def /Hi /Helvetica-Narrow-Oblique def /Hb /Helvetica-Narrow-Bold def /Hx /Helvetica-Narrow-BoldOblique def /KR /Bookman-Light def /KI /Bookman-LightItalic def /KB /Bookman-Demi def /KX /Bookman-DemiItalic def /AR /AvantGarde-Book def /AI /AvantGarde-BookOblique def /AB /AvantGarde-Demi def /AX /AvantGarde-DemiOblique def /NR /NewCenturySchlbk-Roman def /NI /NewCenturySchlbk-Italic def /NB /NewCenturySchlbk-Bold def /NX /NewCenturySchlbk-BoldItalic def /ZD /ZapfDingbats def /ZI /ZapfChancery-MediumItalic def /S /S def /S1 /S1 def /GR /Symbol def /inch {72 mul} bind def /min {2 copy gt {exch} if pop} bind def /setup { counttomark 2 idiv {def} repeat pop landscape {/orientation 90 orientation add def} if /scaling 72 resolution div def linewidth setlinewidth 1 setlinecap pagedimensions xcenter ycenter translate orientation rotation mul rotate width 2 div neg height 2 div translate xoffset inch yoffset inch neg translate margin 2 div dup neg translate magnification dup aspectratio mul scale scaling scaling scale addmetrics 0 0 moveto } def /pagedimensions { useclippath userdict /gotpagebbox known not and { /pagebbox [clippath pathbbox newpath] def roundpage currentdict /roundpagebbox known and {roundpagebbox} if } if pagebbox aload pop 4 -1 roll exch 4 1 roll 4 copy landscape {4 2 roll} if sub /width exch def sub /height exch def add 2 div /xcenter exch def add 2 div /ycenter exch def userdict /gotpagebbox true put } def /landscapepage { landscape not { 0 height scaling div neg translate % not quite 90 rotate } if } bind def /portraitpage { landscape { width scaling div 0 translate % not quite -90 rotate } if } bind def /addmetrics { /Symbol /S null Sdefs cf /Times-Roman /S1 StandardEncoding dup length array copy S1defs cf } def /pagesetup { /page exch def currentdict /pagedict known currentdict page known and { page load pagedict exch get cvx exec } if } def /decodingdefs [ {counttomark 2 idiv {y moveto show} repeat} {neg /y exch def counttomark 2 idiv {y moveto show} repeat} {neg moveto {2 index stringwidth pop sub exch div 0 32 4 -1 roll widthshow} repeat} {neg moveto {spacewidth sub 0.0 32 4 -1 roll widthshow} repeat} {counttomark 2 idiv {y moveto show} repeat} {neg setfunnytext} ] def /setdecoding {/t decodingdefs 3 -1 roll get bind def} bind def /w {neg moveto show} bind def /m {neg dup /y exch def moveto} bind def /done {/lastpage where {pop lastpage} if} def /f { dup /font exch def findfont exch dup /ptsize exch def scaling div dup /size exch def scalefont setfont linewidth ptsize mul scaling 10 mul div setlinewidth /spacewidth ( ) stringwidth pop def } bind def /changefont { /fontheight exch def /fontslant exch def currentfont [ 1 0 fontheight ptsize div fontslant sin mul fontslant cos div fontheight ptsize div 0 0 ] makefont setfont } bind def /sf {f} bind def /cf { dup length 2 idiv /entries exch def /chtab exch def /newencoding exch def /newfont exch def findfont dup length 1 add dict /newdict exch defFID Ne{NexDATT 3 1 Loopt}{POP POPO} IFOR}所有编码类型/阵列类型EQ {NeXDIT/编码新编码放置}如果新Next/度量条目DICT放置NeWDITT/MULTION得到开始{ CHAPTA OLAD POP } 1个1个条目{POPDEF}为NeWordNeXDITT DEFECONPOT POP端}绑定DEF%(%)%用于在某些百分之打印机驻留字体中调整参考点和字符宽度的数组。{ 1指数/If square roots are too high try changing % the lines describing /radical and /radicalex to, % % /radical [0 -75 550 0] % /radicalex [-50 -75 500 0] % % Move braceleftbt a bit - default PostScript character is off a bit. % /Sdefs [ /bracketlefttp [201 500] /bracketleftbt [201 500] /bracketrighttp [-81 380] /bracketrightbt [-83 380] /braceleftbt [203 490] /bracketrightex [220 -125 500 0] /radical [0 0 550 0] /radicalex [-50 0 500 0] /parenleftex [-20 -170 0 0] /integral [100 -50 500 0] /infinity [10 -75 730 0] ] def /S1defs [ /underscore [0 80 500 0] /endash [7 90 650 0] ] def end %%EndProlog %%BeginSetup DpostDict begin mark /rotation 1 def /gotpagebbox true def /linewidth 0.5 def /xoffset 0 def /yoffset 0 def /#copies 1 store /magnification 1 def %%FormsPerPage: 1 /formsperpage 1 def %%Patch from lp statusdict /setduplexmode known { statusdict begin true setduplexmode end } if %%EndPatch from lp /landscape false def /resolution 720 def setup 2 setdecoding /build_12 { pop /optsize ptsize def /osize size def /ofont font def optsize 2 div dup R exch R f 0 size 2 mul 3 div dup neg exch 0 exch rmoveto (1) show rmoveto optsize R f (\244) show f (2) show optsize ofont f } def end %%EndSetup %%Page: 1 1 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 1 pagesetup %INFO[COMPANY: AT&T Labs - Research] %INFO[DATE: November 8, 2001] %INFO[AUTHOR: NAME = N. J. A. Sloane, LOC = xX, AFFILIATION = Mathematical Sciences Research Center] %INFO[DOCUMENT STYLE: CODE = RP, TYPE = Released Paper] %INFO[TITLE:〔13 B f〕(图的研究中产生的图解中未解决的问题)10、3950、1、888、1040、T、%信息[题目:(12)I FF(N.J.A.斯隆)3,713,1,2523,1235,T,11,R,F(数学科学研究中心)3 1762 1 1999 1999 1430 T(AT&T贝尔实验室)标题=编码理论中的两个基本问题〉11 B F(编码理论中的基本问题)5(1892)(1)。Two)1 347 2 720 2015 t 11 R f (Two of the most basic questions in coding theory are: \(i\) what is)12 2836 1 885 2340 t 11 I f (A)3749 2340 w 11 R f (\()3825 2340 w 11 I f (n)3870 2340 w 11 R f (,)3934 2340 w 11 I f (d)3971 2340 w 11 R f (\), the maximal number)3 1005 1 4035 2340 t (of binary vectors of length)4 1229 1 720 2600 t 11 I f (n)1993 2600 w 11 R f (with Hamming distance)2 1083 1 2092 2600 t 11 I f (d)3219 2600 w 11 R f ( is)1 117( what)1 241(apart?, and \(ii\))2 674 3 3318 2600 t 11 I f (A)4393 2600 w 11 R f (\()4469 2600 w 11 I f (n)4514 2600 w 11 R f (,)4578 2600 w 11 I f (d)4615 2600 w 11 R f (,)4679 2600 w 11 I f (w)4716 2600 w 11 R f (\), the)1 241 1 4799 2600 t (maximal number of binary vectors of length)6 1959 1 720 2860 t 11 I f (n)2710 2860 w 11 R f (, Hamming distance)2 889 1 2765 2860 t 11 I f (d)3685 2860 w 11 R f ( where each vector)3 834(apart, and)1 435 2 3771 2860 t (contains precisely)1 789 1 720 3120 t 11 I f (w)1537 3120 w 11 R f (ones?)1639 3120 w (The)885 3445 w 11 I f (Hamming distance)1 885 1 1141 3445 t 11 R f (dist \()1 205 1 2112 3445 t 11 I