%啊!PS-ADOBE-2.0%%Curror:DVIPPSK5.58F版权1986, 1994激进眼软件:%.VI %%%页面:145 %%PosialOrth: Axt%%DeungBox:0、0、612、792、%%EntEffy%%DVIPSRealdLoad:DVIPS-DIV.DVI %DVIPS参数:DPI=300,压缩,comments removed %DVIPSSource: TeX output 1998.05.19:1415 %%BeginProcSet: texc.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 /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]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 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known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /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 {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{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 /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}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 %%BeginProcSet: special.pro TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 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/li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 40258431 52099146 1000 300 300 (self.dvi) @start /Fa 1 60 df<126012F0A212701210A31220A21240A2040B7D830B>59 D E /Fb 2 51 df<120C121C12EC120CAFEAFFC00A137D9211>49 D<121FEA60C01360EAF07013301260EA0070A2136013C012011380EA02005AEA08101210 EA2020EA7FE012FF0C137E9211>I E /Fc 1 117 df<1204120CA35AEAFF80EA1800A25A A45A1261A212621264123809127F910D>116 D E /Fd 1 49 df48 D E /Fe 2 89 df<903801F80890380E0618903838013890386000F04848 1370485A48C71230481420120E5A123C15005AA35AA2EC7FF0EC07801500A31270140E12 3012386C131E6C136438070184D800FEC7FC1D1E7E9C21>71 D<3A01FFC0FF803A001E00 3C00011C13306D13205D010F5B6D48C7FC1482EB038414CCEB01D814F05C130080EB0170 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Fv(7.1)46 b(F)l(amily)16 b(2)407 1698 y Fs(I)421 1691 y Fv(:)k(Binary)c(self-dual)h(co)q(des)28 b(.)22 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.) h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(47)128 1748 y(7.2)46 b(F)l(amily)16 b(2)407 1755 y Fs(I)q(I)435 1748 y Fv(:)k(Doubly-ev)o(en)c(binary)g(self-dual)h(co)q(des)41 b(.)22 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.) h(.)58 b(48)128 1804 y(7.3)46 b(F)l(amily)16 b(3:)j(T)l(ernary)c(co)q (des)25 b(.)d(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(49)128 1860 y(7.4)46 b(F)l(amily)16 b(4)407 1844 y Fs(H)435 1860 y Fv(:)k(Self-dual)d(co)q(des)f(o)o(v)o(er)e Ft(F)900 1867 y Fs(4)936 1860 y Fv(with)i(Hermitian)g(inner)g(pro)q (duct)29 b(.)23 b(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(50)128 1917 y(7.5)46 b(F)l(amily)16 b(4)407 1900 y Fs(E)432 1917 y Fv(:)k(Self-dual)d(co)q(des)f(o)o(v)o(er)e Ft(F)897 1924 y Fs(4)934 1917 y Fv(with)h(Euclidean)i(inner)g(pro)q(duct)37 b(.)23 b(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(52)128 1973 y(7.6)46 b(F)l(amily)16 b(4)407 1954 y Fs(H+)407 1987 y(I)462 1973 y Fv(:)k(Additiv)o(e)d(self-dual)g(co)q(des)e(o)o(v)o(er)g Ft(F)1106 1980 y Fs(4)1142 1973 y Fv(using)h(trace)f(inner)h(pro)q (duct)g(.)23 b(.)f(.)g(.)h(.)58 b(54)128 2030 y(7.7)46 b(F)l(amily)16 b(4)407 2011 y Fs(H+)407 2043 y(I)q(I)462 2030 y Fv(:)k(Additiv)o(e)d(ev)o(en)e(self-dual)i(co)q(des)f(o)o(v)o (er)e Ft(F)1209 2037 y Fs(4)1245 2030 y Fv(using)i(trace)f(inner)h(pro) q(duct)j(.)k(.)58 b(56)128 2086 y(7.8)46 b(F)l(amily)16 b Fq(q)406 2070 y Fs(H)434 2086 y Fv(:)k(Co)q(des)15 b(o)o(v)o(er)g Ft(F)724 2093 y Fr(q)744 2086 y Fv(,)g Fq(q)i Fv(a)e(square,)f(with)i(Hermitian)g(inner)g(pro)q(duct)34 b(.)22 b(.)h(.)f(.)g(.)h(.)58 b(57)128 2143 y(7.9)46 b(F)l(amily)16 b Fq(q)406 2126 y Fs(E)432 2143 y Fv(:)j(Co)q(des)d(o)o (v)o(er)e Ft(F)721 2150 y Fr(q)757 2143 y Fv(with)h(Euclidean)i(inner)g (pro)q(duct)37 b(.)23 b(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.) 58 b(58)128 2199 y(7.10)23 b(F)l(amily)16 b(4)407 2183 y Fp(Z)407 2212 y Fs(I)428 2199 y Fv(:)k(Self-dual)d(co)q(des)e(o)o(v)o (er)g Ft(Z)899 2206 y Fs(4)955 2199 y Fv(.)23 b(.)f(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.) 58 b(61)128 2256 y(7.11)23 b(F)l(amily)16 b(4)407 2239 y Fp(Z)407 2268 y Fs(I)q(I)435 2256 y Fv(:)k(T)o(yp)q(e)15 b(I)q(I)h(self-dual)h(co)q(des)f(o)o(v)o(er)e Ft(Z)1066 2263 y Fs(4)1097 2256 y Fv(.)22 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(65)128 2312 y(7.12)23 b(F)l(amily)16 b Fq(m)424 2296 y Fp(Z)445 2312 y Fv(:)k(Self-dual)d(co)q(des)f(o)o(v)o(er)e Ft(Z)917 2319 y Fr(m)991 2312 y Fv(.)22 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.) g(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(67)60 2414 y Fu(8)42 b(W)l(eigh)o(t)18 b(en)o(umerators)e(of)h (maximally)i(self-orthogonal)g(co)q(des)467 b(67)60 2516 y(9)42 b(Upp)q(er)18 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b(.)22 b(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.) f(.)g(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(91)128 741 y(12.2)23 b(F)l(amily)16 b(3:)j(T)l(ernary)c(co)q(des)25 b(.)d(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(94)128 797 y(12.3)23 b(F)l(amily)16 b(4)407 781 y Fs(H)435 797 y Fv(:)k(Hermitian)c(self-dual)h(co)q(des)e(o)o(v)o(er)g Ft(F)1109 804 y Fs(4)1167 797 y Fv(.)23 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(95)128 854 y(12.4)23 b(F)l(amily)16 b(4)407 837 y Fs(H+)462 854 y Fv(:)k(Additiv)o(e)d(self-dual)g(co)q(des)e(o)o(v)o(er)g Ft(F)1106 861 y Fs(4)1167 854 y Fv(.)23 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)58 b(95)128 910 y(12.5)23 b(F)l(amily)16 b(4)407 894 y Fp(Z)428 910 y Fv(:)k(Self-dual)d(co)q(des)e(o)o(v)o(er)g Ft(Z)899 917 y Fs(4)955 910 y Fv(.)23 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f 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Fs(5)741 1020 y Fq(S)j Fv(of)c(a)h(self-orthogonal)h(binary)f (co)q(de)h Fq(C)i Fv(is)573 1167 y Fq(S)d Fv(=)664 1081 y Fm(8)664 1118 y(<)664 1193 y(:)722 1124 y Fq(C)758 1107 y Fl(?)755 1135 Y fs(0)797 1124 y fn(n)10 b Fq(c)866μy y fl(?)937 1124 Y FV(IF)16 B Fq(C)H Fv(IS)F(单EV)O(EN)722π1210 Y FQ(C)758 1194 Y FL(?)937 1210 y Fv(if)g Fq(C)h Fv(is)f(doubly-ev)o(en)1335 1081 y Fm(9)1335 1118 y(=)1335 1193 y(;)1394 1167 y Fq(:)60 1314 y Fv(The)f(w)o(eigh)o(t)g(en)o (umerator)g(of)g(the)g(shado)o(w)f(of)h Fq(C)j Fv(will)f(usually)g(b)q (e)e(denoted)h(b)o(y)f Fq(S)1498 1321 y Fr(C)1528 1314 y Fv(\()p Fq(x;)8 b(y)r Fv(\).)60 1463 y Fu(Examples.)45 b Fv(\(i\))16 b(If)g Fq(C)j Fv(is)d(the)g(rep)q(etition)h(co)q(de)g Fn(f)p Fv(0)987 1446 y Fr(n)1010 1463 y Fq(;)8 b Fv(1)1054 1446 y Fr(n)1076 1463 y Fn(g)16 b Fv(of)f(ev)o(en)h(length)h Fq(n)p Fv(,)f(then)g(if)g Fq(n)e Fn(\021)h Fv(0)g(\()r(mo)q(d)j(4\),)60 1547 y Fq(S)g Fv(=)e Fq(C)193 1531 y Fl(?)238 1547 y Fv(=)i(all)g(ev)o(en)f(w)o(eigh)o(t)g(v)o(ectors,)f(but)h(if)h Fq(n)e Fn(\021)g Fv(2)h(\()q(mo)q(d)h(4\),)f Fq(S)h Fv(=)f(all)h(o)q (dd)g(w)o(eigh)o(t)e(v)o(ectors.)25 b(\(ii\))18 b(If)60 1632 y Fq(C)e Fv(=)f Fq(i)175 1639 y Fs(2)205 1632 y Fn(\010)10 b Fq(i)266 1639 y Fs(2)296 1632 y Fn(\010)h(\001)d(\001)g (\001)h(\010)i Fq(i)468 1639 y Fs(2)503 1632 y Fv(then)16 b Fq(S)i Fv(is)f(the)f(translate)f(of)h Fq(C)i Fv(b)o(y)e(1010)8 b Fq(:)g(:)g(:)t Fv(10.)21 b(\(iii\))d(Let)e Fq(C)i Fv(b)q(e)f(the)f ([22)p Fq(;)8 b Fv(11)p Fq(;)g Fv(6)o(])60 1717 y(shorter)16 b(Gola)o(y)g(co)q(de)i Fq(g)478 1724 y Fs(22)515 1717 y Fv(,)e(obtained)i(b)o(y)f(\\subtracting")f(\(see)h(Section)g(11.3\))f Fq(i)1470 1724 y Fs(2)1506 1717 y Fv(from)g Fq(g)1637 1724 y Fs(24)1674 1717 y Fv(,)h(so)f(that)g Fq(g)1883 1724 y Fs(22)60 1801 y Fv(consists)i(of)e(all)j(w)o(ords)d(of)h Fq(g)557 1808 y Fs(24)611 1801 y Fv(that)g(b)q(egin)h(00)f(or)g(11,)g (with)g(these)h(t)o(w)o(o)e(co)q(ordinates)h(deleted.)28 b(Then)18 b Fq(S)60 1886 y Fv(consists)d(of)g(the)g(remaining)i(w)o (ords)d(of)h Fq(g)774 1893 y Fs(24)826 1886 y Fv(with)h(the)f(same)g(t) o(w)o(o)f(co)q(ordinates)h(deleted.)60 2008 y Fu(Theorem)i(6.)22 b Fv([69)o(])16 b Fw(The)g(shadow)h Fq(S)i Fw(has)d(the)h(fol)r(lowing) f(pr)n(op)n(erties:)131 2093 y(\(i\))f Fq(S)k Fw(is)c(the)i(set)f(of)g (\\p)n(arity)h(ve)n(ctors")f(for)h Fq(C)s Fw(;)f(that)h(is,)450 2223 y Fq(S)e Fv(=)e Fn(f)p Fq(u)g Fn(2)g Ft(F)670 2204 y Fr(n)670 2234 y Fs(2)707 2223 y Fv(:)f(\()p Fq(u;)c(v)r Fv(\))j Fn(\021)903 2192 y Fv(1)p 903 2213 23 2 v 903 2254 a(2)931 2223 y Fq(w)q(t)p Fv(\()p Fq(v)r Fv(\))h(mo)q(d)29 b(2)46 b Fw(for)17 b(al)r(l)32 b Fq(v)14 b Fn(2)f Fq(C)s Fn(g)309 b Fv(\(53\))131 2354 y Fw(\(ii\))15 b Fq(S)k Fw(is)c(a)i(c)n(oset)f(of)g Fq(C)549 2337 y Fl(?)131 2438 y Fw(\(iii\))633 2523 y Fq(S)661 2530 y Fr(C)691 2523 y Fv(\()p Fq(x;)8 b(y)r Fv(\))j(=)881 2492 y(1)p 862 2512 61 2 v 862 2554 a Fn(j)p Fq(C)s Fn(j)928 2523 y Fq(W)971 2530 y Fr(C)1000 2523 y Fv(\()p Fq(x)f Fv(+)h Fq(y)r(;)d(i)p Fv(\()p Fq(x)g Fn(\000)i Fq(y)r Fv(\)\))16 b Fq(:)492 b Fv(\(54\))p 60 2596 744 2 v 112 2623 a Fk(5)129 2639 y Fj(A)14 b(somewhat)h(more)f(general)i(de\014nition)h(of)d(shado) o(w)h(has)g(b)q(een)g(prop)q(osed)h(in)f([35],)f(but)h(since)h(it)e (fails)i(to)e(p)q(ossess)i(the)60 2685 y(crucial)f(prop)q(erties)f (\(i\))g(and)f(\(iii\))i(of)e(Theorem)g(6)g(w)o(e)g(shall)h(not)g (discuss)g(it)g(here.)967 2853 y Fv(25)p eop %%Page: 26 30 26 29 bop 60 74 a Fu(Pro)q(of.)45 b Fv(If)10 b Fq(C)j Fv(is)e(doubly-ev)o(en)h(then)e(\(i\))g(and)g(\(ii\))h(are)f (immediate,)i(and)e(\(iii\))h(follo)o(ws)g(from)e(the)h(MacWilliams)60 159 y(transform)j(and)h(the)g(fact)g(that)f(the)h(w)o(eigh)o(ts)g(are)g (divisible)j(b)o(y)d(4.)19 b(Supp)q(ose)c Fq(C)i Fv(is)e(singly-ev)o (en,(G)(双EV)o(En)q(de),f(f)g(f)fq(c)766,798,798,244,y,Fv(=)e Fq(c)g fn(n)d fq(c),244,y,f,(Fv),(b),(q)α,y,fn(α)y,fn(α)y,b,Fq(c)i fn(α)e fq(c)αy y fl(?)g(LEG)G Fq(c)1840 166 y fs(0)1873 159 y Fv(b)q(e)60×244 y1063 374 y fn(022)g fq(c)1147 355 y fl(?)