f (u)2326 3445 w 11 R f (,)2390 3445 w 11 I f (v)2427 3445 w 11 R f (\) between two binary vectors)4 1509 1 2484 3445 t 11 I f (u)4080 3445 w 11 S f (=)4162 3445 w 11 R f (\()4241 3445 w 11 I f (u)4286 3445 w 8 R f (1)4354 3467 w 11 R f (,)4412 3445 w 11 I f (...)4476 3445 w 11 R f (,)4569 3445 w 11 I f (u)4633 3445 w 8 I f (n)4701 3467 w 11 R f (\) and)1 281 1 4759 3445 t 11 I f (v)720 3705 w 11 S f (=)795 3705 w 11 R f (\()874 3705 w 11 I f (v)919 3705 w 8 R f (1)980 3727 w 11 R f (,)1038 3705 w 11 I f (...)1102 3705 w 11 R f (,)1195 3705 w 11 I f (v)1259 3705 w 8 I f (n)1320 3727 w 11 R f ( of)1 136(\) is the number)3 717 2 1378 3705 t 11 I f (i)2276 3705 w 11 R f (such that)1 411 1 2352 3705 t 11 I f (u)2808 3705 w 8 I f (i)2876 3727 w 11 S f (\271)2943 3705 w 11 I f (v)3040 3705 w 8 I f (i)3101 3727 w 11 R f (; the)1 210 1 3132 3705 t 11 I f (weight)3387 3705 w 11 R f (wt \()1 155 1 3726 3705 t 11 I f (u)3890 3705 w 11 R f (\) of)1 172 1 3954 3705 t 11 I f (u)4171 3705 w 11 R f (is the number of)3 769 1 4271 3705 t (nonzero)720 3965 w 11 I f (u)1113 3965 w 8 I f (i)1181 3987 w 11 R f ( \()1 45(\(so that dist)2 541 2 1253 3965 t 11 I f (u)1848 3965 w 11 R f (,)1912 3965 w 11 I f (v)1949 3965 w 11 R f (\))2006 3965 w 11 S f (=)2060 3965 w 11 R f (wt \()1 155 1 2139 3965 t 11 I f (u)2303 3965 w 11 S f (-)2385 3965 w 11 I f (v)2464 3965 w 11 R f ( \()1 45( dist)1 202(\)\); and a code in which)5 1074 3 2521 3965 t 11 I f (u)3851 3965 w 11 R f (,)3915 3965 w 11 I f (v)3952 3965 w 11 R f (\))4009 3965 w 11 S f (\263)4090 3965 w 11 I f (d)4187 3965 w 11 R f (for every pair of)3 756 1 4284 3965 t (distinct vectors)1 669 1 720 4225 t 11 I f (u)1417 4225 w 11 R f (,)1481 4225 w 11 I f (v)1518 4225 w 11 R f (can correct)1 481 1 1594 4225 t 11 S f (\351)2700 4547 w (\357)2700 4657 w (\353)2700 4767 w 11 R f (2)2863 4720 w 11 I f (d)2783 4577 w 11 S f (-)2865 4577 w 11 R f (1)2944 4577 w 11 S1 f (_ ____)1 249 1 2767 2767 T 4610 s 11(f)371(3027)4547 W(357)3027 4657 4657(\373)4657 v* f f(可能总是假定(不损失一般性))。We)1 499 2 720 5083 t 11 I f (d)3652 5083 w 11 R f ( [25] for further)3 723( See)1 225(is even.)1 347 3 3745 5083 t (information about codes.)2 1096 1 720 5343 t ( recently made an extensive)4 1274( Smith and the author have)5 1248( D.)1 135( Shearer, W.)2 574( B.)1 130( Brouwer, J.)2 564(A. E.)1 230 7 885 5668 t ( functions)1 438(study [4] of the)3 681 2 720 5928 t 11 I f (A)1868 5928 w 11 R f (\()1944 5928 w 11 I f (n)1989 5928 w 11 R f (,)2053 5928 w 11 I f (d)2090 5928 w 11 R f (\) and)1 223 1 2154 5928 t 11 I f (A)2406 5928 w 11 R f (\()2482 5928 w 11 I f (n)2527 5928 w 11 R f (,)2591 5928 w 11 I f (d)2628 5928 w 11 R f (,)2692 5928 w 11 I f (w)2729 5928 w 11 R f (\), and in particular have computed a table of lower)9 2228 1 2812 5928 t (bounds on these functions for)4 1301 1 720 6188 t 11 I f (n)2049 6188 w 11 S f (\243)2140 6188 w 11 R f ( of these tables are shown in Tables 1 and 2 below.)11 2241(28. Portions)1 562 2 2237 6188 t (In the course of this work a number of unsolved graph theory problems were encountered.)14 3960 1 720 6448 t 9 S1 f (________________)720 6899 w 9 R f ( paper appeared in)3 664(* This)1 296 2 720 7029 t 9 I f (Graph Theory Notes of New York)5 1210 1 1703 7029 t 9 R f (, Vol.)1 204 1 2913 7029 t 9 B f (18)3140 7029 w 9 R f (, 1989, pp. 11-20.)3 641 1 3230 7029 t cleartomark showpage saveobj restore end %%PageBoundingBox: 61 74 514 711 %%EndPage: 1 1 %%Page: 2 2 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 2 pagesetup 11 R f (- 2 -)2 183 1 2788 520 t %INFO[SECTION: LEVEL = 1, NUMBER = 2. ,方向=寻找最大团(11)B f(最大团)2(790)(2)。Finding)1 509 2 720 1040 t 11 R f (The)885 1365 w 11 I f (Hamming graph H)2 853 1 1094 1365 t 11 R f (\()1956 1365 w 11 I f (n)2001 1365 w 11 R f (,)2065 1365 w 11 I f (d)2102 1365 w 11 R f (\) has 2)2 315 1 2166 1365 t 8 I f (n)2487 1321 w 11 R f ( of length)2 446(vertices labeled by the binary vectors)5 1691 2 2575 1365 t 11 I f (n)4752 1365 w 11 R f (, two)1 233 1 4807 1365 t ( joined by an edge if and only if the Hamming distance between the corresponding)14 3702(vertices being)1 618 2 720 1625 t (vectors is at least)3 754 1 720 1885 t 11 I f (d)1502 1885 w 11 R f (. Then)1 309 1 1557 1885 t 11 I f (A)1894 1885 w 11 R f (\()1970 1885 w 11 I f (n)2015 1885 w 11 R f (,)2079 1885 w 11 I f (d)2116 1885 w 11 R f (\) is simply the size of a maximal clique in)9 1845 1 2180 1885 t 11 I f (H)4053 1885 w 11 R f (\()4141 1885 w 11 I f (n)4186 1885 w 11 R f (,)4250 1885 w 11 I f (d)4287 1885 w 11 R f (\).)4351 1885 w ( which is taken from [4], shows the best lower bounds presently known on)13 3288(Table 1,)1 360 2 885 2210 t 11 I f (A)4562 2210 w 11 R f (\()4638 2210 w 11 I f (n)4683 2210 w 11 R f (,)4747 2210 w 11 I f (d)4784 2210 w 11 R f (\) for)1 192 1 4848 2210 t 11 I f (n)720 2470 w 11 S f (\243)811 2470 w 11 R f ( an entry indicates that this is the exact value of)10 2189( period after)2 557(28. a)1 284 3 3 908 2470 T 11 i(a)3976 2470 W 11 r f(\)4052 2470瓦特i f f(n)11,w fzπf(,),αw i i f(d)αw w f r f(特别是it)α()。确切的值是已知的)10 2104 2104 720 2730 2730 T 11 11 n(n)2852 f f((x)),ωw(f)(f)。(已知))(例如,它)in)1 221,2,4262,2470 T(可以看到)In)1 303 2 2785 3055 t ( formed from the rows of a Hadamard matrix of order 16 and its)13 2906(Reed-Muller code of length 16,)4 1414 2 720 3315 t ( the other hand for)4 811( On)1 190(negative [25].)1 609 3 720 3575 t 11 I f (A)2358 3575 w 11 R f ( it is known only that)5 936(\( 17 , 8 \))4 301 2 2434 3575 t (36)2412 3900 w 11 S f (\243)2558 3900 w 11 I f (A)2655 3900 w 11 R f (\( 17 , 8 \))4 301 1 2731 3900 t 11 S f (\243)3077 3900 w 11 R f (37)3174 3900 w 11 I f (.)3320 3900 w 11 R f (\(Upper bounds on)2 793 1 720 4225 t 11 I f (A)1541 4225 w 11 R f (\()1617 4225 w 11 I f (n)1662 4225 w 11 R f (,)1726 4225 w 11 I f (d)1763 4225 w 11 R f (\) can be found for example in [25] or [5], p. 248.\))11 2178 1 1827 4225 t (The \256rst undetermined value is)4 1368 1 885 4550 t 11 I f (A)2281 4550 w 11 R f ( for which it is known only that)7 1387(\( 11 , 4 \),)4 329 2 2357 4550 t (72)2412 4875 w 11 S f (\243)2558 4875 w 11 I f (A)2655 4875 w 11 R f (\( 11 , 4 \))4 301 1 2731 4875 t 11 S f (\243)3077 4875 w 11 R f (79)3174 4875 Wi 11 i f(…)3320 4875 W(h)720 5200 W 11 r f(只有2048个顶点:一些图论理论家,请确定大小)11 3240(是图)4 4(\(ω))α(t)(算法组数-256nDin最近已发表)(a)(最大)吗?4,720,5460,t(不遵循一个周期是一个未解决的)8 1847(1)1 83(表),83(每一个)α(年[Y],[Y],[Y],[Y],[Y])(这个类型的问题)。4(868)(1)16815 4259 1 720 6565 t cleartomark showpage saveobj restore end %%PageBoundingBox: 61 119 514 761 %%EndPage: 2 2 %%Page: 3 3 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 3 pagesetup 11 R f (- 3 -)2 183 1 2788 520 t (TABLE 1)1 437 1 2661 1040 t (Lower bounds on)2 769 1 2334 1170 t 11 I f (A)3131 1170 w 11 R f (\()3198 1170 w 11 I f (n)3234 1170 w 11 R f (,)3298 1170 w 11 I f (d)3335 1170 w 11 R f (\))3390 1170 w 11 I f (n)776 1430 w 11 R f (,)840 1430 w 11 I f (d)877 1430 w 11 R f ( 20)1 275( 18)1 309( 16)1 351( 14)1 391( 12)1 453( 10)1 535( 8)1 562(4 6)1 653 8 1289 1430 t 11 S f (_ ____________________________________________________________________________)1 4207 1 776 1450 t 11 R f ( 1.)1 275( 1.)1 309( 1.)1 350( 1.)1 392( 1.)1 452( 1.)1 508( 1.)1 562( 1.)1 598(5 2.)1 504 9 854 1580 t ( 1.)1 275( 1.)1 309( 1.)1 350( 1.)1 392( 1.)1 452( 1.)1 508( 1.)1 562( 2.)1 598(6 4.)1 504 9 854 1710 t ( 1.)1 275( 1.)1 309( 1.)1 350( 1.)1 392( 1.)1 452( 1.)1 508( 1.)1 562( 2.)1 598(7 8.)1 504 9 854 1840 t(8)16(1)504 1 1 854(1970)8,r(1)1364,1926,1926,f,(α)。2)1,645,7,1873,1970,T(9,20),1 504,1,1,854,π,α,α,α,α,α,α,α,α,π,π,π,ε,α,α,α,α,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,(?)2)1,645,7,1873,2100,T(10,40),1 559,1,1,799,π,α,α,α,α,α,α,α,α,π,π,π,ε,α,α,α,α,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,(?)(1)1(275)(1)。1(309)2)1、645、7、1873、2230、t2)1,591,6,2435,2490,T(32)1(520)(13 256)。2)1,591,6,2435,2620,T(1),1,275(1)1 309(309),α(α)。4)(2)1015(128)1(570)(15 1024)1 642 642 799(α),α(α),α(α),α(α)。2)1,535,5,2943,3010,T(17,2720)1 600 600,1,799,r,(f),α,α,α,α,α,α,α,α,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,π,2.)1 475 4 3395 3140 t ( 1.)1 275( 2.)1 309( 2.)1 350( 2.)1 392( 4.)1 425( 10.)1 521( 64)1 535( 512)1 570(18 5248)1 628 9 799 3270 t (19 10496)1 628 1 799 3400 t 8 R f (6)1433 3356 w 11 R f ( 1.)1 275( 2.)1 309( 2.)1 350( 2.)1 392( 4.)1 425( 20.)1 494(1024 128)1 755 7 1804 3400 t (20 20480)1 628 1 799 3530 t 8 R f (7)1433 3486 w 11 R f (2048)1757 3530 w 8 R f (10)1983 3486 w 11 R f (256 40.)1 632 1 2394 3530 t 8 R f (8)3032 3486 w 11 R f (6.)3348 3530 w 8 R f (17)3437 3486 w 11 R f ( 2.)1 275( 2.)1 309(2. 2)1,433,3,3787,3530,T(2560)1,523(21 36864),1,36864,α,α,α,α,ε,α,ε,α,ε,α,α;42)1,612,1,2380,3660,T,8,R(18)2998,3616,11,11,r,f(8)。2)1,433,3,3787,3660,T(48)1,391(1024)1 577(577),α(α),α,α,π,α,α,α,α,α,α,π,π4)1,502,4,3368,3790,T(68)1,391(2048)1 577(577),α(α),α,α,π,α,α,α,α,α,α,π,πW.Y.R.F*(α),ω,ω,ω,α,ω,α,ω,α,ω,α,ω,ω,α,ω,α,ω,α,ω,ω,α,ω,α,ω,α,α,ω,α,α,ω,α,α,ω,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,α,β4)1,502,4,3368,3920,T(24,294912)1 655 1 1,t,r,(f)2)1,392,2,4137,4050,T(151)1,432(4096)1 535(535)2)1,392,2,4137,4180,T(2),1,275(2)1 309(309),α(α),α(α),α(α),α(α),α,α,α,α,β,π,α,α,β,β,π,α,β,π,π,α,α,β,π,α,β,π,α,β,π,α,β,π,α,β,π,α,β,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,π,α,β,π,π?6)1,414,1,3759,4440,T,8,R(17)4179 4396,11 11 R f(4)。)(α),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),α,w,(f),(f),(f)。2)1、358、1、4446、4440、4440(28 4194304)1 683 1 7992.)1 358 1 4446 4570 t 11 S f (\347)1014 4570 w (\347)1014 4490 w (\347)1014 4380 w (\347)1014 4270 w (\347)1014 4160 w (\347)1014 4050 w (\347)1014 3940 w (\347)1014 3830 w (\347)1014 3720 w (\347)1014 3610 w (\347)1014 3500 w (\347)1014 3390 w (\347)1014 3280 w (\347)1014 3170 w (\347)1014 3060 w (\347)1014 2950 w (\347)1014 2840 w (\347)1014 2730 w (\347)1014 2620 w (\347)1014 2510 w (\347)1014 2400 w (\347)1014 2290 w (\347)1014 2180 w (\347)1014 2070 w (\347)1014 1960 w (\347)1014 1850 w (\347)1014 1740 w (\347)1014 1630 w (\347)1014 1520 w (\347)1014 1410 w cleartomark showpage saveobj restore end %%PageBoundingBox: 67 319 508 761 %%EndPage: 3 3 %%Page: 4 4 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 4 pagesetup 11 R f (- 4 -)2 183 1 2788 520 t (KEY TO TABLE 1)3 864 1 2448 1040 t (Unmarked entries are either trivial or are obtained by shortening the code below.)12 3545 1 1108 1170 t (An entry followed by a period is known to be exact.)10 2287 1 1736 1300 t (1)1163 1690 w 11 S f (=)1273 1690 w 11 R f (Extended Hamming code \([25], p. 23\).)5 1703 1 1426 1690 t (2)1163 1820 w 11 S f (=)1273 1820 w 11 R f (Conference matrix code \([25], p.585\).)4 1666 1 1426 1820 t (3)1163 1950 w 11 S f (=)1273 1950 w 11 R f (Found by M. Best \([5], p. 140\).)6 1385 1 1426 1950 t (4)1163 2080 w 11 S f (=)1273 2080 w 11 R f (From the Steiner system)3 1074 1 1426 2080 t 11 I f (S)2528 2080 w 11 R f (\(5,6,12\) \([5], p. 139, [25], p. 585\).)6 1505 1 2583 2080 t (5)1163 2210 w 11 S f (=)1273 2210 w 11 R f (Romanov \261 see Sect. VI of [4].)6 1368 1 1426 2210 t (6)1163 2340 w 11 S f (=)1273 2340 w 11 R f (From Hamming code over GF\(5\) [16].)5 1697 1 1426 2340 t (7)1163 2470 w 11 S f (=)1273 2470 w 11 R f (From the)1 401 1 1426 2470 t 11 I f (u)1855 2470 w 11 S f (\357)1938 2470 w 11 I f (u)2020 2470 w 11 S f (+)2102 2470 w 11 I f (v)2181 2470 w 11 R f (construction \([25], p. 76\).)3 1130 1 2257 2470 t (8)1163 2600 w 11 S f (=)1273 2600 w 11 R f (Hadamard matrix code \([25], p. 49\).)5 1591 1 1426 2600 t (8)1108 2730 w 11 I f (a)1163 2730 w 11 S f (=)1273 2730 w 11 R f (``Nadler'' code \([25], pp. 75, 79\).)5 1483 1 1426 2730 t (9)1163 2860 w 11 S f (=)1273 2860 w 11 R f (Nordstrom-Robinson code \([25], p. 73\).)4 1756 1 1426 2860 t (10)1108 2990 w 11 S f (=)1273 2990 w 11 R f (Nonlinear code from Construction X \([25], p. 583\).)7 2259 1 1426 2990 t (11)1108 3120 w 11 S f (=)1273 3120 w 11 R f (From Construction X4 \([25], p. 585, Example 7\).)7 2169 1 1426 3120 t (12)1108 3250 w 11 S f (=)1273 