(1144)385 y fs(0)1191 y y fq(:)635 b Fv(\(55))60 504 y(17)b(\014rST)f(and)h(最后)f(夹杂物)j(HA)o(v)o(e)d(index)i(2)e(so)g fq(c)1011 y y fl(?)1008 516 Y fs(0)1055 504 y Fv(=)f fq(c)1141 488 y fl(?)1181 504 y fn(()d Fv(\)p fq(a)eωfv(+)i fq(c)1358 488 y fl(?)1387 504 y Fv(\),)k(sa)o(y)l(,)g(where)h (\()p Fq(a;)8 b(u)p Fv(\))13 b(=)i(0)i(for)60 589 y Fq(u)c Fn(2)g Fq(C)175 596 y Fs(0)194 589 y Fv(,)h(\()p Fq(a;)8 b(v)r Fv(\))j(=)i(1)h(for)g Fq(v)h Fn(2)e Fq(C)604 596 y Fs(1)623 589 y Fv(.)19 b(Th)o(us)c Fq(S)g Fv(=)e Fq(C)897 573 y Fl(?)894 601 y Fs(0)935 589 y Fn(n)c Fq(C)1000 596 y Fs(0)1032 589 y Fv(=)k Fq(a)c Fv(+)g Fq(C)1190 596 y Fs(0)1224 589 y Fv(has)15 b(the)f(prop)q(erties)i(stated)e(in)h (\(i\))g(and)60 674 y(\(ii\).)20 b(Also,)377 804 y Fq(W)420 811 y Fr(C)445 816 y Fk(0)464 804 y Fv(\()p Fq(x;)8 b(y)r Fv(\))40 b(=)693 773 y(1)p 693 794 23 2 v 693 835 a(2)728 804 y Fn(f)p Fq(W)794 811 y Fr(C)824 804 y Fv(\()p Fq(x;)8 b(y)r Fv(\))g(+)j Fq(W)1028 811 y Fr(C)1057 804 y Fv(\()p Fq(x;)d(iy)r Fv(\))p Fn(g)21 b Fq(;)366 912 y(W)409 925 y Fr(C)436 914 y Fh(?)434 936 y Fk(0)464 912 y Fv(\()p Fq(x;)8 b(y)r Fv(\))40 b(=)712 881 y(1)p 693 901 61 2 v 693 943 a Fn(j)p Fq(C)s Fn(j)767 912 y(f)o Fq(W)832 919 y Fr(C)862 912 y Fv(\()p Fq(x)10 b Fv(+)g Fq(y)r(;)e(x)h Fn(\000)i Fq(y)r Fv(\))e(+)i Fq(W)1227 919 y Fr(C)1256 912 y Fv(\()p Fq(x)f Fv(+)h Fq(y)r(;)d(i)p Fv(\()p Fq(x)g Fn(\000)j Fq(y)r Fv(\))p Fn(g)22 b Fq(;)60 1042 y Fv(so)447 1127 y Fq(S)475 1134 y Fr(C)504 1127 y Fv(\()p Fq(x;)8 b(y)r Fv(\))j(=)i Fq(W)713 1140 y Fr(C)740 1129 y Fh(?)(738)1151 y FK(0)779 1127 y fn(000)d fq(w)867 1138 y Fr(c)894 1129 1129 y h(?)936 1127 y Fv(=)1008 1096 y(1)p 988 1117 V 988 1158 a Fn(j)p Fq(C)s Fn(j)1054 1127 y Fq(W)1097 1134 y Fr(C)1127 1127 y Fv(\()p Fq(x)g Fv(+)g Fq(y)r(;)e(i)p Fv(\()p Fq(x)h Fn(\000)h Fq(y)r Fv(\)\))15 b Fq(:)p 1518 1125 16 16 v 131 1239 a Fv(If)g Fq(C)k Fv(is)d(a)f(singly-ev)o(en)j (self-dual)f(co)q(de)f(with)g(doubly-ev)o(en)h(sub)q(co)q(de)g Fq(C)1399 1246 y Fs(0)1419 1239 y Fv(,)e(then)h Fq(C)1587 1222 y Fl(?)1584 1250 y Fs(0)1632 1239 y Fv(is)g(the)g(union)g(of)60 1323 y(four)f(translates)g(of)f Fq(C)447 1330 y Fs(0)466 1323 y Fv(,)h(sa)o(y)g Fq(C)606 1330 y Fs(0)625 1323 y Fv(,)g Fq(C)686 1330 y Fs(1)705 1323 y Fv(,)g Fq(C)766 1330 y Fs(2)785 1323 y Fv(,)g Fq(C)846 1330 y Fs(3)865 1323 y Fv(,)g(with)695 1454 y Fq(C)g Fv(=)e Fq(C)824 1461 y Fs(0)853 1454 y Fn([)e Fq(C)927 1461 y Fs(2)946 1454 y Fq(;)53 b(S)15 b Fv(=)e Fq(C)1136 1461 y Fs(1)1165 1454 y Fn([)d Fq(C)1238 1461 y Fs(3)1273 1454 y Fq(:)553 b Fv(\(56\))131 1584 y(When)14 b Fq(n)h Fv(is)f(a)g(m)o(ultiple)i(of)e (8)g(then)h Fq(C)789 1568 y Fl(0)813 1584 y Fv(=)e Fq(C)894 1591 y Fs(0)921 1584 y Fn([)c Fq(C)993 1591 y Fs(1)1026 1584 y Fv(and)15 b Fq(C)1150 1568 y Fl(00)1183 1584 y Fv(=)e Fq(C)1264 1591 y Fs(0)1292 1584 y Fn([)8 b Fq(C)1363 1591 y Fs(3)1397 1584 y Fv(are)13 b(b)q(oth)i(T)o(yp)q(e)f(I)q(I)i(co)q (des)e(\(in)60 1669 y(the)i(notation)f(of)h(Chapter)g(xx)f(\(Pless\),)h Fq(C)815 1652 y Fl(0)842 1669 y Fv(and)g Fq(C)967 1652 y Fl(00)1004 1669 y Fv(are)g Fw(neighb)n(ors)f Fv(of)g Fq(C)s Fv(\).)21 b(If)c Fq(C)h Fv(has)e(a)f(w)o(eigh)o(t)h(2)g(w)o(ord) 60 1753 y(then)g Fq(C)200 1737 y Fl(0)226 1753 y Fv(and)f Fq(C)350 1737 y Fl(00)387 1753 y Fv(are)f(equiv)m(alen)o(t.)131 1838 y(Similar)20 b(de\014nitions)i(for)c(the)i(shado)o(w)e(can)i(b)q (e)g(giv)o(en)g(in)g(the)f(other)g(t)o(w)o(o)f(cases)i(men)o(tioned.)33 b(If)19 b Fq(C)60 1923 y Fv(is)e(an)g(additiv)o(e)h(trace-Hermitian)g (self-orthogonal)f(co)q(de)h(o)o(v)o(er)e Ft(F)1219 1930 y Fs(4)1240 1923 y Fv(,)h(let)g Fq(C)1370 1930 y Fs(0)1406 1923 y Fv(b)q(e)h(the)f(sub)q(co)q(de)h(with)g(ev)o(en)60 2008 y(Hamming)c(w)o(eigh)o(ts,)g(and)g(secondly)l(,)i(if)e Fq(C)j Fv(is)e(a)f(self-orthogonal)g(co)q(de)h(o)o(v)o(er)e Ft(Z)1429 2015 y Fr(m)1473 2008 y Fv(\()p Fq(m)h Fv(ev)o(en\))g(let)h Fq(C)1763 2015 y Fs(0)1796 2008 y Fv(b)q(e)g(the)60 2092 y(sub)q(co)q(de)h(with)g(Euclidean)h(norms)e(divisible)j(b)o(y)d(2)p Fq(m)p Fv(.)20B(in)c(b)q(Of)f(G)G(Syo)O(W)G(IS)H(De 014Nd)G(B)O(Y)685 Fy(2237)y Fq(s)f Fv(=)776 2151 y y FM(8)776 2188 y(<)776 2263 y(:)834 2194 y Fq(c)870 2177 y y(?)867 2205 y fs(0)909 y y fn(n)10 b Fq(c)975 y 2201 y fs(0)1036 2194 y v fv(if)31 b Fq(c)31 b fn(α)p fv(=)e fq(c)y y fs(α)y fq(c)y y fl(?)(1036)2280 y Fq(c)15 b Fv(=)e Fq(c)1222 2287 y fs(0)1256 2390 y fq(:)60 2390 y fv(if)h fq(c)i fv(is)e(自对偶)h(from)e(族)h(4)4 y fs(h +)y fy y(f)y(f)G(f)o(t)h(组)f fq(c)y y fl(?)1259 1259 y y FS(0)1348 2390 y f v(IS)f fq(z)s fv(\(2))7 b fn(\002)g fq(z)s fv(\(α))y y(if)b b Fq(c)i fv(is)e(自对偶)h(from)e fq(m)y fy(z)αy y fv(())g fq(c)y y fl(?)1227 2401 Y fs(0)845 2486 y Fs(0)877 2474 y Fq(=C)933 2481 y Fs(0)968 2474 y Fv(is)g(isomorphic)g(to)f Fq(Z)s Fv(\(2\))10 b Fn(\002)g Fq(Z)s Fv(\(2\))15 b(if)h Fq(n)g Fv(is)g(ev)o(en)g(and)g(to)60 2559 y Fq(Z)s Fv(\(4\))f(if)g Fq(n)h Fv(is)f(o)q(dd.)131 2644 y(There)g(are)g(analogues)g(of)g(Theorem)g(6.)967 2853 y(26)p eop %%Page: 27 31 27 30 bop 60 74 a Fu(Theorem)17 b(7.)22 b Fw(L)n(et)16 b Fq(C)j Fw(b)n(e)d(a)g(self-ortho)n(gonal)g(additive)h(c)n(o)n(de)e (over)i Ft(F)1270 81 y Fs(4)1291 74 y Fw(,)f(with)h(shadow)g Fq(S)s Fw(.)131 159 y(\(i\))31 b Fq(S)15 b Fv(=)e Fn(f)p Fq(u)g Fn(2)g Ft(F)434 143 y Fr(n)434 171 y Fs(4)471 159 y Fv(:)f(\()p Fq(u;)c(v)r Fv(\))j Fn(\021)i Fq(w)q(t)p Fv(\()p Fq(v)r Fv(\))j(\()s(mo)q(d)i(2\))46 b Fw(for)16 b(al)r(l)g Fq(v)f Fn(2)d Fq(C)s Fn(g)131 244 y Fw(\(ii\))j Fq(S)k Fw(is)c(a)i(c)n(oset)f(of)g Fq(C)549 227 y Fl(?)131 328 y Fw(\(iii\))647 413 y Fq(S)675 420 y Fr(C)705 413 y Fv(\()p Fq(x;)8 b(y)r Fv(\))j(=)895 382 y(1)p 876 403 61 2 v 876 444 a Fn(j)p Fq(C)s Fn(j)942 413 y Fq(W)985 420 y Fr(C)1014 413 y Fv(\()p Fq(x)f Fv(+)h(3)p Fq(y)r(;)d(y)j Fn(\000)f Fq(x)p Fv(\))16 b Fq(;)423 539 y(sw)q(e)499 546 y Fr(S)524 539 y Fv(\()p Fq(x;)8 b(y)r(;)g(z)r Fv(\))j(=)758 509 y(1)p 739 529 V 739 570 a Fn(j)p Fq(C)s Fn(j)821 539 y Fq(sw)q(e)897 546 y Fr(C)927 539 y Fv(\()p Fq(x)f Fv(+)g Fq(y)i Fv(+)f(2)p Fq(z)r(;)d Fn(\000)p Fq(x)h Fn(\000)i Fq(y)h Fv(+)e(2)p Fq(z)r(;)e(y)j Fn(\000)g Fq(x)p Fv(\))102 665 y Fq(cw)q(e)177 672 y Fr(S)202 665 y Fv(\()p Fq(x;)d(y)r(;)g(z)r(;)g(t)p Fv(\))i(=)472 635 y(1)p 453 655 V 453 697 a Fn(j)p Fq(C)s Fn(j)519 665 y Fq(cw)q(e)594 672 y Fr(C)623 665 y Fv(\()p Fq(x)g Fv(+)g Fq(y)i Fv(+)f Fq(z)h Fv(+)e Fq(t;)e Fn(\000)p Fq(x)i Fn(\000)h Fq(y)h Fv(+)e Fq(z)i Fv(+)f Fq(t;)d Fn(\000)p Fq(x)i Fv(+)g Fq(y)i Fn(\000)f Fq(z)h Fv(+)f Fq(t;)d Fn(\000)p Fq(x)i Fv(+)g Fq(y)i Fv(+)f Fq(z)h Fn(\000)e Fq(t)p Fv(\))17 b Fq(:)60 811 y Fu(Remark.)44 b Fv(It)17 b(follo)o(ws)g(from)g(Theorem)f(7)h(that)f(there)h(is)h(a)f(co)q(de)g (equiv)m(alen)o(t)i(to)d Fq(C)k Fv(that)c(has)h(1)1792 795 y Fr(n)1831 811 y Fn(2)f Fq(S)s Fv(.)60 896 y(F)l(or)f(the)g(n)o (um)o(b)q(er)g(of)g(v)o(ectors)g(of)f(w)o(eigh)o(t)h Fq(n)h Fv(in)g Fq(S)h Fv(is)703 1011 y Fq(S)731 1018 y Fr(C)760 1011 y Fv(\(0)p Fq(;)8 b Fv(1\))j(=)946 980 y(1)p 927 1001 V 927 1042 a Fn(j)p Fq(C)s Fn(j)993 1011 y Fq(W)1036 1018 y Fr(C)1065 1011 y Fv(\(3)p Fq(;)d Fv(1\))j Fq(>)i Fv(0)i Fq(:)60 1126 y Fv(All)h(v)o(ectors)f(of)g(full)h(w)o (eigh)o(t)f(are)g(equiv)m(alen)o(t.)60 1236 y Fu(Theorem)i(8.)22 b Fw(L)n(et)16 b Fq(C)j Fw(b)n(e)d(a)g(self-ortho)n(gonal)g(line)n(ar)f (c)n(o)n(de)h(over)h Ft(Z)1234 1243 y Fs(4)1251 1236 y Fw(,)f(with)h(shadow)g Fq(S)s Fw(.)131 1320 y(\(i\))e Fq(S)g Fv(=)e Fn(f)p Fq(u)f Fn(2)h Ft(Z)424 1304 y Fr(n)424 1332 y Fs(4)457 1320 y Fv(:)f(\()p Fq(u;(a)(v)(b)Fw(for)(b),(b),(b),(b),(b),f(q)d fn(v)d fn(f)f fq(c)r fn(g)αfy(f(ii))i fq(s)k fw(is)c(a)i(c)n(OSET)f(of)g fq(c)y y fl(?)c(v)r fv(\)j j fn(021)653,1303 y fs(1)p 653 1310 18 18 V131 1490 y Fw(\(iii\))f Fq(sw)q(e)302 1497 y Fr(S)327 1490 y Fv(\()p Fq(x;)8 b(y)r(;)g(z)r Fv(\))j(=)557 1472 y Fs(1)p 542 1479 48 2 v 542 1506 a Fl(j)p Fr(C)r Fl(j)594 1490 y Fq(sw)q(e)670 1497 y Fr(C)700 1490 y Fv(\()p Fq(x)e 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74 a Fx(6.)50 b(In)n(v)l(arian)n(t)25 b(theory)60 178 y Fo(6.1.)41 b(An)19 b(in)n(tro)r(duction)f(to)h(in)n(v)m(arian)n (t)f(theory)131 292 y Fv(If)e Fq(C)i Fv(is)f(self-dual)g(then)g(its)f (w)o(eigh)o(t)g(en)o(umerator)f(m)o(ust)g(b)q(e)i(unc)o(hanged)g(b)o(y) e(the)h(appropriate)g(trans-)60 376 y(formation)k(from)f(Theorem)i(4.) 35 b(As)21 b(w)o(e)f(will)i(see,)g(this)f(imp)q(oses)g(strong)e (restrictions)i(on)g(the)f(w)o(eigh)o(t)60 461 y(en)o(umerator.)131 546 y(W)l(e)d(b)q(egin)h(b)o(y)f(discussing)i(the)e(particular)h(case)f (of)f(the)h(w)o(eigh)o(t)g(en)o(umerator)f Fq(W)6 b Fv(\()p Fq(x;)i(y)r Fv(\))16 b(of)h(a)f(binary)60 630 y(doubly-ev)o(en)h (self-dual)g(co)q(de)e Fq(C)s Fv(.)20 b(Since)d Fq(C)h Fv(is)d(self-dual,)i(Theorem)e(4)g(implies)650 758 y Fq(W)6 b Fv(\()p Fq(x;)i(y)r Fv(\))40 b(=)957 727 y(1)p 928 747 82 2 v 928 790 a(2)951 777 y Fr(n=)p Fs(2)1014 758 y Fq(W)6 b Fv(\()p Fq(x)k Fv(+)h Fq(y)r(;)d(x)h Fn(\000)i Fq(y)r Fv(\))846 868 y(=)42 b Fq(W)980 809 y Fm(\022)1015 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Fq(f)f Fn(\000)12 b Fq(g)21 b Fv(and)e(the)g(pro)q(duct)g Fq(f)5 b(g)r Fv(.)30 b(Also)19 b(it)h(is)f(enough)g(to)f(study)h(the)g Fw(homo)n(gene)n(ous)60 842 y Fv(in)o(v)m(arian)o(ts)d(\(in)f(whic)o(h) h(all)h(terms)d(ha)o(v)o(e)h(the)g(same)g(degree\).)131 927 y(So)i(the)h(righ)o(t)g(question)g(to)g(ask)f(is:)26 b(ho)o(w)17 b(man)o(y)h(linearly)h(indep)q(enden)o(t,)i(homogeneous)c (in)o(v)m(arian)o(ts)60 1012 y(are)e(there)g(of)g(eac)o(h)g(degree)h Fq(d)p Fv(?)k(Let's)15 b(call)h(this)g(n)o(um)o(b)q(er)g Fq(a)1087 1019 y Fr(d)1107 1012 y Fv(.)131 1097 y(A)f(con)o(v)o(enien)o (t)i(w)o(a)o(y)e(to)g(handle)i(the)f(n)o(um)o(b)q(ers)g Fq(a)989 1104 y Fs(0)1008 1097 y Fq(;)8 b(a)1053 1104 y Fs(1)1072 1097 y Fq(;)g(a)1117 1104 y Fs(2)1136 1097 y Fq(;)g(:)g(:)g(:)13 b Fv(is)k(b)o(y)e(com)o(bining)i(them)f(in)o(to)g (a)f(p)q(o)o(w)o(er)60 1181 y(series)h(or)f(generating)g(function)683 1312 y(\010\()p Fq(\025)p Fv(\))c(=)i Fq(a)862 1319 y Fs(0)892 1312 y Fv(+)d Fq(a)961 1319 y Fs(1)981 1312 y 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b(=)i Fq(a)650 283 y Fs(0)680 276 y Fv(+)11 b Fq(a)750 283 y Fs(1)769 276 y Fq(\025)f Fv(+)g Fq(a)875 283 y Fs(2)895 276 y Fq(\025)922 257 y Fs(2)951 276 y Fv(+)h Fn(\001)d(\001)g(\001)549 373 y Fv(=)42 b(\(1)10 b(+)g Fq(\025)749 354 y Fs(8)779 373 y Fv(+)g Fq(\025)851 354 y Fs(16)898 373 y Fv(+)g Fq(\025)970 354 y Fs(24)1017 373 y Fv(+)h Fn(\001)d(\001)g(\001)n Fv(\)\(1)h(+)h Fq(\025)1256 354 y Fs(24)1303 373 y Fv(+)h Fq(\025)1376 354 y Fs(48)1423 373 y Fv(+)f Fn(\001)e(\001)g(\001)n Fv(\))15 b Fq(:)272 b Fv(\(62\))131 489 y(W)l(e)15 b(can)h(deduce)g(one)g(fact)f (immediately:)22 b Fq(a)919 496 y Fr(d)955 489 y Fv(is)16 b(zero)f(unless)i Fq(d)e Fv(is)h(a)f(m)o(ultiple)i(of)e(8,)g(i.e.)21 b(the)16 b(degree)60 574 y(of)g(a)h(homogeneous)g(in)o(v)m(arian)o(t)g (m)o(ust)f(b)q(e)i(a)e(m)o(ultiple)j(of)d(8.)25 b(\(This)17 b(already)g(pro)o(v)o(es)f(that)g(the)h(length)g(of)60 658 y(a)j(doubly-ev)o(en)h(binary)g(self-dual)g(co)q(de)g(m)o(ust)e(b)q (e)i(a)f(m)o(ultiple)i(of)d(8.\))34 b(But)20 b(w)o(e)f(can)i(sa)o(y)e (more.)34 b(The)60 743 y(righ)o(t-hand)17 b(side)h(of)e(\(62\))g(is)h (exactly)g(what)f(w)o(e)h(w)o(ould)g(\014nd)g(if)h(there)e(w)o(ere)h(t) o(w)o(o)e(\\basic")i(in)o(v)m(arian)o(ts,)g(of)60 828 y(degrees)e(8)g(and)h(24,)e(suc)o(h)h(that)g(all)h(in)o(v)m(arian)o(ts) g(are)f(formed)g(from)f(sums)h(and)h(pro)q(ducts)f(of)g(them.)131 912 y(This)i(is)g(b)q(ecause)h(t)o(w)o(o)e(in)o(v)m(arian)o(ts,)h Fq(\022)q Fv(,)h(of)e(degree)h(8,)g(and)g Fq(\036)p Fv(,)g(of)f(degree) i(24,)e(w)o(ould)h(giv)o(e)g(rise)h(to)e(the)60 997 y(follo)o(wing)g (in)o(v)m(arian)o(ts.)p 630 1012 720 2 v 651 1051 a(Degree)f(d)43 b(In)o(v)m(arian)o(ts)f(Num)o(b)q(er)16 b Fq(a)1309 1058 y Fr(d)p 630 1070 V 728 1109 a Fv(0)206 b(1)228 b(1)728 1166 y(8)206 b Fq(\022)230 b Fv(1)716 1222 y(16)185 b Fq(\022)969 1206 y Fs(2)1208 1222 y Fv(1)716 1279 y(24)161 b Fq(\022)945 1262 y Fs(3)966 1279 y Fq(;)8 b(\036)194 b Fv(2)716 1335 y(32)150 b Fq(\022)934 1319 y Fs(4)954 1335 y Fq(;)8 b(\022)q(\036)184 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y Fs(32)1006 1870 y Fv(+)g(2)p Fq(\025)1102 1851 y Fs(40)1149 1870 y Fv(+)f(3)p Fq(\025)1244 1851 y Fs(48)1291 1870 y Fv(+)g Fn(\001)e(\001)g(\001)551 1967 y Fv(=)13 b(\(1)d(+)g Fq(\025)722 1948 y Fs(8)751 1967 y Fv(+)h Fq(\025)824 1948 y Fs(16)871 1967 y Fv(+)f Fq(\025)943 1948 y Fs(24)990 1967 y Fv(+)h Fn(\001)d(\001)g(\001)n Fv(\)\(1)h(+)h Fq(\025)1229 1948 y Fs(24)1276 1967 y Fv(+)h Fq(\025)1349 1948 y Fs(48)1396 1967 y Fv(+)f Fn(\001)e(\001)g (\001)n Fv(\))551 2064 y(=)761 2033 y(1)p 604 2054 338 2 v 604 2095 a(\(1)i Fn(\000)g Fq(\025)727 2082 y Fs(8)746 2095 y Fv(\)\(1)f Fn(\000)i Fq(\025)887 2082 y Fs(24)924 2095 y Fv(\))962 2064 y Fq(;)864 b Fv(\(64\))60 2181 y(whic)o(h)15 b(agrees)e(with)h(\(61\).)k(So)c(if)g(w)o(e)g(can)g (\014nd)g(t)o(w)o(o)f(algebraically)i(indep)q(enden)o(t)i(in)o(v)m (arian)o(ts)d(of)f(degrees)h(8)60 2265 y(and)i(24,)f(w)o(e)g(will)i(ha) o(v)o(e)e(solv)o(ed)h(our)g(problem.)21 b(The)16 b(answ)o(er)f(will)i 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Fv(42)979 2720 y(=)i Fq(x)1053 2702 y Fs(4)1072 2720 y Fq(y)1096 2702 y Fs(4)1116 2720 y Fv(\()p Fq(x)1160 2702 y Fs(4)1189 2720 y Fn(\000)e Fq(y)1259 2702 y Fs(4)1278 2720 y Fv(\))1296 2702 y Fs(4)1839 2720 y Fv(\(65\))967 2853 y(30)p eop %%Page: 31 35 31 34 bop 60 74 a Fv(rather)12 b(than)g Fq(\036)324 81 y Fs(24)373 74 y Fv(itself.)20 b(So)12 b(w)o(e)g(ha)o(v)o(e)g(pro)o(v)o (ed)f(the)i(follo)o(wing)f(theorem,)g(disco)o(v)o(ered)h(b)o(y)f (Gleason)g(in)h(1970.)60 190 y Fu(Theorem)k(11.)22 b Fw(A)o(ny)16 b(invariant)g(of)g(the)h(gr)n(oup)g Fn(G)946 197 y Fs(1)982 190 y Fw(is)f(a)g(p)n(olynomial)g(in)g Fq(\036)1382 197 y Fs(8)1418 190 y Fw(and)g Fq(\036)1533 174 y Fl(0)1533 202 y Fs(24)1570 190 y Fw(.)131 306 y Fv(This)f(also)h(giv)o(es)f(us)g(the)g(solution)h(to)f(our)g(original)h (problem:)60 422 y Fu(Theorem)h(12.)22 b Fw(A)o(ny)17 b(p)n(olynomial)f(which)i(satis\014es)d(Equations)i Fv(\()p Fw(57)p Fv(\))g Fw(and)g Fv(\()p Fw(58)p Fv(\))g Fw(is)f(a)h(p)n (olynomial)g(in)f Fq(\036)1900 429 y Fs(8)60 506 y Fw(and)g Fq(\036)175 490 y Fl(0)175 518 y Fs(24)213 506 y Fw(.)131 622 y Fv(Finally)l(,)23 b(w)o(e)d(ha)o(v)o(e)h(c)o(haracterized)g(the)g (w)o(eigh)o(t)f(en)o(umerator)g(of)g(a)h(doubly-ev)o(en)h(binary)f (self-dual)60 707 y(co)q(de.)60 822 y Fu(Theorem)c(13.)22 b Fv(\(Gleason)14 b([105)o(].\))19 b Fw(The)14 b(weight)h(enumer)n (ator)h(of)f(any)f(T)m(yp)n(e)g(II)g(binary)h(self-dual)f(c)n(o)n(de)g (is)60 907 y(a)i(p)n(olynomial)g(in)g Fq(\036)411 914 y Fs(8)447 907 y Fw(and)g Fq(\036)562 890 y Fl(0)562 918 y Fs(24)600 907 y Fw(.)131 1023 y Fv(Alternativ)o(e)21 b(pro)q(ofs)f(of)g(this)h(astonishing)g(theorem)g(are)f(giv)o(en)h(b)o (y)g(Berlek)m(amp)h(et)e(al.)37 b([14)o(],)21 b(and)60 1107 y(Brou)o(\023)-21 b(e)14 b(and)g(Enguehard)h([33)o(])f(\(see)g (also)g(Assm)o(us)h(and)f(Mattson)f([4)o(]\).)19 b(But)c(the)f(pro)q (of)g(giv)o(en)h(here)g(seems)60 1192 y(to)g(b)q(e)g(the)h(most)e (informativ)o(e,)h(and)g(the)h(easiest)f(to)g(understand)g(and)h(to)e 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Fu(Finding)i(the)f(basic)g(in)o(v)m(arian)o(ts.)45 b Fv(In)15 b(general,)f(\014nding)h(the)e(basic)i(in)o(v)m(arian)o(ts)f (is)g(a)f(simpler)i(problem)60 2017 y(than)21 b(\014nding)i(\010\()p Fq(\025)p Fv(\).)37 b(In)22 b(our)g(applications)h(w)o(e)e(can)g(often) g(use)h(the)g(w)o(eigh)o(t)f(en)o(umerators)f(of)h(co)q(des)60 2101 y(ha)o(ving)c(the)g(appropriate)f(prop)q(erties,)i(as)e(in)h(the)g (ab)q(o)o(v)o(e)f(example,)i(or)e(basic)h(in)o(v)m(arian)o(ts)h(can)e (b)q(e)i(found)60 2186 y(b)o(y)d Fw(aver)n(aging)p Fv(,)f(using)i(the)g (follo)o(wing)f(simple)i(result)f(\(pro)o(v)o(ed)e(in)i(Section)g (6.2\).)60 2302 y Fu(Theorem)h(14.)22 b Fw(If)e Fq(f)5 b Fv(\()p Fu(x)p Fv(\))19 b(=)h Fq(f)5 b Fv(\()p Fq(x)659 2309 y Fs(1)679 2302 y Fq(;)j(:)g(:)g(:)t(;)g(x)806 2309 y Fr(m)839 2302 y Fv(\))20 b Fw(is)f(any)h(p)n(olynomial)g(in)g Fq(m)g Fw(variables,)h(and)f Fn(G)j