3250 w 11 R f (Wagner [34].)1 583 1 13250 t(13)1108 3380 W 11 s f=(1273)3380 W 11 r f(缩短的非本原BCH码长度32([25),p 586))4263,11, [25], Chap. 20\).)7 2029 1 1426 3640 t (16)1108 3770 w 11 S f (=)1273 3770 w 11 R f (Self-dual double circulant code \([5], p. 189, [25], p. 509\).)9 2531 1 1426 3770 t (17)1108 3900 w 11 S f (=)1273 3900 w 11 R f (From Hadamard matrices using Levenshtein's construction \([25], p. 50\).)8 3189 1 1426 3900 t (18)1108 4030 w 11 S f (=)1273 4030 w 11 R f (Extended quasi-cyclic code [18].)3 1443 1 1426 4030 t (19)1108 4160 w 11 S f (=)1273 4160 w 11 R f (Extended cyclic code [19] \(see Table 11 of [4]\).)8 2103 1 1426 4160 t (20)1108 4290 w 11 S f (=)1273 4290 w 11 R f (Hashim-Pozdniakov linear code [17].)3 1646 1 1426 4290 t (21)1108 4420 w 11 S f (=)1273 4420 w 11 R f (Cyclic code \(see Table 11 of [4]\).)6 1477 1 1426 4420 t (22)1108 4550 w 11 S f (=)1273 4550 w 11 R f (Piret [29].)1 446 1 1426 4550 t (23)1108 4680 w 11 S f (=)1273 4680 w 11 R f (Linear code \(Eq. \(51\) of [4], [25], p. 593\).)8 1851 1 1426 4680 t ( The)1 232(Similar problems arise in studying constant weight codes.)7 2586 2 885 5070 t 11 I f (Johnson graph J)2 747 1 3737 5070 t 11 R f (\()4493 5070 w 11 I f (n)4538 5070 w 11 R f (,)4602 5070 w 11 I f (d)4639 5070 w 11 R f (,)4703 5070 w 11 I f (w)4740 5070 w 11 R f (\) has)1 217 1 4823 5070 t 11 S f (\354)729 5327 w (\356)729 5437 w 11 I f (w)783 5434 w (n)792 5324 w 11 S f (\374)857 5327 w (\376)857 5437 w 11 R f ( length)1 307(vertices labeled by binary vectors of)5 1618 2 944 5368 t 11 I f (n)2901 5368 w 11 R f (and weight)1 489 1 2988 5368 t 11 I f (w)3509 5368 w 11 R f (, two vertices being joined by an)6 1457 1 3583 5368 t ( vectors is at least)4 890(edge if and only if the Hamming distance between the corresponding)10 3292 2 720 5688 t 11 I f (d)4957 5688 w 11 R f (.)5012 5688 w (Equivalently, the vertices represent)3 1563 1 720 5948 t 11 I f (w)2315 5948 w 11 R f (-subsets of an)2 612 1 2389 5948 t 11 I f (n)3033 5948 w 11 R f (-set, two vertices being joined by an edge if)8 1952 1 3088 5948 t (and only if the corresponding subsets intersect in at most)9 2575 1 720 6208 t 11 I f (W)3331 6208 W 11 S F(-)3432 6208 W 11 11 S1 F()3511 3511 6208 W 11 S1 F 3511 3511 6208 m 83建造12π6208μd(D)αW F F(点)。Then)1 587 1 3694 6208 t 11 I f (A)4317 6208 w 11 R f (\()4393 6208 w 11 I f (n)4438 6208 w 11 R f (,)4502 6208 w 11 I f (d)4539 6208 w 11 R f (,)4603 6208 w 11 I f (w)4640 6208 w 11 R f ( the)1 171(\) is)1 146 2 4723 6208 t (size of a maximal clique in)5 1188 1 720 6468 t 11 I f (J)1936 6468 w 11 R f (\()1993 6468 w 11 I f (n)2038 6468 w 11 R f (,)2102 6468 w 11 I f (d)2139 6468 w 11 R f (,)2203 6468 w 11 I f (w)2240 6468 w 11 R f (\).)2323 6468 w ( also taken from [4], shows the best lower bounds on)10 2405(Table 2,)1 360 2 885 6793 t 11 I f (A)3686 6793 w 11 R f (\()3762 6793 w 11 I f (n)3807 6793 w 11 R f (, 10 ,)2 184 1 3871 6793 t 11 I f (w)4064 6793 w 11 R f (\) for)1 199 1 4147 6793 t 11 I f (n)4382 6793 w 11 S f (\243)4473 6793 w 11 R f (28. Again)1 470 1 4570 6793 t ( \256rst open case is)4 818( The)1 241( an open problem.)3 839(every entry not followed by a period is)7 1818 4 720 7053 t 11 I f (A)4479 7053 w 11 R f (\( 20 , 10 , 9 \),)6 485 1 4555 7053 t (where we know only that)4 1110 1 720 7313 t cleartomark showpage saveobj restore end %%PageBoundingBox: 61 44 514 761 %%EndPage: 4 4 %%Page: 5 5 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 5 pagesetup 11 R f (- 5 -)2 183 1 2788 520 t (20)2334 1040 w 11 S f (\243)2480 1040 w 11 I f (A)2577 1040 w 11 R f (\( 20 , 10 , 9 \))6 457 1 2653 1040 t 11 S f (\243)3155 1040 w 11 R f (24)3252 1040 w 11 I f (.)3398 1040 w 11 R f (The lower bound is a cyclic code: take all cyclic shifts of the vector)13 2968 1 720 1365 t (00010001001010011111)2330 1690 w (\(see Table 11 of [4]\), while the upper bound comes from [3].)11 2674 1 720 2015 t ( of the upper bounds)4 924(It should be said at this point that there is a tiny bit of doubt about some)16 3231 2 885 2340 t (on)720 2600 w 11 I f (A)869 2600 w 11 R f (\()945 2600 w 11 I f (n)990 2600 w 11 R f (,)1054 2600 w 11 I f (d)1091 2600 w 11 R f (,)1155 2600 w 11 I f (w)1192 2600 w 11 R f (\) in [3] in the case)5 851 1 1275 2600 t 11 I f (d)2165 2600 w 11 S f (=)2247 2600 w 11 R f ( it is usually)3 568( While)1 334( to recompute them.)3 910(10; we are planning)3 902 4 2326 2600 t ( a lower bound on)4 902(easy to verify)2 649 2 720 2860 t 11 I f (A)2326 2860 w 11 R f (\()2402 2860 w 11 I f (n)2447 2860 w 11 R f (,)2511 2860 w 11 I f (d)2548 2860 w 11 R f (\) or)1 182 1 2612 2860 t 11 I f (A)2849 2860 w 11 R f (\()2925 2860 w 11 I f (n)2970 2860 w 11 R f (,)3034 2860 w 11 I f (d)3071 2860 w 11 R f (,)3135 2860 w 11 I f (w)3172 2860 w 11 R f (\) \(by checking the Hamming distance)5 1785 1 3255 2860 t (between the codewords\) upper bounds are much harder to verify.)9 2856 1 720 3120 t (TABLE 2)1 437 1 2661 3445 t (Lower bounds on)2 769 1 2264 3575 t 11 I f (A)3061 3575 w 11 R f (\()3128 3575 w 11 I f (n)3164 3575 w 11 R f (,10,)3219 3575 w 11 I f (w)3385 3575 w 11 R f (\))3459 3575 w 11 I f (n)934 3835 w 11 R f (,)998 3835 w 11 I f (w)1035 3835 w 11 R f ( 11 12 13 14)4 1680( 10)1 447( 9)1 405( 8)1 396(6 7)1 459 5 1366 3835 t 11 S f (_ ______________________________________________________________________)1 3892 1 934 3855 t 11 R f ( 1 1 1 0 0)5 2100( 1)1 405( 1)1 396( 2)1 390(12 2.)1 469 5 966 3985 t ( 1 1 1 1 0)5 2100( 1)1 405( 2)1 396( 2)1 390(13 2.)1 469 5 966 4115 t ( 1 1 1 1 1)5 2100( 2)1 405( 2)1 382( 2.)1 404(14 2.)1 469 5 966 4245 t (15 3.)1 447 1 966 4375 t 8 I f (j)1426 4331 w 11 R f (3.)1734 4375 w 8 I f (j)1830 4331 w 11 R f ( 3 1 1 1 1)5 2100(3 3)1 460 2 2166 4375 t ( 4.)1 382(16 3.)1 469 2 966 4505 t 8 I f (j)1830 4461 w 11 R f (4.)2130 4505 w 8 I f (j)2226 4461 w 11 R f (4 3 3 1 1 1)5 2155 1 2571 4505 t ( 5.)1 382(17 3.)1 469 2 966 4635 t 8 I f (j)1830 4591 w 11 R f (6.)2130 4635 w 8 I f (j)2226 4591 w 11 R f (6 5 3 3 1 1)5 