Fw(is)d(a)g(\014nite)60 2386 y(gr)n(oup)d(of)f Fq(m)10 b Fn(\002)h Fq(m)16 b Fw(matric)n(es,)g(then)p 755 2472 28 2 v 755 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Fv(if)714 2600 y Fq(A)748 2607 y Fr(\013)783 2600 y Fn(\016)10 b Fq(f)5 b Fv(\()p Fu(x)p Fv(\))11 b(=)i Fq(f)5 b Fv(\()p Fq(A)1045 2607 y Fr(\013)1070 2600 y Fu(x)p Fv(\))12 b(=)h Fq(f)5 b Fv(\()p Fu(x)p Fv(\))572 b(\(82\))60 2729 y(for)15 b(all)h Fq(\013)d Fv(=)g(1)p Fq(;)8 b(:)g(:)g(:)t(;)g(g)r Fv(.)967 2853 y(37)p eop %%Page: 38 42 38 41 bop 60 74 a Fu(Example.)45 b Fv(Let)660 159 y Fn(G)687 166 y Fs(4)719 159 y Fv(=)767 100 y Fm(\032\022)836 131 y Fv(1)h(0)836 187 y(0)g(1)935 100 y Fm(\023)973 159 y Fq(;)1016 100 y Fm(\022)1054 131 y Fn(\000)p Fv(1)63 b(0)1072 187 y(0)g Fn(\000)p Fv(1)1223 100 y Fm(\023\033)1308 159 y Fq(;)60 271 y Fv(a)15 b(group)g(of)g(order)f Fq(g)g Fv(=)f(2.)20 b(Then)c Fq(x)682 254 y Fs(2)701 271 y Fv(,)f Fq(xy)i Fv(and)f Fq(y)907 254 y Fs(2)941 271 y Fv(are)f(homogeneous)g (in)o(v)m(arian)o(ts)h(of)f(degree)g(2.)131 355 y(Ev)o(en)g(if)h Fq(f)5 b Fv(\()p Fq(x)p Fv(\))14 b(isn't)h(an)h(in)o(v)m(arian)o(t,)f (its)g(a)o(v)o(erage)f(o)o(v)o(er)h(the)g(group,)p 761 454 28 2 v 761 491 a Fq(f)5 b Fv(\()p Fu(x)p Fv(\))11 b(=)917 460 y(1)p 916 480 24 2 v 916 522 a Fq(g)977 435 y Fr(g)956 450 y Fm(X)952 541 y Fr(\013)p Fs(=1)1028 491 y Fq(A)1062 498 y Fr(\013)1096 491 y Fn(\016)f Fq(f)5 b Fv(\()p Fu(x)p Fv(\))619 b(\(83\))60 626 y(is,)17 b(as)g(w)o(as)e (already)i(stated)f(in)i(Theorem)f(14.)23 b(T)l(o)17 b(pro)o(v)o(e)f(this,)h(observ)o(e)f(that)g(an)o(y)h Fq(A)1578 633 y Fr(\014)1617 626 y Fn(2)e(G)k Fv(transforms)60 711 y(the)c(righ)o(t-hand)h(side)g(of)f(\(83\))f(in)o(to)785 770 y(1)p 784 790 V 784 832 a Fq(g)845 745 y Fr(g)824 760 y Fm(X)820 851 y Fr(\013)p Fs(=1)888 801 y Fv(\()p Fq(A)940 808 y Fr(\013)964 801 y Fq(A)998 808 y Fr(\014)1022 801 y Fv(\))c Fn(\016)g Fq(f)5 b Fv(\()p Fu(x)p Fv(\))14 b Fq(;)638 b Fv(\(84\))60 920 y(b)o(y)15 b(\(81\).)k(As)c Fq(A)338 927 y Fr(\013)378 920 y Fv(runs)g(through)g Fn(G)s Fv(,)g(so)f(do)q(es)i Fq(A)901 927 y Fr(\013)926 920 y Fq(A)960 927 y Fr(\014)983 920 y Fv(,)f(if)h Fq(A)1087 927 y Fr(\014)1126 920 y 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Fq(h)p Fv(\()p Fu(x)p Fv(\),)k Fq(f)5 b Fv(\()p Fu(x)p Fv(\))p Fq(h)p Fv(\()p Fu(x)p Fv(\),)16 b(and)i Fq(cf)5 b Fv(\()p Fu(x)p Fv(\))16 b(\()p Fq(c)60 1547 y Fv(complex\);)f(or)e(in)i(other)f(w)o(ords)f(the)i(set)e (of)h(in)o(v)m(arian)o(ts)h(of)f Fn(G)s Fv(,)f(whic)o(h)i(w)o(e)f (denote)h(b)o(y)f Fn(J)8 b Fv(\()p Fn(G)s Fv(\),)13 b(forms)g(a)h (ring.)131 1632 y(One)d(of)g(the)g(main)h(problems)g(of)e(in)o(v)m (arian)o(t)i(theory)f(is)g(to)g(describ)q(e)i Fn(J)8 b Fv(\()p Fn(G)s Fv(\).)18 b(Since)12 b(the)f(transformations)60 1717 y(in)22 b Fn(G)h Fv(do)e(not)f(c)o(hange)h(the)g(degree)g(of)f(a)h (p)q(olynomial,)i(it)e(is)h(enough)f(to)f(describ)q(e)i(the)f (homogeneous)60 1801 y(in)o(v)m(arian)o(ts)16 b(\(for)e(an)o(y)h(in)o (v)m(arian)o(t)h(is)f(a)g(sum)g(of)g(homogeneous)g(in)o(v)m(arian)o (ts\).)60 1950 y Fu(Basic)i(in)o(v)m(arian)o(ts.)45 b Fv(Our)15 b(goal)f(is)g(to)g(\014nd)h(a)e(\\basis")h(for)g(the)g(in)o (v)m(arian)o(ts)h(of)e Fn(G)s Fv(,)h(that)f(is,)i(a)f(set)g(of)f(basic) 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Fv(\()p Fu(x)p Fv(\))17 b(are)h Fw(algebr)n(aic)n(al)r(ly)h(indep)n(endent)p Fv(.)28 b(A)19 b(fundamen)o(tal)60 2521 y(result)d(from)e(algebra)h(is) h(\(Jacobson)f([154)o(],)f(v)o(ol.)h(3,)g(p.)g(154\):)60 2643 y Fu(Theorem)i(17.)22 b Fw(A)o(ny)16 b Fq(m)10 b Fv(+)g(1)16 b Fw(p)n(olynomials)g(in)g Fq(m)g Fw(variables)g(ar)n(e)g (algebr)n(aic)n(al)r(ly)f(dep)n(endent.)967 2853 y Fv(38)p eop %%Page: 39 43 39 42 bop 131 74 a Fv(The)15 b(\014rst)f(t)o(yp)q(e)h(of)g(basis)g(w)o (e)g(migh)o(t)f(lo)q(ok)h(for)g(is)g(a)g(set)f(of)h Fq(m)f Fv(algebraically)j(indep)q(enden)o(t)g(in)o(v)m(arian)o(ts)60 159 y Fq(f)82 166 y Fs(1)102 159 y Fv(\()p Fu(x)p Fv(\))p Fq(;)8 b(:)g(:)g(:)t(f)268 166 y Fr(m)301 159 y Fv(\()p Fu(x)p Fv(\).)18 b(Suc)o(h)12 b(a)g(set)f(is)h(indeed)i(a)d(\\basis,")h (for)f(b)o(y)h(Theorem)f(17)g(an)o(y)h(in)o(v)m(arian)o(t)g(is)g (algebraically)60 244 y(dep)q(enden)o(t)j(on)e Fq(f)361 251 y Fs(1)381 244 y Fq(;)8 b(:)g(:)g(:)d(;)j(f)505 251 y Fr(m)551 244 y Fv(and)14 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Fv(.)k(F)l(or)15 b(example,)738 1770 y Fq(x)764 1754 y Fs(2)825 1770 y Fv(=)907 1752 y Fs(1)p 907 1759 18 2 v 907 1786 a(4)938 1735 y Fm(\000)957 1736 y Fn(p)p 994 1736 42 2 v 994 1770 a Fq(f)1016 1777 y Fs(1)1047 1770 y Fv(+)1092 1736 y Fn(p)p 1130 1736 V 34 x Fq(f)1152 1777 y Fs(2)1172 1735 y Fm(\001)1191 1745 y Fs(2)1233 1770 y Fq(;)734 1857 y(xy)43 b Fv(=)907 1839 y Fs(1)p 907 1846 18 2 v 907 1872 a(4)930 1857 y Fv(\()p Fq(f)970 1864 y Fs(1)1000 1857 y Fn(\000)10 b Fq(f)1067 1864 y Fs(2)1087 1857 y Fv(\))15 b Fq(;)60 1936 y Fv(and)g(so)g(on.)131 2021 y(Ho)o(w)o(ev)o(er,)21 b(b)o(y)h(far)f(the)g(most)g(con)o(v)o(enien)o (t)h(description)h(of)e(the)h(in)o(v)m(arian)o(ts)g(is)g(a)f(set)h Fq(f)1706 2028 y Fs(1)1726 2021 y Fq(;)8 b(:)g(:)g(:)t(;)g(f)1849 2028 y Fr(l)1883 2021 y Fv(of)60 2106 y(in)o(v)m(arian)o(ts)15 b(with)g(the)f(prop)q(ert)o(y)h(that)e(an)o(y)h(in)o(v)m(arian)o(t)h (is)g(a)g Fw(p)n(olynomial)f Fv(in)h Fq(f)1390 2113 y Fs(1)1410 2106 y Fq(;)8 b(:)g(:)g(:)d(;)j(f)1534 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Fs(1)883 2529 y Fq(f)905 2536 y Fs(2)935 2529 y Fn(\000)c Fq(f)1008 2510 y Fs(2)1003 2540 y(2)1040 2529 y Fv(=)i(0)i Fq(:)60 2641 y Fv(The)g(existence)h(of) e(a)g(p)q(olynomial)i(basis,)f(and)f(a)h(metho)q(d)f(of)g(\014nding)i (it,)f(is)g(giv)o(en)g(b)o(y)f(the)h(next)g(theorem.)967 2853 y(39)p eop %%Page: 40 44 40 43 bop 60 74 a Fu(Theorem)17 b(19.)22 b Fv(\(No)q(ether)e([205)n(];) h([325)o(,)f(p.)f(275].\))31 b Fw(The)19 b(ring)h(of)g(invariants)g(of) g(a)g(\014nite)f(gr)n(oup)i Fn(G)i Fw(of)60 159 y(c)n(omplex)15 b Fq(m)9 b Fn(\002)g Fq(m)16 b Fw(matric)n(es)f(has)h(a)f(p)n (olynomial)h(b)n(asis)e(c)n(onsisting)g(of)i(not)f(mor)n(e)h(than)1577 125 y Fm(\000)1596 139 y Fr(m)p Fs(+)p Fr(g)1618 175 y(m)1672 125 y Fm(\001)1707 159 y Fw(invariants,)60 244 y(of)i(de)n(gr)n(e)n(e)f(not)h(exc)n(e)n(e)n(ding)e Fq(g)r Fw(,)i(wher)n(e)g Fq(g)h Fw(is)f(the)g(or)n(der)g(of)h Fn(G)s Fw(.)25 b(F)m(urthermor)n(e)19 b(this)e(b)n(asis)g(may)i(b)n(e)e (obtaine)n(d)60 328 y(by)f(taking)g(the)h(aver)n(age)f(over)g Fn(G)j Fw(of)e(al)r(l)f(monomials)859 458 y Fq(x)885 438 y Fr(b)900 443 y Fk(1)885 471 y Fs(1)920 458 y Fq(x)946 438 y Fr(b)961 443 y Fk(2)946 471 y Fs(2)988 458 y Fn(\001)8 b(\001)g(\001)d Fq(x)1074 440 y Fr(b)1089 444 y Fi(m)1074 470 y Fr(m)60 588 y Fw(of)16 b(total)h(de)n(gr)n(e)n(e)353 556 y Fm(P)405 588 y Fq(b)425 595 y Fr(i)455 588 y Fw(not)f(exc)n(e)n (e)n(ding)e Fq(g)r Fw(.)60 736 y Fu(Pro)q(of.)45 b Fv(Let)15 b(the)h(group)f Fn(G)i Fv(consist)f(of)f(the)g(transformations)f (\(79\).)k(Supp)q(ose)657 866 y Fq(f)5 b Fv(\()p Fq(x)728 873 y Fs(1)747 866 y Fq(;)j(:)g(:)g(:)d(;)j(x)875 873 y Fr(m)908 866 y Fv(\))k(=)986 826 y Fm(X)1008 913 y Fr(e)1054 866 y Fq(c)1074 873 y Fr(e)1092 866 y Fq(x)1118 846 y Fr(e)1134 851 y Fk(1)1118 879 y Fs(1)1161 866 y Fn(\001)c(\001)g(\001)e Fq(x)1248 848 y Fr(e)1264 852 y Fi(m)1248 878 y Fr(m)1310 866 y Fq(;)60 1001 y(c)80 1008 y Fr(e)115 1001 y Fv(complex,)18 b(is)g(an)o(y)e(in)o(v)m(arian)o (t)i(of)e 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Fs(1)1295 1951 y Fn(\001)8 b(\001)g(\001)e Fq(u)1382 1935 y Fr(e)1398 1939 y Fi(m)1382 1963 y Fr(m)1445 1951 y Fv(in)648 2087 y Fq(P)677 2094 y Fr(e)709 2087 y Fv(=)782 2032 y Fr(g)761 2047 y Fm(X)757 2137 y Fr(\013)p Fs(=1)824 2087 y Fv(\()p Fq(u)868 2094 y Fs(1)888 2087 y Fq(x)914 2063 y Fs(\()p Fr(\013)p Fs(\))914 2100 y(1)976 2087 y Fv(+)11 b Fn(\001)d(\001)g(\001)g Fv(+)i Fq(u)1156 2094 y Fr(m)1189 2087 y Fq(x)1215 2068 y Fs(\()p Fr(\013)p Fs(\))1215 2098 y Fr(m)1268 2087 y Fv(\))1286 2068 y Fr(e)1319 2087 y Fq(;)507 b Fv(\(85\))60 2222 y(where)15 b Fq(e)e Fv(=)g Fq(e)294 2229 y Fs(1)324 2222 y Fv(+)e Fn(\001)d(\001)g(\001)g Fv(+)i Fq(e)499 2229 y Fr(m)533 2222 y Fv(.)20 b(In)15 b(other)g(w)o(ords,)f(the)h Fq(P)991 2229 y Fr(e)1025 2222 y Fv(are)g(the)g(p)q(o)o(w)o(er)g(sums)g(of)g (the)g Fq(g)i Fv(quan)o(tities)510 2352 y Fq(u)536 2359 y Fs(1)556 2352 y Fq(x)582 2328 y Fs(\(1\))582 2365 y(1)639 2352 y Fv(+)10 b Fn(\001)e(\001)g(\001)g Fv(+)j Fq(u)819 2359 y Fr(m)852 2352 y Fq(x)878 2334 y Fs(\(1\))878 2364 y Fr(m)925 2352 y Fq(;)d(:)g(:)g(:)d(;)j(u)1053 2359 y Fs(1)1072 2352 y Fq(x)1098 2328 y Fs(\()p Fr(g)q Fs(\))1098 2365 y(1)1156 2352 y Fv(+)i Fn(\001)e(\001)g(\001)g Fv(+)j Fq(u)1336 2359 y Fr(m)1369 2352 y Fq(x)1395 2334 y Fs(\()p Fr(g)q Fs(\))1395 2364 y Fr(m)1458 2352 y Fq(:)60 2482 y Fv(An)o(y)17 b(p)q(o)o(w)o(er)g(sum)g Fq(P)421 2489 y Fr(e)440 2482 y Fv(,)g Fq(e)f Fv(=)h(1)p Fq(;)8 b Fv(2)p Fq(;)g(:)g(:)f(:)m Fv(,)17 b(can)g(b)q(e)h(written)f(as)g(a)g(p)q (olynomial)i(with)f(rational)f(co)q(e\016cien)o(ts)h(in)60 2567 y(the)d(\014rst)g Fq(g)i Fv(p)q(o)o(w)o(er)d(sums)h Fq(P)546 2574 y Fs(1)566 2567 y Fv(,)g Fq(P)623 2574 y Fs(2)643 2567 y Fq(;)8 b(:)g(:)g(:)d(;)j(P)774 2574 y Fr(g)793 2567 y Fv(.)20 b(Therefore)15 b(an)o(y)g Fq(J)1143 2574 y Fr(e)1177 2567 y Fv(for)856 2697 y Fq(e)e Fv(=)952 2644 y Fr(m)938 2656 y Fm(X)939 2747 y Fr(i)p Fs(=1)1005 2697 y Fq(e)1026 2704 y Fr(i)1053 2697 y Fq(>)g(g)967 2853 y Fv(40)p eop %%Page: 41 45 41 44 bop 60 74 a Fv(\(whic)o(h)16 b(is)f(a)g(co)q(e\016cien)o(t)h(of)f Fq(P)584 81 y Fr(e)603 74 y Fv(\))g(can)g(b)q(e)h(written)f(as)g(a)g(p) q(olynomial)i(in)f(the)f(sp)q(ecial)i(in)o(v)m(arian)o(ts)707 191 y Fq(J)732 198 y Fr(e)796 191 y Fv(with)46 b Fq(e)951 198 y Fs(1)981 191 y Fv(+)10 b Fn(\001)e(\001)g(\001)g Fv(+)j Fq(e)1156 198 y Fr(m)1202 191 y Fn(\024)i Fq(g)60 308 y Fv(\(whic)o(h)g(are)g(the)f(co)q(e\016cien)o(ts)i(of)e Fq(P)660 315 y Fs(1)680 308 y Fq(;)c(:)g(:)g(:)d(;)j(P)811 315 y Fr(g)830 308 y Fv(\).)19 b(Th)o(us)13 b(an)o(y)f(in)o(v)m(arian)o (t)h(can)g(b)q(e)g(written)g(as)f(a)h(p)q(olynomial)h(in)60 392 y(those)h Fq(J)204 399 y Fr(e)238 392 y Fv(with)341 360 y Fm(P)385 374 y Fr(m)385 404 y(i)p Fs(=1)452 392 y Fq(e)473 399 y Fr(i)500 392 y Fn(\024)e Fq(g)r Fv(.)19 b(The)c(n)o(um)o(b)q(er)h(of)f(suc)o(h)g Fq(J)1042 399 y Fr(e)1076 392 y Fv(is)h(the)f(n)o(um)o(b)q(er)h(of)f Fq(e)1439 399 y Fs(1)1458 392 y Fv(,)g Fq(e)1507 399 y Fs(2)1527 392 y Fq(;)8 b(:)g(:)g(:)d(;)j(e)1650 399 y Fr(m)1698 392 y Fv(with)15 b Fq(e)1822 399 y Fr(i)1849 392 y Fn(\025)e Fv(0)60 477 y(and)k Fq(e)171 484 y Fs(1)203 477 y Fv(+)12 b Fn(\001)c(\001)g(\001)h Fv(+)j Fq(e)382 484 y Fr(m)432 477 y Fn(\024)k Fq(g)r Fv(,)h(whic)o(h)h(is)717 443 y Fm(\000)736 457 y Fr(m)p Fs(+)p Fr(g)759 493 y(m)813 443 y Fm(\001)832 477 y Fv(.)26 b(Finally)l(,)19 b(deg)8 b Fq(J)1138 484 y Fr(e)1173 477 y Fn(\024)17 b Fq(g)r Fv(,)f(and)i Fq(J)1394 484 y Fr(e)1430 477 y Fv(is)g(obtained)g(b)o(y)f (a)o(v)o(eraging)60 562 y Fq(x)86 542 y Fr(e)102 547 y Fk(1)86 574 y Fs(1)129 562 y Fn(\001)8 b(\001)g(\001)e Fq(x)216 545 y Fr(e)232 549 y Fi(m)216 573 y Fr(m)279 562 y Fv(o)o(v)o(er)14 b(the)h(group.)p 626 560 16 16 v 60 708 a Fu(Molien's)h(theorem.)44 b Fv(Since)16 b(w)o(e)e(kno)o(w)f (from)g(Theorem)h(19)g(that)f(a)h(p)q(olynomial)h(basis)g(alw)o(a)o(ys) e(exists,)60 793 y(w)o(e)g(can)g(go)f(ahead)h(with)h(con\014dence)g (and)f(try)g(to)f(\014nd)i(it,)f(using)h(the)f(metho)q(ds)g(describ)q 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b(pro)q(of)g(dep)q(ends)i(on)e(the)g (follo)o(wing)h(theorem.)60 1549 y Fu(Theorem)h(20.)22 b Fv([199)o(,)17 b(p.)g(258],)e([276)o(,)i(p.)g(17])f Fw(The)i(numb)n(er)f(of)h(line)n(arly)f(indep)n(endent)f(invariants)h (of)h Fn(G)60 1633 y Fw(of)e(de)n(gr)n(e)n(e)g(1)g(is)769 1718 y Fq(a)793 1725 y Fs(1)825 1718 y Fv(=)879 1687 y(1)p 878 1708 24 2 v 878 1749 a Fq(g)939 1663 y Fr(g)918 1678 y Fm(X)914 1768 y Fr(\013)p Fs(=1)989 1718 y Fv(trace\()p Fq(A)1140 1725 y Fr(\013)1165 1718 y Fv(\))f Fq(:)60 1870 y Fu(Pro)q(of.)45 b Fv(Let)843 1954 y Fq(S)15 b Fv(=)939 1924 y(1)p 939 1944 V 939 1986 a Fq(g)1000 1899 y Fr(g)979 1914 y Fm(X)975 2004 y Fr(\013)p Fs(=1)1050 1954 y Fq(A)1084 1961 y Fr(\013)1124 1954 y Fq(:)60 2067 y Fv(Changing)j(the)g(v)m(ariables)g(on)g(whic)o(h)g Fn(G)j Fv(acts)c(from)g Fq(x)1016 2074 y Fs(1)1035 2067 y Fq(;)8 b(:)g(:)g(:)d(;)j(x)1163 2074 y Fr(m)1213 2067 y Fv(to)17 b Fq(y)1293 2074 y Fs(1)1313 2067 y Fq(;)8 b(:)g(:)g(:)d(;)j(y)1437 2074 y Fr(m)1470 2067 y Fv(,)17 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FV(1244)2588 2588(Y)1261 2600 y()1322 1322 Y FQ(!)1352 2642 Y FR(D)P FL(000)P FS(1)1350 2673 2673 Y(1)1417 2661 Y FQ(!)1445 2668 y Fs(2)1142 2740 y Fv(0)1513 2711 y(.)1531 2724 y(.)1549 2736 y(.)1571 2446 y Fm(3)1571 2519 y(7)1571 2544 y(7)1571 2568 y(7)1571 2593 y(7)1571 2618 y(7)1571 2643 y(7)1571 2670 y(5)1622 2605 y Fq(;)967 2853 y Fv(42)p eop %%Page: 43 47 43 46 bop 60 78 a Fv(and)16 b(trace)f Fq(A)297 54 y Fs([)p Fr(d)p Fs(])297 83 y Fr(\013)350 78 y Fv(=)h(sum)f(of)g(the)h(pro)q (ducts)g(of)f Fq(!)897 85 y fs(1)917 78 y fq(;)8 b(:)g(:):d(())j(!) 1047 85 y Fr(m)1095 78 y Fv(tak)o(en)15 b Fq(d)g Fv(at)g(a)g(time.)21 b(But)16 b(this)g(is)g(exactly)g(the)60 162 y(co)q(e\016cien)o(t)g(of)f Fq(\025)351 146 y Fr(d)386 162 y Fv(in)h(the)f(expansion)h(of)f (\(87\).)p 919 160 16 16 v 131 247 a(It)i(is)h(w)o(orth)e(remarking)h (that)g(the)h(Molien)g(series)g(do)q(es)g(not)f(determine)h(the)f (group.)26 b(F)l(or)17 b(example)60 332 y(there)e(are)g(t)o(w)o(o)f (groups)h(of)g(2)9 b Fn(\002)i 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b(\013)778 2610 y Fs(12)828 2603 y Fv(=)13 b Fq(a)p Fv(\()p Fq(a)942 2584 y Fs(3)971 2603 y Fv(+)e(8)p Fq(p)1063 2584 y Fs(3)1082 2603 y Fv(\))p Fq(;)534 2700 y(\014)560 2707 y Fs(6)592 2700 y Fv(=)i Fq(a)664 2681 y Fs(12)712 2700 y Fn(\000)d Fv(12)p Fq(b;)22 b(\031)884 2707 y Fs(9)916 2700 y Fv(=)13 b(\()p Fq(x)1008 2681 y Fs(3)1037 2700 y Fn(\000)e Fq(y)1107 2681 y Fs(3)1126 2700 y Fv(\)\()p Fq(y)1186 2681 y Fs(3)1215 2700 y Fn(\000)g Fq(z)1284 2681 y Fs(3)1304 2700 y Fv(\)\()p Fq(z)1363 2681 y Fs(3)1392 2700 y Fn(\000)g Fq(x)1464 2681 y Fs(3)1483 2700 y Fv(\))k Fq(:)967 2853 y Fv(49)p eop %%Page: 50 54 50 53 bop 60 74 a Fu(cw)o(e)17 b(of)g Fq(C)s Fu(,)g(not)h(requiring)f (that)i Fv(1)742 58 y Fr(n)778 74 y Fn(2)13 b Fq(C)48 b Fv([198)o(])14 b(\(No)o(w)h(the)g(length)h(is)g(just)e(a)h(m)o (ultiple)i(of)e(4.\))399 234 y Fq(G)e Fv(=)496 162 y Fm(*)553 203 y Fv(1)p 534 224 61 2 v 534 232 a Fn(p)p 572 232 23 2 v 38 x Fv(3)607 137 y Fm(2)607 210 y(6)607 237 y(4)656 178 y Fv(1)45 b(1)j(1)656 234 y(1)41 b Fq(!)p 792 210,30,2 V 44 W(!)656 291 Y FV(1)P 720 720 V 41 W FQ(!)J(!)(842)237 y(5)877 y y(y)y y(y)y y(α)y v(α)y(y),y(y)G(α)G((α))y(α)G(α)y(y)y,y(y)y y,(y)y fq(α)y(y)y,(y)y y(y)y fv(α)y y fq(!)842 137 Y FM(3)842 842 Y(7)1433 291 Y(!)1484 137 y Fm(3)1484 210 y(7)1484 237 y(5)1512 162 y(+)1568 234 y Fq(;)60 391 y Fv(order)15 b(96)746 476 y(\010)d(=)881 445 y(1)d(+)i(4)p Fq(\025)1009 429 y Fs(12)1056 445 y Fv(+)f Fq(\025)1128 429 y Fs(24)p 844 465 358 2 v 844 507 a Fv(\(1)g Fn(\000)g Fq(\025)967 494 y Fs(4)986 507 y Fv(\)\(1)g Fn(\000)g Fq(\025)1127 494 y Fs(12)1164 507 y Fv(\))1182 494 y Fs(2)1222 476 y Fq(:)870 628 y(R)j Fv(:)964 575 y Fs(6)943 588 y Fm(X)944 679 y Fr(i)p Fs(=0)1011 628 y Fq(f)1038 609 y Fs(\()p Fr(i)p Fs(\))1079 628 y Fq(S)60 756 y Fv(where)h Fq(S)h Fv(=)e Ft(C)6 b Fv([)p Fq(\022)342 763 y Fs(4)365 756 y Fq(;)i(\022)408 739 y Fs(2)407 767 y(6)428 756 y Fq(;)g(t)465 739 y Fs(12)502 756 y Fv(],)13 b Fq(s)g Fv(=)g Fq(y)d Fv(+)e Fq(z)r Fv(,)14 b 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Fs(\(4\))1276 840 y Fv(=)e Fq(t)1340 824 y Fs(8)1360 840 y Fq(\036)1387 847 y Fs(4)1407 840 y Fv(,)i Fq(f)1462 824 y Fs(\(5\))1522 840 y Fv(=)e Fq(t)1586 824 y Fs(10)1623 840 y Fq(\036)1650 824 y Fs(2)1650 852 y(4)1670 840 y Fq(\022)1691 847 y Fs(6)1711 840 y Fv(.)60 925 y Fu(Co)q(des:)p 403 931 1174 2 v 424 970 a Fv(Ring)52 b(Co)q(des)p 403 989 V 424 1028 a(\(102\))40 b Fq(t)585 1035 y Fs(4)620 1028 y Fv(\(30\),)14 b Fq(g)751 1035 y Fs(12)803 1028 y Fv(\(31\))424 1086 y(\(103\))40 b Fq(e)590 1067 y Fs(4+)590 1098 y(3)652 1086 y Fv(\()p Fn(x)p Fv(11.4\),)14 b Fq(g)839 1093 y Fs(12)891 1086 y Fv(\(31\),)f Fq(S)s Fv(\(36\))h(\()p Fn(x)p Fv(12.2\);)f Fq(X)t(Q)1368 1093 y Fs(23)1420 1086 y Fv(\()p Fn(x)p Fv(12.2\))1816 1027 y(\(104\))60 1211 y Fo(7.4.)41 b(F)-5 b(amily)17 b Fg(4)397 1193 y Fs(H)426 1211 y Fo(:)24 b(Self-dual)18 b(co)r(des)g(o)n(v)n(er)h Ft(F)996 1218 y Fs(4)1036 1211 y Fo(with)f(Hermitian)e(inner)i(pro)r (duct)60 1325 y Fu(h)o(w)o(e)e(of)i Fq(C)48 b Fv(\([187)n(];)15 b([189)o(,)f(p.)i(621]\):)155 1460 y Fq(G)c Fv(=)251 1388 y Fm(*)289 1429 y Fv(1)p 289 1450 23 2 v 289 1491 a(2)325 1388 y Fm(378 1432 Y fv(1)59 b(3)378 1488 y(1)41μb fn(000)p fv(1)521 521 y fM(!)562 1460 Y FQ(;)605×1388 y FM()658 658 1432 y Fv(1)59 b(0)658 1488 y(0)0 b fn(α)p fv(α)y y fm(!)