2155 1 2571 4635 t (18 4.)1 447 1 966 4765 t 8 I f (j)1426 4721 w 11 R f (6.)1734 4765 w 8 I f (j)1830 4721 w 11 R f (9.)2102 4765 w 8 I f (q)2191 4721 w 8 R f (2)2237 4721 w 11 R f (10.)2506 4765 w 8 I f (s)2650 4721 w 11 R f (9 6 4 3 1)4 1735 1 2991 4765 t ( 8.)1 379(19 4.)1 469 2 966 4895 t 8 I f (x)1820 4851 w 11 R f (12.)2082 4895 w 8 I f (sb)2226 4851 w 11 R f (19.)2504 4895 w 8 I f (c)2648 4851 w 11 R f ( 4 3)2 840( 8)1 393(19 12)1 530 3 2963 4895 t (20 5.)1 446 1 966 5025 t 8 I f (s)1418 4981 w 11 R f (10.)1678 5025 w 8 I f (q)1822 4981 w 8 R f (2)1868 4981 w 11 R f (17.)2088 5025 w 8 I f (m)2232 4981 w 11 R f (20)2518 5025 w 8 I f (c)2634 4981 w 11 R f (38.)2893 5025 w 8 I f (hm)3037 4981 w 11 R f ( 5)1 393(20 17 10)2 950 2 3383 5025 t (21 7.)1 441 1 966 5155 t 8 I f (a)1413 5111 w 11 R f (13.)1683 5155 w 8 I f (xh)1827 5111 w 11 R f (21.)2100 5155 w 8 I f (c)2244 5111 w 11 R f (27)2498 5155 w 8 I f (pc)2614 5111 w 11 R f (38 38 27 21 13)4 1790 1 2963 5155 t ( 16.)1 386(22 7.)1 469 2 966 5285 t 8 I f (pc)1827 5241 w 11 R f (24)2096 5285 w 8 I f (sd)2212 5241 w 11 R f (35)2498 5285 w 8 I f (pc)2614 5241 w 11 R f (42)2921 5285 w 8 I f (ec)3037 5241 w 11 R f (46)3358 5285 w 8 I f (c)3474 5241 w 11 R f (42 35 24)2 950 1 3803 5285 t (23 8.)1 421 1 966 5415 t 8 I f (x)1393 5371 w 8 R f (2)1434 5371 w 11 R f (20)1717 5415 w 8 I f (y)1833 5371 w 11 R f (33)2094 5415 w 8 I f (pc)2210 5371 w 11 R f (45)2498 5415 w 8 I f (pc)2614 5371 w 11 R f (54)2918 5415 w 8 I f (pc)3034 5371 w 11 R f (63)3338 5415 w 8 I f (pc)3454 5371 w 11 R f (63 54 45)2 950 1 3803 5415 t (24 9.)1 421 1 966 5545 t 8 I f (x)1393 5501 w 8 R f (2)1434 5501 w 11 R f (24)1717 5545 w 8 I f (c)1833 5501 w 11 R f (38)2094 5545 w 8 I f (pc)2210 5501 w 11 R f (56)2518 5545 w 8 I f (c)2634 5501 w 11 R f (72)2938 5545 w 8 I f (c)3054 5501 w 11 R f (90)3338 5545 w 8 I f (pc)3454 5501 w 11 R f (96)3778 5545 w 8 I f (c)3894 5501 w 11 R f (90 72)1 530 1 4223 5545 t (25 10.)1 473 1 966 5675 t 8 I f (s)1445 5631 w 11 R f (28)1700 5675 w 8 I f (ec)1816 5631 w 11 R f (48)2096 5675 w 8 I f (ec)2212 5631 w 11 R f (72)2501 5675 w 8 I f (ec)2617 5631 w 11 R f (100)2911 5675 w 8 I f (c)3082 5631 w 11 R f (125)3331 5675 w 8 I f (c)3502 5631 w 11 R f (130)3733 5675 w 8 I f (ec)3904 5631 w 11 R f (130 125)1 585 1 4196 5675 t (26 13.)1 446 1 966 5805 t 8 I f (q)1418 5761 w 8 R f (2)1464 5761 w 11 R f (28 54)1 462 1 1742 5805 t 8 I f (pc)2210 5761 w 11 R f (84)2498 5805 w 8 I f (pc)2614 5761 w 11 R f (130)2911 5805 w 8 I f (c)3082 5761 w 11 R f (168)3311 5805 w 8 I f (pc)3482 5761 w 11 R f (185)3751 5805 w 8 I f (y)3922 5761 w 11 R f (191)4171 5805 w 8 I f (y)4342 5761 w 11 R f (185)4616 5805 w (27 14.)1 446 1 966 5935 t 8 I f (q)1418 5891 w 8 R f (9)1464 5891 w 11 R f (36)1692 5935 w 8 I f (q)1808 5891 w 8 R f (3)1854 5891 w 11 R f (66)2094 5935 w 8 I f (pc)2210 5891 w 11 R f (111)2491 5935 w 8 I f (c)2662 5891 w 11 R f (159)2891 5935 w 8 I f (pc)3062 5891 w 11 R f (213)3311 5935 w 8 I f (ya)3482 5891 w 11 R f (257)3751 5935 w 8 I f (y)3922 5891 w 11 R f (283)4151 5935 w 8 I f (ya)4322 5891 w 11 R f (283)4616 5935 w (28 16.)1 460 1 966 6065 t 8 I f (m)1432 6021 w 11 R f (37)1692 6065 w 8 I f (q)1808 6021 w 8 R f (4)1854 6021 w 11 R f (78)2094 6065 w 8 I f (pc)2210 6021 w 11 R f (132)2471 6065 w 8 I f (pc)2642 6021 w 11 R f (195)2891 6065 w 8 I f (yd)3062 6021 w 11 R f (280)3311 6065 w 8 I f (ya)3482 6021 w 11 R f (356)3731 6065 w 8 I f (ya)3902 6021 w 11 R f (414)4151 6065 w 8 I f (ya)4322 6021 w 11 R f (435)4571 6065 w 8 I f (yd)4742 6021 w 11 S f (\347)1191 6065 w (\347)1191 6015 w (\347)1191 5905 w (\347)1191 5795 w (\347)1191 5685 w (\347)1191 5575 w (\347)1191 5465 w (\347)1191 5355 w (\347)1191 5245 w (\347)1191 5135 w (\347)1191 5025 w (\347)1191 4915 w (\347)1191 4805 w (\347)1191 4695 w (\347)1191 4585 w (\347)1191 4475 w (\347)1191 4365 w (\347)1191 4255 w (\347)1191 4145 w (\347)1191 4035 w (\347)1191 3925 w (\347)1191 3815 w cleartomark showpage saveobj restore end %%PageBoundingBox: 61 169 514 761 %%EndPage: 5 5 %%Page: 6 6 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 6 pagesetup 11 R f (- 6 -)2 183 1 2788 520 t (KEY TO TABLE 2)3 864 1 2547 1040 t (An eNTIt之后是一个精确的周期。)10 2287,1,1835,1170 T(截面和表的参考文献是[4 ])6 6 1 2135 2135 1300 t 11 11 f(a)11αw f=(=)αf f(从一个平凡的设计或它的对偶)。III\).)8 1871 1 1025 1690 t 11 I f (c)806 1820 w 11 S f (=)909 1820 w 11 R f (Cyclic code \(Table 11\).)3 1036 1 1025 1820 t 11 I f (ec)758 1950 w 11 S f (=)909 1950 w 11 R f (Extended cyclic code \(Table 12\).)4 1452 1 1025 1950 t 11 I f (hm)720 2080 w 11 S f (=)909 2080 w 11 R f (Hadamard matrix code \(Theorem 10\).)4 1665 1 1025 2080 t 11 I f (j)823 2210 w 11 S f (=)909 2210 w 11 R f (Juxtaposing \(Eq. \(1\) of Sect. Ⅲ)5 1459,1,1025,2210,T 11,I,F(m)775,2340,2340,11 s,F=909=2340 2340 11 f(杂项结构)。XI\).)3 1693 1 1025 2340 t 11 I f (pc)751 2470 w 11 S f (=)909 2470 w 11 R f (Orbits under a single permutation \(Table 14\).)6 1993 1 1025 2470 t 11 I f (qi)768 2600 w 11 S f (=)909 2600 w 11 R f (Quasi-cyclic code, for 2)3 1053 1 1025 2600 t 11 S f (\243)2114 2600 w 11 I f (i)2211 2600 w 11 S f (\243)2278 2600 w 11 R f (9 \261 \256xed by a permutation containing)6 1651 1 2375 2600 t 11 I f (i)4054 2600 w 11 R f (cycles of length)2 695 1 4113 2600 t 11 I f (n / i)2 135 1 4836 2600 t 11 R f (\(Table 13\).)1 487 1 1025 2730 t 11 I f (s)811 2860 w 11 S f (=)909 2860 w 11 R f (Section of code below or diagonally down to right, obtained from \(5\) of Sect. III.)14 3570,1,1025,2860,T,11,I,F(SB)756,2990,2990,11,S,F(=)909 2990 2990 11 R f(下面的代码段,通过代码的直接检查获得)(SeCT)。iii)(12)3395,1,1025,2990,T,11,I,F(SD)756,3120,3120,11,S,F(=)909 3120 3120 11 r f(代码对角线向右向下的部分,通过直接检查代码获得)。(iii))15 4213,1,1025,3120,T,11,I,F(x)806,3250,3250,11,S,F(=)909 3250 3250 11 R f(字典代码)。