+)880 1460 y Fv(=)13 b(W)l(eyl)j(group)f (of)g(t)o(yp)q(e)g Fq(G)1362 1467 y Fs(2)1394 1448 y Fn(\030)1394 1462 y Fv(=)1445 1460 y(dihedral)i(group)d Fq(D)1788 1467 y Fs(12)60 1595 y Fv(\(Shephard)i(&)f(T)l(o)q(dd)h (#2b\))778 1680 y(\010)d(=)1025 1649 y(1)p 877 1669 320 2 v 877 1711 a(\(1)c Fn(\000)i Fq(\025)1000 1698 y Fs(2)1019 1711 y Fv(\)\(1)e Fn(\000)i Fq(\025)1160 1698 y Fs(6)1179 1711 y Fv(\))725 1813 y Fq(R)i Fv(:)1015 1783 y(1)p 803 1803 447 2 v 803 1844 a Fq(x)829 1831 y Fs(2)859 1844 y Fv(+)d(3)p Fq(y)951 1831 y Fs(2)971 1844 y Fq(;)22 b(y)1030 1831 y Fs(2)1050 1844 y Fv(\()p Fq(x)1094 1831 y Fs(2)1123 1844 y Fn(\000)11 b Fq(y)1193 1831 y Fs(2)1212 1844 y Fv(\))1230 1831 y Fs(2)1816 1813 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Fv(+)e(12)p Fq(xy)591 1351 y Fs(3)620 1367 y Fv(+)f(3)p Fq(y)712 1351 y Fs(4)747 1367 y Fq(:)131 1513 y(c)151 1520 y Fs(6)183 1513 y Fv(=)231 1416 y Fm(2)231 1489 y(6)231 1516 y(4)279 1456 y Fv(1)45 b(1)j(1)d(1)g(1)j(1)279 1513 y(0)d(0)j(0)d(1)c Fq(!)p 622 1488 V 44 W(!)279(1569)y fv(1)g fq(!)p 415 1545 V 44 W(!)i Fv(0)i(0)j(0)672 1416 y Fm(3)672 1489 y(7)672 1516 y(5)708 1513 y Fq(;)252 1628 y(cw)q(e)12 b Fv(=)h Fq(x)413 1612 y Fs(6)443 1628 y Fv(+)d Fn(\001)e(\001)g(\001)e Fv(\(4)14 b(terms\))c(+)g(6)p Fq(x)838 1612 y Fs(3)858 1628 y Fq(y)r(z)r(t)g Fv(+)h Fn(\001)d(\001)g(\001)d Fv(\(4)15 b(terms\))9 b(+)h(9)p Fq(x)1326 1612 y Fs(2)1346 1628 y Fq(y)1370 1612 y Fs(2)1390 1628 y Fq(z)1413 1612 y Fs(2)1442 1628 y Fv(+)h Fn(\001)d(\001)g(\001)e Fv(\(4)14 b(terms\))o Fq(;)252 1685 y(hw)q(e)f Fv(=)g Fq(x)420 1668 y Fs(6)449 1685 y Fv(+)e(6)p Fq(x)544 1668 y Fs(3)563 1685 y Fq(y)587 1668 y Fs(3)617 1685 y Fv(+)f(27)p Fq(x)734 1668 y Fs(2)754 1685 y Fq(y)778 1668 y Fs(4)807 1685 y 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1590 y(1)p 809 1611 408 2 v 809 1652 a Fq(x)835 1639 y Fs(2)865 1652 y Fv(+)d(3)p Fq(y)957 1639 y Fs(2)977 1652 y Fq(;)22 b(y)r Fv(\()p Fq(x)1080 1639 y Fs(2)1109 1652 y Fn(\000)11 b Fq(y)1179 1639 y Fs(2)1198 1652 y Fv(\))1236 1621 y Fq(:)567 b Fv(\(124\))60 1769 y Fu(cw)o(e)26 b(of)h Fq(S)s Fu(,)h Fv(1)336 1753 y Fr(n)385 1769 y Fn(2)e Fq(C)s Fu(:)45 b Fv(Belongs)24 b(to)e(image)h(of)g(\(117\))f (under)i(the)f(map)g(that)f(sends)i(\()p Fq(x;)8 b(y)r(;)g(z)r(;)g(t)p Fv(\))21 b(to)60 1854 y(\()p Fq(x;)8 b(y)r(;)g(z)r(;)g(t)p Fv(\))p Fq(\014)274 1861 y Fs(4)291 1854 y Fq(M)335 1861 y Fs(4)355 1854 y Fv(:)712 1939 y Fq(R)k Fv(:)942 1908 y(1)p Fq(;)23 b(AB)r(D)p 789 1928 475 2 v 789 1970 a(C)q(;)f(A)892 1957 y Fs(2)922 1970 y Fv(+)10 b Fq(B)1003 1957 y Fs(2)1024 1970 y Fq(;)22 b(D)1098 1957 y Fs(2)1118 1970 y Fq(;)g(A)1187 1957 y Fs(2)1207 1970 y Fq(B)1243 1957 y Fs(2)1816 1939 y Fv(\(125\))60 2087 y Fu(cw)o(e)17 b(of)h Fq(W)270 2070 y Fs(\(1\))328 2087 y Fn(\000)10 b Fq(W)422 2070 y Fs(\(3\))470 2087 y Fu(,)17 b Fv(1)525 2070 y Fr(n)561 2087 y Fn(2)c Fq(C)s Fu(:)45 b Fq(G)13 b Fv(=)g Fn(h)p Fq(M)859 2094 y Fs(4)879 2087 y Fq(;)8 b(\013)929 2094 y Fs(4)948 2087 y Fq(;)g(\014)995 2094 y Fs(4)1013 2087 y Fn(i)15 b Fv(with)h(c)o (haracter)f Fq(\037)p Fv(\()p Fq(M)1439 2094 y Fs(4)1459 2087 y Fv(\))d(=)i(1,)h Fq(\037)p Fv(\()p Fq(\013)1664 2094 y Fs(4)1683 2087 y Fv(\))e(=)g Fq(\037)p Fv(\()p Fq(\014)1834 2094 y Fs(4)1854 2087 y Fv(\))g(=)60 2172 y(\()p Fn(\000)p Fv(1\))154 2155 y Fr(n)177 2172 y Fv(,)i(order)g(48) 628 2256 y(\010)e(=)1025 2226 y(1)p 727 2246 620 2 v 727 2288 a(\(1)c Fn(\000)i Fq(\025)p Fv(\)\(1)d Fn(\000)j Fq(\025)990 2274 y Fs(2)1009 2288 y Fv(\)\(1)e Fn(\000)i Fq(\025)1150 2274 y Fs(3)1169 2288 y Fv(\)\(1)e Fn(\000)i Fq(\025)1310 2274 y Fs(4)1329 2288 y Fv(\))547 2390 y Fq(R)i Fv(:)1015 2359 y(1)p 625 2380 803 2 v 625 2421 a Fq(D)q(;)22 b(A)733 2408 y Fs(2)763 2421 y Fv(+)11 b Fq(B)845 2408 y Fs(2)875 2421 y Fv(+)g Fq(C)957 2408 y Fs(2)976 2421 y Fq(;)22 b(AB)r(C)q(;)h(A)1185 2408 y Fs(4)1215 2421 y Fv(+)10 b Fq(B)1296 2408 y Fs(4)1327 2421 y Fv(+)g Fq(C)1408 2408 y Fs(4)1816 2390 y Fv(\(126\))60 2502 y(where)536 2586 y Fq(A)j Fv(=)g Fq(x)d Fv(+)g Fq(y)r(;)23 b(B)15 b Fv(=)e Fq(x)d Fn(\000)h Fq(y)r(;)22 b(C)15 b Fv(=)e Fq(z)f Fv(+)f Fq(t;)23 b(D)13 b Fv(=)g Fq(z)f Fn(\000)f Fq(t:)967 2853 y Fv(55)p eop %%Page: 56 60 56 59 bop 60 74 a Fu(sw)o(e)16 b(of)i Fq(W)267 58 y Fs(\(1\))324 74 y Fn(\000)11 b Fq(W)419 58 y Fs(\(3\))466 74 y Fu(,)17 b Fv(1)521 58 y Fr(n)557 74 y Fn(2)c Fq(C)s Fu(:)698 187 y Fv(\010)g(=)1025 157 y(1)p 797 177 480 2 v 797 219 a(\(1)c Fn(\000)i Fq(\025)920 206 y Fs(2)939 219 y Fv(\)\(1)e Fn(\000)i Fq(\025)1080 206 y Fs(3)1099 219 y Fv(\)\(1)e Fn(\000)i Fq(\025)1240 206 y Fs(4)1259 219 y Fv(\))584 323 y Fq(R)i Fv(:)1015 292 y(1)p 662 312 729 2 v 662 354 a Fq(A)696 341 y Fs(2)726 354 y Fv(+)d Fq(B)807 341 y Fs(2)838 354 y Fv(+)h Fq(C)920 341 y Fs(2)939 354 y Fq(;)22 b(AB)r(C)q(;)h(A)1148 341 y Fs(4)1178 354 y Fv(+)10 b Fq(B)1259 341 y Fs(4)1290 354 y Fv(+)g Fq(C)1371 341 y Fs(4)1816 323 y Fv(\(127\))60 468 y Fu(cw)o(e)17 b(of)g Fq(S)s Fu(,)g Fv(1)306 452 y Fr(n)342 468 y Fn(2)c Fq(S)s Fu(:)44 b Fv(Belongs)16 b(to)e(image)i(of)f(\(121\))e(under)j (the)f(map)g Fq(x)e Fn(!)gfq(y)r fv(,)i fq(y)f fn(!)ffq(x)p fv(,)i fq(z)g fn(!)e Fq(t)p fv(,)i fq(t)e fn(!)g Fq(z)r Fv(:)474 581 y Fq(R)g Fv(:)1015 550 y(1)p 552 571 950 2 v 552 612 a Fn(\000)p Fq(D)q(;)22 b(A)11 b Fn(\000)f Fq(B)j Fv(+)d Fq(C)q(;)22 b(A)946 599 y Fs(2)976 612 y Fv(+)10 b Fq(B)1057 599 y Fs(2)1088 612 y Fv(+)g Fq(C)1169 599 y Fs(2)1189 612 y Fq(;)22 b(A)1258 599 y Fs(3)1288 612 y Fn(\000)10 b Fq(B)1369 599 y Fs(3)1400 612 y Fv(+)h Fq(C)1482 599 y Fs(3)1816 581 y Fv(\(128\))60 727 y Fu(sw)o(e)16 b(of)i Fq(S)s Fu(,)e Fv(1)303 710 y Fr(n)339 727 y Fn(2)d Fq(S)s Fu(:)45 b Fv(Set)15 b Fq(D)e Fv(=)g(0)i(in)h(\(128\).)60 872 y Fu(h)o(w)o(e)g(of)i Fq(S)s Fu(,)e Fv(1)310 856 y Fr(n)346 872 y Fn(2)d Fq(S)s Fu(:)44 b Fv(Same)16 b(as)e(\(123\).)60 1018 y Fu(cw)o(e)f(of)g Fq(W)261 1001 y Fs(\(1\))312 1018 y Fn(\000)s Fq(W)399 1001 y Fs(\(3\))447 1018 y Fu(,)g Fv(1)498 1001 y Fr(n)534 1018 y Fn(2)g Fq(S)s Fu(:)45 b Fq(G)12 b Fv(=)h Fn(h)p Fq(M)826 1025 y Fs(4)846 1018 y Fq(;)8 b(\014)893 1025 y Fs(4)924 1018 y Fv(=)13 b(diag)q Fn(f)p Fv(1)p Fq(;)8 b Fn(\000)p Fv(1)p Fq(;)g Fn(\000)p Fv(1)p Fq(;)g Fn(\000)p Fv(1)p Fn(gi)o Fv(,)h(with)j(c)o(haracter)f Fq(\037)p Fv(\()p Fq(M)1786 1025 y Fs(4)1806 1018 y Fv(\))i(=)g(1,)60 1103 y Fq(\037)p Fv(\()p Fq(\014)132 1110 y Fs(4)152 1103 y Fv(\))f(=)h(\()p Fn(\000)p Fv(1\))324 1086 y Fr(n)347 1103 y Fv(,)i(order)f(12)698 1187 y(\010)f(=)1025 1157 y(1)p 797 1177 480 2 v 797 1218 a(\(1)c Fn(\000)i Fq(\025)p Fv(\))938 1205 y Fs(2)957 1218 y Fv(\(1)e Fn(\000)i Fq(\025)1080 1205 y Fs(2)1099 1218 y Fv(\)\(1)e Fn(\000)i Fq(\025)1240 1205 y Fs(3)1259 1218 y Fv(\))492 1312 y Fq(R)h Fv(:)1015 1282 y(1)p 569 1302 914 2 v 569 1344 a Fq(D)q(;)23 b(A)10 b Fn(\000)g Fq(B)j Fn(\000)e Fq(C)q(;)21 b(A)928 1330 y Fs(2)958 1344 y Fv(+)11 b Fq(B)1040 1330 y Fs(2)1070 1344 y Fv(+)g Fq(C)1152 1330 y Fs(2)1171 1344 y Fq(;)22 b(A)1240 1330 y Fs(3)1270 1344 y Fn(\000)11 b Fq(B)1352 1330 y Fs(3)1382 1344 y Fn(\000)g Fq(C)1464 1330 y Fs(3)1816 1312 y Fv(\(129\))60 1458 y Fu(Remark.)44 b Fv(W)l(e)21 b(ma)o(y)e(obtain)i Fq(W)684 1441 y Fs(\(1\))745 1458 y Fn(\000)14 b Fq(W)843 1441 y Fs(\(3\))911 1458 y Fv(b)o(y)20 b(applying)i Fq(\013)1199 1465 y Fs(4)1239 1458 y Fv(to)e Fq(W)1349 1441 y Fs(\(0\))1410 1458 y Fn(\000)14 b Fq(W)1508 1441 y Fs(\(2\))1555 1458 y Fv(,)22 b(whic)o(h)f(in)g(turn)f(is)60 1543 y(obtained)c(b)o(y)f(applying)i Fq(\014)522 1550 y Fs(4)556 1543 y Fv(to)e Fq(W)661 1526 y Fs(\(0\))718 1543 y Fv(+)c Fq(W)813 1526 y Fs(\(2\))860 1543 y Fv(.)60 1662 y Fo(7.7.)41 b(F)-5 b(amily)26 b Fg(4)406 1641 y Fs(H+)406 1674 y(I)q(I)462 1662 y Fo(:)41 b(Additiv)n(e)27 b(ev)n(en)f(self-dual)g(co)r(des)h(o)n(v)n(er)g Ft(F)1441 1669 y Fs(4)1489 1662 y Fo(using)g(trace)g(inner)189 1720 y(pro)r(duct)60 1834 y Fu(h)o(w)o(e)16 b(of)i Fq(C)s Fu(:)45 b Fv(Same)15 b(as)g(family)g(4)659 1817 y Fs(H)688 1834 y Fv(,)f(see)i(\(105\).)60 1979 y Fu(cw)o(e)h(of)g Fq(C)s Fu(,)g Fv(1)311 1963 y Fr(n)347 1979 y Fn(2)c Fq(C)s Fu(:)702 2064 y Fq(G)g Fv(=)g Fn(h)p Fq(M)861 2071 y Fs(4)880 2064 y Fq(;)8 b(\013)930 2071 y Fs(4)949 2064 y Fq(;)g(\014)996 2071 y Fs(4)1015 2064 y Fn(i)15 b Fq(;)52 b Fv(order)15 b(48)689 2167 y(\010)d(=)974 2136 y(1)e(+)h Fq(\025)1080 2120 y Fs(4)p 787 2156 500 2 v 787 2198 a Fv(\(1)f Fn(\000)g Fq(\025)910 2185 y Fs(2)929 2198 y Fv(\))947 2185 y Fs(2)967 2198 y Fv(\(1)f Fn(\000)i Fq(\025)1090 2185 y Fs(4)1109 2198 y Fv(\)\(1)e Fn(\000)h Fq(\025)1249 2185 y Fs(6)1269 2198 y Fv(\))449 2294 y Fq(R)j Fv(:)925 2263 y(1)p Fq(;)22 b(AB)r(C)s(D)p 527 2283 999 2 v 527 2325 a(D)566 2312 y Fs(2)586 2325 y Fq(;)g(A)655 2312 y Fs(2)685 2325 y Fv(+)11 b Fq(B)767 2312 y Fs(2)797 2325 y Fv(+)g Fq(C)879 2312 y Fs(2)898 2325 y