VIII\).)3 1426 1 1025 3250 t 11 I f (xh)751 3380 w 11 S f (=)909 3380 w 11 R f (Lexicode with seed \(Table 8\).)4 1313 1 1025 3380 t 11 I f (x)742 3510 w 11 R f (2)799 3510 w 11 S f (=)909 3510 w 11 R f (Complement of lexicode with sum constraint \(Table 7\).)7 2444 1 1025 3510 t 11 I f (y)806 3640 w 11 S f (=)909 3640 w 11 R f (No known structure \(Table 16\).)4 1387 1 1025 3640 t 11 I f (ya)751 3770 w 11 S f (=)909 3770 w 11 R f (Obtained by extending the code above it in the table; no other structure.)12 3154 1 1025 3770 t 11 I f (yd)751 3900 w 11 S f (=)909 3900 w 11 R f (Obtained by extending the code diagonally above it to left; no other structure.)12 3417 1 1025 3900 t %INFO[SECTION: LEVEL = 1, NUMBER = 3. ,方向=寻找最大加权群(11)B f(最大加权群)3(1239)(3)。Finding)1 509 2 720 4355 t 11 R f ( many cases a code containing the)6 1560( In)1 158( in the following way.)4 1014(Weighted clique problems arise)3 1423 4 885 4680 t (maximal number \()2 896 1 720 4940 t 11 I f (A)1625 4940 w 11 R f (\()1701 4940 w 11 I f (n)1746 4940 w 11 R f (,)1810 4940 w 11 I f (d)1847 4940 w 11 R f (,)1911 4940 w 11 I f (w)1948 4940 w 11 R f ( example)1 440( For)1 250( has a nontrivial symmetry group.)5 1691( of vectors)2 547(\) \))1 81 5 2031 4940 t 11 I f (A)720 5200 w 11 R f (\( 18 , 8 , 6 \))6 402 1 796 5200 t 11 S f (=)1216 5200 w 11 R f (21 is realized by the code)5 1119 1 1295 5200 t (\(110100\)\(100000\)\(110000\),)2263 5525 w (\(000010\)\(110100\)\(100001\),)2263 5655 w (\(000011\)\(100001\)\(010100\),)2263 5785 w (\(010101\)\(010101\)\(000000\),)2263 5915 w (\(000000\)\(000000\)\(111111\).)2263 6045 w (The parentheses indicate that the permutation)5 1997 1 720 6370 t (\( 1 , 2 , 3 , 4 , 5 , 6 \) \( 7 , 8 , 9 , 10 , 11 , 12 \) \( 13 , 14 , 15 , 16 , 17 , 18 \))38 2463 1 1648 6695 t ( three vectors each give rise to six)7 1540( \256rst)1 206( The)1 232(of order six is to be applied to the indicated vectors.)10 2342 4 720 7020 t ( codewords and the last vector to a single codeword \(since it)11 2717(codewords, the fourth vector to two)5 1603 2 720 7280 t cleartomark showpage saveobj restore end %%PageBoundingBox: 61 48 534 761 %%EndPage: 6 6 %%Page: 7 7 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 7 pagesetup 11 R f (- 7 -)2 183 1 2788 520 t (is \256xed by the permutation\).)4 1245 1 720 1040 t ( we choose some permutation group)5 1605(To generalize this construction,)3 1391 2 885 1365 t 11 I f (G)3912 1365 w 11 R f (on)4022 1365 w 11 I f (n)4163 1365 w 11 R f (letters, and divide)2 791 1 4249 1365 t (the vectors of length)3 918 1 720 1625 t 11 I f (n)1672 1625 w 11 R f (into)1761 1625 w 11 I f (orbits)1967 1625 w 11 R f (under)2259 1625 w 11 I f (G)2542 1625 w 11 R f (\(two vectors are in the same orbit if and only if some)11 2385 1 2655 1625 t (permutation in)1 656 1 720 1885 t 11 I f (G)1415 1885 w 11 R f ( vertices)1 380( The)1 238( form a weighted graph as follows.)6 1601( We)1 219( the other\).)2 503(sends one to)2 566 6 1533 1885 t ( distinct vectors is)3 814(represent good orbits \(orbits in which the Hamming distance between any two)11 3506 2 720 2145 t (at least)1 315 1 720 2405 t 11 I f (d)1071 2405 w 11 R f (\), and two vertices are joined by an edge if and only if the Hamming distance between)16 3914 1 1126 2405 t ( vector in the other orbit is at least)8 1528(every vector in one orbit and every)6 1559 2 720 2665 t 11 I f (d)3838 2665 w 11 R f ( vertex is weighted)3 842(. (1)287(按相应轨道的大小)。6 1700 3 3 t(是最大的恒权码不变)。每个)1,305,2,3893,2665,t(图中最大加权团的大小)10 2333标题=图着色问题〉11 B F(着色问题)2(881)(4)。Graph)1 452 2 720 4160 t 11 R f ( be)1 136(Besides looking for group-invariant codes we used many other constructions in [1], as can)13 4019 2 885 4485 t ( particularly powerful construction, applicable to codes with)7 2655( A)1 136( above.)1 318( 2)1 83(seen in Table)2 589 5 720 4745 t 11 I f (d)4530 4745 w 11 S f (=)4612 4745 w 11 R f (4, is the)2 349 1 4691 4745 t 11 I f (partitioning construction)1 1137 1 720 5005 t 11 R f ( good colorings of the graphs)5 1446( this one needs)3 744( For)1 240(\(see [4], [13]\).)2 694 4 1916 5005 t 11 I f (H)720 5265 w 11 R f (\()808 5265 w 11 I f (n)853 5265 w 11 R f (, 4 ,)2 129 1 917 5265 t 11 I f (w)1055 5265 w 11 R f (\).)1138 5265 w (Let)885 5590 w 11 S f (P)1065 5590 w 11 R f (\()1159 5590 w 11 I f (n)1204 5590 w 11 R f (,)1268 5590 w 11 I f (w)1305 5590 w 11 R f (\))1388 5590 w 11 S f (=)1442 5590 w 11 R f (\()1521 5590 w 11 I f (X)1566 5590 w 8 R f (1)1646 5612 w 11 R f (,)1704 5590 w 11 I f (...)1768 5590 w 11 R f (,)1861 5590 w 11 I f (X)1925 5590 w 8 I f (m)2005 5612 w 11 R f ( a collection of disjoint sets or)6 1370(\) be)1 173 2 2081 5590 t 11 I f (color classes X)2 680 1 3659 5590 t 8 R f (1)4352 5612 w 11 R f (,)4410 5590 w 11 I f (...)4474 5590 w 11 R f (,)4567 5590 w 11 I f (X)4631 5590 w 8 I f (m)4711 5612 w 11 R f (, each)1 262 1 4778 5590 t (of which is a code of length)6 1281 1 720 5850 t 11 I f (n)2039 5850 w 11 R f (, distance 4 and constant weight)5 1455 1 2094 5850 t 11 I f (w)3587 5850 w 11 R f ( whose union contains all)4 1155(, and)1 224 2 3661 5850 t 11 S f (\354)729 6107 w (\356)729 6217 w 11 I f (w)783 6214 w (n)792 6104 w 11 S f (\374)857 6107 w (\376)857 6217 w 11 R f (vectors of weight)2 796 1 956 6148 t 11 I f (w)1797 6148 w 11 R f ( other words)2 583(. W~(i)f~(n),ρ,(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),(f),α,(f),π,f,(f),(f),(f),(f),(f),(f),(f),(f),w,w,f,f,f,(f)。in)1 192,2,1871,6148,t,11,s,f(p)2691,6148,6148,11,r,(()2785 6148(1)289、1、4751、6148 T(假定)720 6468 W 11 s f(357)1101 6486瓦特11 f(x)11(x){f}{f}(ω);(2)140,1,1490,6441,T,11,S,F(263),1694,6468,w(357),1782,6486,6486,f,(x),α,π,(f)(f)(f)(f)。