Fq(;)22 b(A)967 2312 y Fs(4)997 2325 y Fv(+)11 b Fq(B)1079 2312 y Fs(4)1109 2325 y Fv(+)g Fq(C)1191 2312 y Fs(4)1210 2325 y Fq(;)23 b(A)1280 2312 y Fs(6)1309 2325 y Fv(+)11 b Fq(B)1391 2312 y Fs(6)1421 2325 y Fv(+)g Fq(D)1506 2312 y Fs(6)1816 2294 y Fv(\(130\))60 2439 y Fu(sw)o(e)16 b(of)i Fq(C)s Fu(,)f Fv(1)309 2423 y Fr(n)344 2439 y Fn(2)c Fq(C)s Fu(:)45 b Fv(\(Set)15 b Fq(t)e Fv(=)g Fq(z)k Fv(in)f(cw)o(e\))698 2552 y(\010)d(=)1025 2522 y(1)p 797 2542 480 2 v 797 2584 a(\(1)c Fn(\000)i Fq(\025)920 2571 y Fs(2)939 2584 y Fv(\)\(1)e Fn(\000)i Fq(\025)1080 2571 y Fs(4)1099 2584 y Fv(\)\(1)e Fn(\000)i Fq(\025)1240 2571 y Fs(6)1259 2584 y Fv(\))574 2688 y Fq(R)i Fv(:)1015 2657 y(1)p 652 2677 749 2 v 652 2720 a(symmetric)i(p)q(olynomials)i(in) f Fq(A)1213 2703 y Fs(2)1233 2720 y Fv(,)f Fq(B)1297 2703 y Fs(2)1317 2720 y Fv(,)g Fq(C)1381 2703 y Fs(2)1816 2688 y Fv(\(131\))967 2853 y(56)p eop %%Page: 57 61 57 60 bop 60 74 a Fu(cw)o(e)17 b(of)g Fq(C)k Fu(\(not)d(assuming)f Fv(1)636 58 y Fr(n)672 74 y Fn(2)c Fq(C)s Fu(\):)744 205 y Fq(G)g Fv(=)g Fn(h)p Fq(M)903 212 y Fs(4)922 205 y Fq(;)8 b(\014)969 212 y Fs(4)988 205 y Fn(i)p Fq(;)52 b Fv(order)15 b(12)769 335 y(\010)d(=)912 304 y(1)e(+)g Fq(\025)1017 287 y Fs(2)1047 304 y Fv(+)g(2)p Fq(\025)1142 287 y Fs(4)p 867 324 340 2 v 867 366 a Fv(\(1)f Fn(\000)i Fq(\025)990 353 y Fs(2)1009 366 y Fv(\))1027 353 y Fs(3)1047 366 y Fv(\(1)e Fn(\000)i Fq(\025)1170 353 y Fs(6)1189 366 y Fv(\))1816 335 y(\(132\))60 446 y Fu(Co)q(des:)20 b Fv(The)15 b(follo)o(wing)h(co)q(des)g(will)h(b)q(e)e(used:)232 576 y Fq(i)248 583 y Fs(1)309 576 y Fv(=)41 b([1])p Fq(;)52 b(cw)q(e)12 b Fv(=)h Fq(sw)q(e)g Fv(=)g Fq(hw)q(e)g Fv(=)g Fq(x)d Fv(+)g Fq(y)232 673 y(i)248 655 y Fl(0)248 685 y Fs(1)309 673 y Fv(=)41 b([)p Fq(!)R FV())p Fq((52)B(CW)q(E)12 B Fv(=)H FQ(SW)q(E)G Fv(=)G Fq(x)d Fv(+)H fq(z)230 770 y(i)246 752 y fl(00)246 782 y fs(1)309 770 y y fv(=)770 b([ p)αv v fq(!)r fv())p fq(e)52(b)Fv(e)12 fq(x)e Fv(+)f fq(t);23 b(SW)q(e)13μb Fv(=)g fq(x)d Fv(+)g fq(z)232 868 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Fv(1)1(y)b b fn(α)p fv(α)yy y fM(!)979(1628)y fq(;)1045 1556 y Fm()1102 1599 y fq(i)45 b Fv(0)1098 1656 1656(0)g fq(i)1206 1206 y FM(!)+)1280 1628 y Fq(;)53 b Fv(order)15 b(8)849 1788 y(\010)d(=)974 1757 y(1)e(+)h Fq(\025)1080 1741 y Fs(4)p 947 1777 180 2 v 947 1819 a Fv(\(1)e Fn(\000)i Fq(\025)1070 1806 y Fs(4)1089 1819 y Fv(\))1107 1806 y Fs(2)299 1926 y Fq(R)h Fv(:)829 1895 y(1)p Fq(;)22 b(x)913 1878 y Fs(2)933 1895 y Fq(y)957 1878 y Fs(2)987 1895 y Fn(\000)10 b Fv(2)p Fq(xy)1105 1878 y Fs(3)1135 1895 y Fv(+)g Fq(y)1204 1878 y Fs(4)p 376 1915 1300 2 v 376 1957 a Fq(x)402 1944 y Fs(4)432 1957 y Fv(+)h(4\()p Fq(q)g Fn(\000)g Fv(1\))p Fq(xy)687 1944 y Fs(3)716 1957 y Fv(+)f(\()p Fq(q)i Fn(\000)e Fv(1\)\()p Fq(q)h Fn(\000)g Fv(3\))p Fq(y)1057 1944 y Fs(4)1076 1957 y Fq(;)22 b(x)1137 1944 y Fs(3)1157 1957 y Fq(y)12 b Fv(+)e(\()p Fq(q)i Fn(\000)f Fv(3\))p Fq(xy)1423 1944 y Fs(3)1452 1957 y Fn(\000)f Fv(\()p Fq(q)i Fn(\000)e Fv(2\))p Fq(y)1657 1944 y Fs(4)60 2073 y Fo(7.10.)41 b(F)-5 b(amily)17 b Fg(4)425 2054 y Fp(Z)425 2085 y Fs(I)447 2073 y Fo(:)24 b(Self-dual)18 b(co)r(des)g(o)n(v)n(er)h Ft(Z)1024 2080 y Fs(4)60 2187 y Fu(cw)o(e)e(of)g Fq(C)s Fu(:)45 b Fv(\([166)o(]\))272 2363 y Fq(G)13 b Fv(=)369 2291 y Fm(*)402 2363 y Fq(M)446 2370 y Fs(4)478 2363 y Fv(=)531 2332 y(1)p 531 2352 23 2 v 531 2394 a(2)567 2241 y Fm(0)567 2314 y(B)567 2339 y(B)567 2364 y(B)567 2391 y(@)624 2278 y Fv(1)76 b(1)h(1)g(1)624 2335 y(1)83 b Fq(i)42 b Fn(\000)p Fv(1)48 b Fn(\000)p Fq(i)624 2391 y Fv(1)41 b Fn(\000)p Fv(1)77 b(1)41 b Fn(\000)p Fv(1)624 2448 y(1)48 b Fn(\000)p Fq(i)42 b Fn(\000)p Fv(1)84 b Fq(i)966 2241 y Fm(1)966 2314 y(C)966 2339 y(C)966 2364 y(C)966 2391 y(A)1010 2363 y Fq(;)22 b(\013)1074 2370 y Fs(4)1107 2363 y Fv(=)13 b(diag)q Fn(f)p Fv(1)p Fq(;)8 b(i;)g Fv(1)p Fq(;)f(i)p Fn(g)1425 2291 y Fm(+)1463 2363 y Fq(;)67 b Fv(order)15 b(64)698 2574 y(\010)e(=)966 2543 y(1)c(+)i Fq(\025)1071 2527 y Fs(10)p 797 2563 480 2 v 797 2605 a Fv(\(1)e Fn(\000)i Fq(\025)p Fv(\)\(1)e Fn(\000)h Fq(\025)1060 2592 y Fs(4)1079 2605 y Fv(\))1097 2592 y Fs(2)1117 2605 y Fv(\(1)f Fn(\000)i Fq(\025)1240 2592 y Fs(8)1259 2605 y Fv(\))710 2712 y Fq(R)h Fv(:)813 2681 y(1)p Fq(;)22 b Fv(\()p Fq(B)r(C)s(D)q Fv(\))1018 2664 y Fs(2)1037 2681 y Fv(\()p Fq(B)1091 2664 y Fs(4)1121 2681 y Fn(\000)11 b Fq(C)1203 2664 y Fs(4)1222 2681 y Fv(\))p 787 2701 479 2 v 787 2743 a Fq(A;)23 b(B)893 2730 y Fs(4)923 2743 y Fv(+)11 b Fq(C)1005 2730 y Fs(4)1024 2743 y Fq(;)23 b(D)1099 2730 y Fs(4)1118 2743 y Fq(;)g(B)1190 2730 y Fs(4)1210 2743 y Fq(C)1246 2730 y Fs(4)1816 2712 y Fv(\(138\))967 2853 y(61)p eop %%Page: 62 66 62 65 bop 60 74 a Fv(where)529 159 y Fq(A)12 b Fv(=)h Fq(x)d Fv(+)h Fq(z)r(;)22 b(B)16 b Fv(=)d Fq(y)f Fv(+)e Fq(t;)23 b(C)15 b Fv(=)e Fq(x)d Fn(\000)h Fq(z)r(;)22 b(D)14 b Fv(=)f Fq(y)f Fn(\000)e Fq(t)16 b(:)364 b Fv(\(139\))60 307 y Fu(sw)o(e)16 b(of)i Fq(C)s Fu(:)45 b Fv(\(Set)15 b Fq(t)e Fv(=)g Fq(y)k Fv(in)f(cw)o(e\))708 438 y(\010)d(=)1025 407 y(1)p 807 427 460 2 v 807 469 a(\(1)c Fn(\000)i Fq(\025)p Fv(\)\(1)d Fn(\000)j Fq(\025)1070 456 y Fs(4)1089 469 y Fv(\)\(1)e Fn(\000)i Fq(\025)1230 456 y Fs(8)1249 469 y Fv(\))757 590 y Fq(R)h Fv(:)1015 559 y(1)p 834 580 385 2 v 834 621 a Fq(A;)23 b(B)940 608 y Fs(4)970 621 y Fv(+)11 b Fq(C)1052 608 y Fs(4)1071 621 y Fq(;)23 b(B)1143 608 y Fs(4)1163 621 y Fq(C)1199 608 y Fs(4)1816 590 y Fv(\(140\))60 738 y Fu(h)o(w)o(e)16 b(of)i Fq(C)s Fu(:)45 b Fv(\(Set)14 b Fq(t)g Fv(=)e Fq(z)j Fv(=)e Fq(y)k Fv(in)f(cw)o(e\))788 869 y(\010)d(=)974 838 y(1)d(+)h Fq(\025)1080 822 y Fs(8)p 887 858 301 2 v 887 900 a Fv(\(1)e Fn(\000)i Fq(\025)p Fv(\)\(1)d Fn(\000)j Fq(\025)1150 887 y Fs(4)1169 900 y Fv(\))642 1030 y Fq(R)h Fv(:)895 999 y(1)p Fq(;)22 b(y)977 983 y Fs(4)997 999 y Fv(\()p Fq(x)10 b Fn(\000)g Fq(y)r Fv(\))1138 983 y Fs(4)p 719 1019 614 2 v 719 1061 a Fq(x)g Fv(+)h Fq(y)r(;)22 b(y)r Fv(\()p Fq(x)10 b Fn(\000)g Fq(y)r Fv(\)\()p Fq(x)1069 1048 y Fs(2)1098 1061 y Fv(+)h Fq(xy)h Fv(+)e(2)p Fq(y)1296 1048 y Fs(2)1316 1061 y Fv(\))1816 1030 y(\(141\))60 1178 y Fu(cw)o(e,)16 b Fv(1)198 1162 y Fr(n)234 1178 y Fn(2)d Fq(C)518 1315 y(G)f Fv(=)614 1243 y Fm(*)648 1315 y Fq(M)692 1322 y Fs(4)712 1315 y Fq(;)c(\013)762 1322 y Fs(4)781 1315 y Fq(;)802 1194 y Fm(0)801 1267 y(B)801 1292 y(B)801 1316 y(B)801 1343 y(@)858 1231 y Fv(0)41 b(1)g(0)h(0)858 1287 y(0)f(0)g(1)h(0)858 1343 y(0)f(0)g(0)h(1)858 1400 y(1)f(0)g(0)h(0)1094 1194 y Fm(1)1094 1267 y(C)1094 1292 y(C)1094 1316 y(C)1094 1343 y(A)1130 1243 y(+)1187 1315 y Fq(;)52 b Fv(order)15 b(1024)680 1517 y(\010)d(=)859 1486 y(\(1)e(+)g Fq(\025)982 1469 y Fs(12)1019 1486 y Fv(\)\(1)f(+)i Fq(\025)1160 1469 y Fs(16)1197 1486 y Fv(\))p 778 1506 517 2 v 778 1548 a(\(1)f Fn(\000)g Fq(\025)901 1535 y Fs(4)920 1548 y Fv(\)\(1)g Fn(\000)g Fq(\025)1061 1535 y Fs(8)1080 1548 y Fv(\))1098 1535 y Fs(2)1118 1548 y Fv(\(1)f Fn(\000)i Fq(\025)1241 1535 y Fs(16)1278 1548 y Fv(\))370 1659 y Fq(R)h Fv(:)643 1628 y(\(1)p Fq(;)21 b(A)752 1612 y Fs(12)800 1628 y Fv(+)10 b Fq(B)881 1612 y Fs(12)930 1628 y Fv(+)g Fq(C)1011 1612 y Fs(12)1058 1628 y Fv(+)h Fq(D)1143 1612 y Fs(12)1180 1628 y Fv(\))f Fn(\002)g Fv(\(1)p Fq(;)22 b(\033)1355 1635 y Fs(16)1392 1628 y Fv(\))p 447 1648 1159 2 v 447 1690 a Fq(A)481 1677 y Fs(4)511 1690 y Fv(+)11 b Fq(B)593 1677 y Fs(4)623 1690 y Fv(+)g Fq(C)705 1677 y Fs(4)734 1690 y Fv(+)g Fq(D)819 1677 y Fs(4)838 1690 y Fq(;)23 b(A)908 1677 y Fs(8)938 1690 y Fv(+)10 b Fq(B)1019 1677 y Fs(8)1050 1690 y Fv(+)g Fq(C)1131 1677 y Fs(8)1161 1690 y Fv(+)g Fq(D)1245 1677 y Fs(8)1265 1690 y Fq(;)22 b(\033)1326 1697 y Fs(8)1346 1690 y Fq(;)g(A)1415 1677 y Fs(4)1435 1690 y Fq(B)1471 1677 y Fs(4)1491 1690 y Fq(C)1527 1677 y Fs(4)1547 1690 y Fq(D)1586 1677 y Fs(4)1816 1659 y Fv(\(142\))60 1771 y(where)491 1901 y Fq(\033)517 1908 y Fs(8)578 1901 y Fv(=)42 b Fq(A)689 1882 y Fs(4)709 1901 y Fq(D)748 1882 y Fs(4)778 1901 y Fv(+)10 b Fq(B)859 1882 y Fs(4)880 1901 y Fq(C)916 1882 y Fs(4)950 1901 y Fq(;)474 1998 y(\033)500 2005 y Fs(16)578 1998 y Fv(=)42 b(\()p Fq(AB)r(C)s(D)q Fv(\))836 1979 y Fs(2)856 1998 y Fv(\()p Fq(A)908 1979 y Fs(4)927 1998 y Fq(B)963 1979 y Fs(4)994 1998 y Fv(+)10 b Fq(C)1075 1979 y Fs(4)1095 1998 y Fq(D)1134 1979 y Fs(4)1163 1998 y Fn(\000)h Fq(A)1243 1979 y Fs(4)1263 1998 y Fq(C)1299 1979 y Fs(4)1328 1998 y Fn(\000)g Fq(B)1410 1979 y Fs(4)1430 1998 y Fq(D)1469 1979 y Fs(4)1489 1998 y Fv(\))60 2146 y Fu(sw)o(e)16 b(of)i Fq(C)s Fu(,)f Fn(\006)p Fv(1)344 2130 y Fr(n)380 2146 y Fn(2)c Fq(C)48 b Fv(\(Set)15 b