The)1 284 2 2065 6468 t 11 S f (p)2738 6468 w 11 R f (\()2808 6468 w 11 I f (n)2853 6468 w 11 R f (,)2917 6468 w 11 I f (w)2954 6468 w 11 R f (\))3037 6468 w 11 S f (=)3091 6468 w 11 R f (\()3170 6468 w 11 S f (\357)3206 6486 w 11 I f (X)3260 6468 w 8 R f (1)3340 6490 w 11 S f (\357)3389 6486 w 11 R f (,)3443 6468 w 11 I f (...)3507 6468 w 11 R f (,)3600 6468 w 11 S f (\357)3655 6486 w 11 I f (X)3709 6468 w 8 I f (m)3789 6490 w 11 S f (\357)3856 6486 w 11 R f (\) is the)2 358 1 3910 6468 t 11 I f (index vector)1 567 1 4325 6468 t 11 R f (of)4949 6468 w 11 S f (P)720 6728 w 11 R f (\()814 6728 w 11 I f (n)859 6728 w 11 R f (,)923 6728 w 11 I f (w)960 6728 w 11 R f (\), and)1 250 1 1043 6728 t 11 S f (p)2169 7105 w 11 R f (\()2239 7105 w 11 I f (n)2284 7105 w 11 R f (,)2348 7105 w 11 I f (w)2385 7105 w 11 R f (\))2468 7105 w (.)2559 7072 w 11 S f (p)2633 7105 w 11 R f (\()2703 7105 w 11 I f (n)2748 7105 w 11 R f (,)2812 7105 w 11 I f (w)2849 7105 w 11 R f (\))2932 7105 w 11 S f (=)3032 7105 w 8 I f (i)3148 7217 w 8 S f (=)3189 7217 w 8 R f (1)3246 7217 w 16 S f (S)3170 7138 w 8 I f (m)3188 7001 w 11 S f (\357)3332 7123 w 11 I f (X)3386 7105 w 8 I f (i)3466 7127 w 11 S f (\357)3497 7123 w 8 R f (2)3541 7061 w cleartomark showpage saveobj restore end %%PageBoundingBox: 61 55 514 761 %%EndPage: 7 7 %%Page: 8 8 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 8 pagesetup 11 R f (- 8 -)2 183 1 2788 520 t (is its)1 213 1 720 1040 t 11 I f (norm)967 1040 w 11 R f ( several different colorings are known for a given)8 2216(. When)1 351 2 1199 1040 t 11 I f (n)3800 1040 w 11 R f (and)3889 1040 w 11 I f (w)4081 1040 w 11 R f (we denote them by)3 851 1 4189 1040 t 11 S f (P)720 1300 w 8 R f (1)818 1322 w 11 R f (\()876 1300 w 11 I f (n)921 1300 w 11 R f (,)985 1300 w 11 I f (w)1022 1300 w 11 R f (\),)1105 1300 w 11 S f (P)1197 1300 w 8 R f (2)1295 1322 w 11 R f (\()1353 1300 w 11 I f (n)1398 1300 w 11 R f (,)1462 1300 w 11 I f (w)1499 1300 w 11 R f (\) ,)1 82 1 1582 1300 t 11 I f (...)1700 1300 w 11 R f (, and their index vectors by)5 1197 1 1784 1300 t 11 S f (p)3009 1300 w 8 R f (1)3083 1322 w 11 R f (\()3141 1300 w 11 I f (n)3186 1300 w 11 R f (,)3250 1300 w 11 I f (w)3287 1300 w 11 R f (\),)3370 1300 w 11 S f (p)3462 1300 w 8 R f (2)3536 1322 w 11 R f (\()3594 1300 w 11 I f (n)3639 1300 w 11 R f (,)3703 1300 w 11 I f (w)3740 1300 w 11 R f (\) ,)1 82 1 3823 1300 t 11 I f (...)3941 1300 w 11 R f (.)4052 1300 w ( important)1 459( The)1 229( [4] or [13] for details of the partitioning construction.)9 2403(The reader is referred to)4 1064 4 885 1625 t ( the best colorings)3 816(point here is that)3 752 2 720 1885 t 11 S f (P)2320 1885 w 11 R f (\()2414 1885 w 11 I f (n)2459 1885 w 11 R f (,)2523 1885 w 11 I f (w)2560 1885 w 11 R f (\) to use in the construction are those that are maximal)10 2397 1 2643 1885 t ( say that one coloring)4 1275( We)1 289(in the following sense.)3 1243 3 720 2145 t 11 S f (P)3637 2145 w 11 R f (\()3731 2145 w 11 I f (n)3776 2145 w 11 R f (,)3840 2145 w 11 I f (w)3877 2145 w 11 R f ( vector)1 384(\) with index)2 696 2 3960 2145 t 11 S f (p)720 2405 w 11 R f (\()790 2405 w 11 I f (n)835 2405 w 8 R f (1)903 2427 w 11 R f (,)961 2405 w 11 I f (w)1025 2405 w 8 R f (1)1112 2427 w 11 R f (\))1170 2405 w 11 S f (=)1260 2405 w 11 R f (\()1366 2405 w 11 I f (a)1411 2405 w 8 R f (1)1479 2427 w 11 R f (,)1537 2405 w 11 I f (...)1601 2405 w 11 R f (,)1694 2405 w 11 I f (a)1758 2405 w 8 I f (m)1826 2427 w 11 R f (\))1902 2405 w 11 I f (dominates)2007 2405 w 11 R f (another)2528 2405 w 11 S f (P)2925 2405 w 11 R f (\()3019 2405 w 11 I f (n)3064 2405 w 11 S f (\242)3132 2401 w 11 R f (,)3178 2405 w 11 I f (w)3242 2405 w 11 S f (\242)3329 2401 w 11 R f (\) with index vector)3 953 1 3375 2405 t 11 S f (p)4396 2405 w (\242)4463 2401 w 11 R f (\()4509 2405 w 11 I f (n)4554 2405 w 8 R f (1)4622 2427 w 11 R f (,)4680 2405 w 11 I f (w)4744 2405 w 8 R f (1)4831 2427 w 11 R f (\))4889 2405 w 11 S f (=)4979 2405 w 11 R f (\()720 2665 w 11 I f (b)765 2665 w 8 R f (1)833 2687 w 11 R f (,)891 2665 w 11 I f (...)955 2665 w 11 R f (,)1048 2665 w 11 I f (b)1112 2665 w 8 I f (m)1180 2687 w 8 S f (\242)1246 2680 w 11 R f (\) if and only if)4 636 1 1290 2665 t 8 I f (i)2511 3154 w 8 S f (=)2552 3154 w 8 R f (1)2609 3154 w 16 S f (S)2533 3075 w 8 I f (j)2569 2938 w 11 I f (a)2704 3042 w 8 I f (i)2772 3064 w 11 S f (\263)2849 3042 w 8 I f (i)2956 3154 w 8 S f (=)2997 3154 w 8 R f (1)3054 3154 w 16 S f (S)2978 3075 w 8 I f (j)3014 2938 w 11 I f (b)3149 3042 w 8 I f (i)3217 3064 w 11 R f (holds for all)2 532 1 720 3458 t 11 I f (j)1280 3458 w 11 S f(1329)3458 W(11)R f(1),1,92,1,1408,3458,T,11,I,F(…)1536,3458,3458,f,({)α,(x),(x),(f),(f),(f),(f),(f),(f),(f),(f),(f)(f)(f)。A)1 216 2 2243 3458 t 11 I f (maximal)2983 3458 w 11 R f (if it is not dominated by any other.)7 1525 1 3389 3458 t ( of colorings with small values of)6 1492(In [4] we made an extensive investigation)6 1848 2 885 3783 t 11 I f (n)4256 3783 w 11 R f (, and found over)3 729 1 4311 3783 t (a thousand)1 483 1 720 4043 t 11 S f (P)1241 4043 w 11 R f (\()1335 4043 w 11 I f (n)1380 4043 w 11 R f (,)1444 4043 w 11 I f (w)1481 4043 w 11 R f (\) with)1 270 1 1564 4043 t 11 I f (n)1871 4043 w 11 S f (\243)1962 4043 w 11 R f ( portion of this)3 680( A)1 144(16, no one of which is dominated by any other.)9 2157 3 2059 4043 t (list is shown in the following table.)6 1554 1 720 4303 t (TABLE 3)1 437 1 2661 4628 t (Good colorings)1 681 1 2346 4758 t 11 S f (P)3055 4758 w 11 R f (\()3149 4758 w 11 I f (n)3194 4758 w 11 R f (,)3258 4758 w 11 I f (w)3295 4758 w 11 R f (\))3378 4758 w 11 I f ( m)1 271( i)1 208(n w)1 322 3 1333 5148 t 11 R f ( vector of)2 420(Norm Notes Index)2 1091 2 2315 5148 t 11 S f (P)3854 5148 w 8 I f (i)3952 5170 w 11 R f (\()3992 5148 w 11 I f (n)4037 5148 w 11 R f (,)4101 5148 w 11 I f (w)4165 5148 w 11 R f (\))4248 5148 w ( 4,4,4,4,2,2)1 736( *)1 338( 72)1 403( 6)1 275( 1)1 230(6 3)1 284 6 1361 5343 t ( 7,7,6,6,5,4)1 736( *)1 338( 211)1 403( 6)1 275( 1)1 230(7 3)1 284 6 1361 5473 t ( 8,8,8,8,8,8,8)1 819( *)1 338( 448)1 403( 7)1 275( 1)1 230(8 3)1 284 6 1361 5603 t ( 14,14,12,12,10,8)1 1011( *)1 338( 844)1 403( 6)1 275( 1)1 230(8 4)1 284 6 1361 5733 t ( 12,12,12,12,12,12,12)1 1204( *)1 338( 1008)1 403( 7)1 275( 1)1 230(9 3)1 284 6 1361 5863 t ( 18,18,18,18,16,15,15,8)1 1625( 2066)1 403( 8)1 275( 1)1 230(9 4)1 284 5 1361 5993 t ( 18,18,18,18,18,14,13,7,1,1)1 1791( 2036)1 403( 10)1 275( 2)1 230(9 4)1 284 5 1361 6123 t ( 13,13,13,13,13,13,13,13,13,3)1 1563( *)1 338( 1530)1 403( 10)1 275( 1)1 230(10 3)1 339 6 1306 6253 t ( 30,30,30,30,30,22,22,12,2,2)1 1444( [13])1 402( 5620)1 403( 10)1 275( 1)1 230(10 4)1 339 6 1306 6383 t ( 30,30,30,30,26,25,22,15,2)1 1763( 5614)1 403( 9)1 275( 2)1 230(10 4)1 339 5 1306 6511680(8044)1(403)(8)1 275(1)1 230(230),(一般看来,)(α)(α)(一个星号表示着色)。(给定值的唯一最大着色)是α,βi,(f),(n),α,w,f,(f),(f),(f),w,f,f(一些可能的情况)。3 t(36,36,34,34,29,29,27,27)1In)1 178 2 3432 7228 t cleartomark showpage saveobj restore end %%PageBoundingBox: 61 53 514 761 %%EndPage: 8 8 %%Page: 9 9 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 9 pagesetup 11 R f (- 9 -)2 183 1 2788 520 t (color)720 1040 w 11 I f (H)979 1040 w 11 R f (\()1067 1040 w 11 I f (n)1112 1040 w 11 R f (, 4 ,)2 129 1 1176 1040 t 11 I f (w)1314 1040 w 11 R f (\) so that each color class is a)7 1296 1 1397 1040 t 11 I f (t)2727 1040 w 11 R f (-design: such colorings are said to form a)7 1858 1 2758 1040 t 11 I f (large set)1 389 1 4651 1040 t 11 R f ( [6]-[13], [20]-[24], [27], [28], [30]-[33], and other references cited in [4].)11 3240( See)1 214(of designs.)1 477 3 720 1300 t ( \256nding a good coloring is a generalization of usual problem of \256nding a good)14 3470(The problem of)2 685 2 885 1625 t (code.)720 1885 w 11 I f (Each)1025 1885 w 11 R f ( code with)2 486(color class is)2 598 2 1293 1885 t 11 I f (d)2419 1885 w 11 S f (=)2501 1885 w 11 R f (4, so now the goal is to \256nd a small number of large)12 2460 1 2580 1885 t (disjoint codes.)1 637 1 720 2145 t ( not marked with an asterisk)5 1283( 3)1 83( the colorings in Table)4 1022(We would like to know whether any of)7 1767 4 885 2470 t (\(or those in Table 6 of [4]\) can be improved.)9 1958 1 720 2730 t ( at present)2 497(A more important problem, however, is to bring some order into this subject:)12 3658 2 885 3055 t ( mathematical structure, and can be described only by)8 2371(almost all the best colorings known have no)7 1949 2 720 3315 t (listing the vectors in each color class.)6 1646 1 720 3575 t cleartomark showpage saveobj restore end %%PageBoundingBox: 61 418 514 761 %%EndPage: 9 9 %%Page: 10 10 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 10 pagesetup 11 R f (- 10 -)2 238 1 2761 520 t 11 B f (BIBLIOGRAPHY)2445 1040 w 11 R f ( and J. Nesetril, ``On the maximum weighted clique problem'', Math.)10 3087( tal)1 105( \302)1 5( Balas, V. Chva)3 698([1] E.)1 332 5 813 1495 t (of Operations Research, vol. 12 (S)1 118(Balas和C.)Sux-([O] E])。(Soop.,Vo.O.[Y],(ppl 1054-1068))(α),C.(Odlyzko)和N.(Odlyzko),(Odlyzko)和(M.),(MacWiLAMS,A.)(1987),pp.522-535)7,2327,1,1050,1755,T(在任意图'',暹罗j)6 1578(Yu,''发现最大团)5 1539信息理论,第24卷)10 2646(‘二进制码的界限’)4 1344 2 1050 1050 T(\(1978),pp.81-93.)2 798 1 1050 t(史米斯,‘一个新的常量表)αi(d))(N.和W.)J. A.(Shearer),N.(),(b,)信息理论7,2316,1,1050,3770,T(SLAON,‘球面Packings,格和群’,Springer -)7 2785(J.A.)2 234(和N.)2 355(康威)1 410(H.)Theory,)3 827( H. F. Denniston, ``Some maximal arcs in \256nite projective planes'',)10 3061([6] R.)1 339 3 813 4680 t (vol. 6 \(1969\), pp. 317-319.)4 1188 1 1050 4940 t ( Discrete Math., vol. 9)4 1016( Denniston, ``Some packings with Steiner triple systems'',)7 2619( H. F.)2 253([7] R.)1 339 4 813 5265 t (\(1974\), pp. 213-227.)2 908 1 1050 5525 t ( 9)1 83( of the 15 schoolgirls'', Discrete Math., vol.)7 2026( Denniston, ``Sylvester's problem)3 1526( H. F.)2 253([8] R.)1 339 5 813 5850 t (\(1974\), pp. 229-233.)2 908 1 1050 6110 t ( \(1976\), pp.)2 542( 8)1 83( 5-designs'', Bull. 朗德数学Soc., vol.)6 1902( Denniston, ``Some new)3 1108( H. F.)2 253([9] R.)1 339 6 813 6435 t (263-267.)1050 6695 w ( 7)1 83( vol.)1 221( Denniston, ``Four doubly resolvable complete 3-designs'', Ars Comb.,)8 3331( H. F.)2 253([10] R.)1 394 5 758 7020 t (\(1979\), pp. 265-272.)2 908 1 1050 7280 t cleartomark showpage saveobj restore end %%PageBoundingBox: 65 48 514 761 %%EndPage: 10 10 %%Page: 11 11 %%PageBoundingBox: (atend) DpostDict begin /saveobj save def mark 11 pagesetup 11 R f (- 11 -)2 238 1 2761 520 t ( Annals of Discrete)3 895( 9, 3\)'',)2 310( Denniston, ``Enumeration of symmetric designs \(25,)6 2430( H. 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Phelps, ``A class of large sets of Steiner triple systems oF阶15’,预印本)15 3564([28)K.)1 399 2 2 758 3640 t(26)1 138(体积)1(皮雷,‘好线性码的长度γ和’’,IEEE Trac。信息理论,(13)3547((29)p)1 382 4 758 3965 t((1980),p 227)2 652 652(三重系统),离散斯坦纳(TyrLink),''最大不交数的L. ],([Y]),α,π(数学,V.O.(α),pp.29~300)。Theory, vol.)5 1459( Teirlinck, ``On making two Steiner triple systems)7 2219([31] L.)1 387 4 758 5135 t (\(1977\), pp. 349-350.)2 908 1 1050 5395 t ( \(1988\),)1 369( 25)1 138( Comb., vol.)2 593( Teirlinck, ``On large sets of disjoint ordered designs'', Ars)9 2795([32] L.)1 387 5 758 5720 t (pp. 31-37.)1 450 1 1050 5980 t ( Teirlinck, ``A completion of Lu's determination of the spectrum for large sets of)13 3895([33] L.)1 387 2 758 6305 t (disjoint Steiner triple systems'', preprint.)4 1811 1 1050 6565 t ( \(1966\),)1 355( 9)1 83( Wagner, ``A search technique for quasi-perfect codes'', Info. 控制,卷(10)3358(j)1 99([34)T.)1 387 5 758 6890 t(pp.94-99)1 1 450·t } CultMark OK展示页.SaveObj还原结束%%%Page BoosikBox:γ-%%%PEGE:γ%%%Page BoeDangBox:(AtEnter)dPosiDT开始/SaveObj保存DEF标记MyPa Page SePUP %%信息[标题:〔13 B f〕(图的研究中产生的图解中未解决的问题)10、3950、1、888、1040、T、%信息[题目:3,713,1,2523,1235,T,11,R,F(数学科学研究中心)3 1762 1 1999 1999 t t(AT&T贝尔实验室),Murray Hill,NJ(),(B),(B),F(抽象),α,W,F,F(在图理论中)。(最近的二进制代码工作揭示了一个未解决的问题),例如:(a:在某些图中,256nDn极大团,\256n鼎极大)α(α类型问题)〉12Ⅰf(N.J.A.斯隆)] (weighted cliques, and \256nding good colorings.)5 2009 1 720 3055 t 11 S1 f (_ __________)1 600 1 720 6880 t 9 R f ( paper appeared in)3 664(* This)1 296 2 720 7010 t 9 I f (Graph Theory Notes of New York)5 1210 1 1703 7010 t 9 R f (, Vol.)1 204 1 2913 7010 t 9 B f (18)3140 7010 w 9 R f (, 1989, pp. 11-20.)3 641 1 3230 7010 t cleartomark showpage saveobj restore end %%PageBoundingBox: 61 76 514 711 %%EndPage: 13 13 %%Trailer DpostDict begin done end %%Pages: 13 %%DocumentFonts: Times-Bold Times-Italic Times-Roman Symbol Times-Roman