Fq(t)e Fv(=)g Fq(y)k Fv(in)f(cw)o(e\))769 2277 y(\010)c(=)966 2246 y(1)d(+)i Fq(\025)1071 2230 y Fs(12)p 867 2266 340 2 v 867 2308 a Fv(\(1)e Fn(\000)i Fq(\025)990 2295 y Fs(4)1009 2308 y Fv(\)\(1)e Fn(\000)i Fq(\025)1150 2295 y Fs(8)1169 2308 y Fv(\))1187 2295 y Fs(2)607 2438 y Fq(R)h Fv(:)901 2407 y(1)p Fq(;)22 b(A)993 2391 y Fs(4)1012 2407 y Fq(B)1048 2391 y Fs(4)1069 2407 y Fq(C)1105 2391 y Fs(4)p 685 2427 656 2 v 685 2469 a Fq(A)719 2456 y Fs(4)749 2469 y Fv(+)10 b Fq(B)830 2456 y Fs(4)861 2469 y Fv(+)g Fq(C)942 2456 y Fs(4)962 2469 y Fq(;)22 b(A)1031 2456 y Fs(8)1066 2469 y Fq(B)1102 2456 y Fs(8)1138 2469 y Fq(C)1174 2456 y Fs(8)1193 2469 y Fq(;)g(B)1264 2456 y Fs(4)1285 2469 y Fq(C)1321 2456 y Fs(4)1360 2438 y Fq(:)443 b Fv(\(143\))60 2549 y(This)14 b(ring)f(ma)o(y)f(also)h(b)q(e) h(describ)q(ed)h(as)e Fq(R)786 2556 y Fs(0)810 2549 y Fn(\010)6 b Fq(B)887 2533 y Fs(4)908 2549 y Fq(C)944 2533 y Fs(4)964 2549 y Fq(R)999 2556 y Fs(0)1023 2549 y Fn(\010)g Fq(B)1100 2533 y Fs(8)1121 2549 y Fq(C)1157 2533 y Fs(8)1177 2549 y Fq(R)1212 2556 y Fs(0)1231 2549 y Fv(,)13 b(where)g Fq(R)1421 2556 y Fs(0)1453 2549 y Fv(is)h(the)f(ring)g(of)g(symmetric)60 2634 y(p)q(olynomials)k(in)f Fq(A)400 2618 y Fs(4)420 2634 y Fv(,)e Fq(B)483 2618 y Fs(4)504 2634 y Fv(,)h Fq(C)568 2618 y Fs(4)587 2634 y Fv(.)967 2853 y(62)p eop %%Page: 63 67 63 66 bop 60 74 a Fu(h)o(w)o(e)16 b(of)i Fq(C)s Fu(,)e Fn(\006)p Fv(1)350 58 y Fr(n)387 74 y Fn(2)d Fq(C)s Fu(:)44 b Fv(\(Set)15 b Fq(t)e Fv(=)g Fq(z)i Fv(=)e Fq(y)k Fv(in)f(cw)o(e\))770 205 y(\010)c(=)868 174 y(\(1)e(+)g Fq(\025)991 158 y Fs(8)1010 174 y Fv(\)\(1)f(+)i Fq(\025)1151 158 y Fs(12)1188 174 y Fv(\))p 868 194 338 2 v 877 236 a(\(1)e Fn(\000)i Fq(\025)1000 223 y Fs(4)1019 236 y Fv(\)\(1)e Fn(\000)i Fq(\025)1160 223 y Fs(8)1179 236 y Fv(\))482 366 y Fq(R)i Fv(:)560 335 y(\(1)p Fq(;)22 b(y)660 319 y Fs(2)679 335 y Fv(\()p Fq(x)723 319 y Fs(2)753 335 y Fv(+)10 b(3)p Fq(y)845 319 y Fs(2)865 335 y Fv(\)\()p Fq(x)927 319 y Fs(2)956 335 y Fn(\000)g Fq(y)1025 319 y Fs(2)1045 335 y Fv(\))1063 319 y Fs(2)1082 335 y Fv(\))g Fn(\002)h Fv(\(1)p Fq(;)21 b(y)1255 319 y Fs(4)1275 335 y Fv(\()p Fq(x)1319 319 y Fs(2)1348 335 y Fn(\000)11 b Fq(y)1418 319 y Fs(2)1438 335 y Fv(\))1456 319 y Fs(4)1475 335 y Fv(\))p 560 355 933 2 v 795 397 a(\()p Fq(x)839 384 y Fs(2)869 397 y Fv(+)f(3)p Fq(y)961 384 y Fs(2)981 397 y Fv(\))999 384 y Fs(2)1018 397 y Fq(;)22 b(y)1077 384 y Fs(4)1097 397 y Fv(\()p Fq(x)10 b Fn(\000)g Fq(y)r Fv(\))1238 384 y Fs(4)1816 366 y Fv(\(144\))60 514 y Fu(cw)o(e)20 b(of)g Fq(C)s Fu(,)g Fv(1)320 498 y Fr(n)359 514 y Fn(2)d Fq(S)s Fu(:)44 b Fv(If)18 b(1)567 498 y Fr(n)607 514 y Fn(2)e Fq(S)s Fv(,)h(P)o(art)g(\(i\))g(of)g(Theorem)g(8) h(implies)h(that)e(if)h(a)f(v)o(ector)g(0)1677 498 y Fr(a)1697 514 y Fv(1)1720 498 y Fr(b)1737 514 y Fv(2)1760 498 y Fr(c)1777 514 y Fq(d)1801 498 y Fr(d)1838 514 y Fn(2)f Fq(C)60 599 y Fv(then)d Fq(b)5 b Fn(\000)g Fq(d)g Fv(+)g(2)p Fq(c)14 b Fn(\021)405 581 y Fs(1)p 405 588 18 2 v 405 614 a(2)427 599 y Fv(\()p Fq(b)5 b Fv(+)g Fq(d)p Fv(\))g(+)g(2)p Fq(c)13 b Fv(\()7 b(mo)q(d)20 b(4\),)13 b(i.e.)19 b Fq(b)12 b Fn(\021)h Fv(3)p Fq(d)g Fv(\()7 b(mo)q(d)20 b(8\),)12 b(and)h(so)f Fq(\014)1432 606 y Fs(4)1464 599 y Fv(=)h(diag)q Fn(f)p Fv(1)p Fq(;)8 b(\021)r(;)g Fv(1)p Fq(;)f(\021)1777 582 y Fs(5)1793 599 y Fn(g)13 b(2)g Fq(G)p Fv(.)741 729 y Fq(G)f Fv(=)h Fn(h)p Fq(M)899 736 y Fs(4)918 729 y Fq(;)8 b(\014)965 736 y Fs(4)984 729 y Fn(i)p Fq(;)37 b Fv(order)15 b(192)620 859 y(\010)d(=)966 829 y(1)d(+)i Fq(\025)1071 812 y Fs(18)p 718 849 638 2 v 718 891 a Fv(\(1)f Fn(\000)g Fq(\025)p Fv(\)\(1)f Fn(\000)h Fq(\025)981 877 y Fs(4)1000 891 y Fv(\)\(1)g Fn(\000)g Fq(\025)1141 877 y Fs(8)1160 891 y Fv(\)\(1)f Fn(\000)i Fq(\025)1301 877 y Fs(12)1338 891 y Fv(\))430 1002 y Fq(R)h Fv(:)605 971 y(1)p Fq(;)22 b(B)699 955 y Fs(2)720 971 y Fq(C)756 955 y Fs(2)775 971 y Fq(D)814 955 y Fs(2)834 971 y Fv(\()p Fq(B)888 955 y Fs(4)918 971 y Fn(\000)10 b Fq(C)999 955 y Fs(4)1019 971 y Fv(\)\()p Fq(B)1091 955 y Fs(4)1121 971 y Fv(+)g Fq(D)1205 955 y Fs(4)1225 971 y Fv(\)\()p Fq(C)1297 955 y Fs(4)1326 971 y Fv(+)g Fq(D)1410 955 y Fs(4)1430 971 y Fv(\))p 507 991 1039 2 v 507 1033 a Fq(A;)23 b(B)613 1020 y Fs(4)643 1033 y Fv(+)11 b Fq(C)725 1020 y Fs(4)754 1033 y Fn(\000)g Fq(D)839 1020 y Fs(4)858 1033 y Fq(;)23 b(B)930 1020 y Fs(8)960 1033 y Fv(+)11 b Fq(C)1042 1020 y Fs(8)1071 1033 y Fv(+)g Fq(D)1156 1020 y Fs(8)1175 1033 y Fq(;)23 b(B)1247 1020 y Fs(12)1295 1033 y Fv(+)10 b Fq(C)1376 1020 y Fs(12)1424 1033 y Fn(\000)g Fq(D)1508 1020 y Fs(12)1816 1002 y Fv(\(145\))60 1150 y Fu(sw)o(e)16 b(and)i(h)o(w)o(e)e(of)i Fq(C)s Fu(,(550)1134 y y fn(2)c fq(s)s fU(s)44 b Fv(相同)16 b(as)e((140))g((140))h((141))f(RESP)q(SEDV)O(El)L(…)60 Fu(CW)O(E)J(Of)G Fq(S)S Fu(S)B B Fv(α)B(图像)H(of)ε(\(α))G(下)I Fq(a)d Fn(?)f f n(006)p fv(1)gfq(b)r fv(,)i fq(b)h fn(n)d Fq(\ 021)R(c)q(;)8 b(c)13 b fn(!)gfq(a)p fv(,)i fq(d)e fn(!)g Fq(\021)1477 1282 y Fs(3)1496 1298 y Fq(D)698 1429 y Fv(\010)g(=)948 1398 y(\(1)c(+)i Fq(\025)1071 1381 y Fs(10)1108 1398 y Fv(\))p 797 1418 480 2 v 797 1460 a(\(1)e Fn(\000)i Fq(\025)p Fv(\)\(1)e Fn(\000)h Fq(\025)1060 1447 y Fs(4)1079 1460 y Fv(\))1097 1447 y Fs(2)1117 1460 y Fv(\(1)f Fn(\000)i Fq(\025)1240 1447 y Fs(8)1259 1460 y Fv(\))655 1590 y Fq(R)i Fv(:)796 1559 y(1)p Fq(;)22 b(A)888 1542 y Fs(2)908 1559 y Fq(C)944 1542 y Fs(2)963 1559 y Fq(D)1002 1542 y Fs(2)1022 1559 y Fv(\()p Fn(\000)p Fq(A)1109 1542 y Fs(4)1139 1559 y Fn(\000)10 b Fq(C)1220 1542 y Fs(4)1239 1559 y Fv(\))p 733 1579 587 2 v 733 1621 a Fq(B)r(;)23 b(A)839 1608 y Fs(4)869 1621 y Fn(\000)11 b Fq(C)951 1608 y Fs(4)970 1621 y Fq(;)32 b Fn(\000)11 b Fq(D)1100 1608 y Fs(4)1120 1621 y Fq(;)32 b Fn(\000)10 b Fq(A)1244 1608 y Fs(4)1264 1621 y Fq(C)1300 1608 y Fs(4)1816 1590 y Fv(\(146\))60 1738 y Fu(sw)o(e)16 b(of)i Fq(S)s Fu(:)44 b Fv(the)16 b(image)f(of)g(\(140\))e(under)j Fq(A)d Fn(!)g fq(b)r fv(,)j fq(b)f fn(!)E FQ(021)R(C)S FV(,)H FQ(C)H FN(!)e Fq(A)708 1868 y Fv(\010)g(=)1025 1838 y(1)p 807 1858 460 2 v 807 1900 a(\(1)c Fn(\000)i Fq(\025)p Fv(\)\(1)d Fn(\000)j Fq(\025)1070 1886 y Fs(4)1089 1900 y Fv(\)\(1)e Fn(\000)i Fq(\025)1230 1886 y Fs(8)1249 1900 y Fv(\))730 2021 y Fq(R)i Fv(:)1015 1990 y(1)p 808 2010 438 2 v 808 2052 a Fq(B)r(;)23 b(A)914 2039 y Fs(4)944 2052 y Fn(\000)10 b Fq(C)1025 2039 y Fs(4)1045 2052 y Fq(;)32 b Fn(\000)11 b Fq(A)1170 2039 y Fs(4)1189 2052 y Fq(C)1225 2039 y Fs(4)1816 2021 y Fv(\(147\))60 2132 y(It)k(follo)o(ws)g(that)g(the)g (norms)g(of)g(v)o(ectors)f(in)i(the)f(shado)o(w)g(are)g(congruen)o(t)g (to)g Fq(n)d Fv(mo)q(d)28 b(8.)60 2281 y Fu(cw)o(e)17 b(of)g Fq(S)s Fu(,)g Fv(1)306 2264 y Fr(n)342 2281 y Fn(2)c Fq(C)s Fu(:)44 b Fv(the)16 b(image)f(of)g(\(143\))e(under)j Fq(A)d Fn(!)gfq(b)r fv(,)j fq(b)f fn(!)E FQ(\ 021)R(c)sv fv(,)h fq(c)h fn(!)e Fq(a)p fv(,)i fq(d)f fn(!)f Fq(\021)1657 2264 y Fs(3)1676 2281 y Fq(D)q Fv(.)60 2429 y Fu(sw)o(e)j(of)i Fq(S)s Fu(,)e Fn(\006)p Fv(1)338 2412 y Fr(n)374 2429 y Fn(2)d Fq(C)s Fu(:)769 2514 y Fv(\010)f(=)966 2483 y(1)d(+)i Fq(\025)1071 2466 y Fs(12)p 867 2503 340 2 v 867 2545 a Fv(\(1)e Fn(\000)i Fq(\025)990 2532 y Fs(4)1009 2545 y Fv(\)\(1)e Fn(\000)i Fq(\025)1150 2532 y Fs(8)1169 2545 y Fv(\))1187 2532 y Fs(2)554 2656 y Fq(R)i Fv(:)887 2625 y(1)p Fq(;)32 b Fn(\000)10 b Fq(A)1034 2609 y Fs(4)1054 2625 y Fq(B)1090 2609 y Fs(4)1111 2625 y Fq(C)1147 2609 y Fs(4)p 632 2645 790 2 v 632 2687 a Fq(A)666 2674 y Fs(4)696 2687 y Fv(+)g Fq(B)777 2674 y Fs(4)808 2687 y Fn(\000)g Fq(C)889 2674 y Fs(4)909 2687 y Fq(;)22 b(A)978 2674 y Fs(8)1008 2687 y Fv(+)10 b Fq(B)1089 2674 y Fs(8)1120 2687 y Fv(+)g Fq(C)1201 2674 y Fs(8)1221 2687 y Fq(;)32 b Fn(\000)11 b Fq(A)1346 2674 y Fs(4)1366 2687 y Fq(C)1402 2674 y Fs(4)1816 2656 y Fv(\(148\))967 2853 y(63)p eop %%Page: 64 68 64 67 bop 60 74 a Fu(cw)o(e)17 b(of)g Fq(S)s Fu(,)g Fv(1)306 58 y Fr(n)342 74 y Fn(2)c Fq(S)s Fu(:)432 205 y Fq(R)g Fv(:)609 174 y(1)p Fq(;)22 b(A)701 158 y Fs(2)721 174 y Fq(C)757 158 y Fs(2)776 174 y Fq(D)815 158 y Fs(2)835 174 y Fv(\()p Fq(A)887 158 y Fs(4)917 174 y Fv(+)10 b Fq(C)998 158 y Fs(4)1018 174 y Fv(\)\()p Fq(C)1090 158 y Fs(4)1119 174 y Fv(+)g Fq(D)1203 158 y Fs(4)1223 174 y Fv(\)\()p Fq(A)1293 158 y Fs(4)1322 174 y Fn(\000)g Fq(D)1406 158 y Fs(4)1426 174 y Fv(\))p 510 194 1033 2 v 510 236 a Fq(B)r(;)23 b(A)616 223 y Fs(4)646 236 y Fn(\000)10 b Fq(C)727 223 y Fs(4)757 236 y Fv(+)h Fq(D)842 223 y Fs(4)861 236 y Fq(;)22 b(A)930 223 y Fs(8)960 236 y Fv(+)11 b Fq(C)1042 223 y Fs(8)1071 236 y Fv(+)g Fq(D)1156 223 y Fs(8)1175 236 y Fq(;)23 b(A)1245 223 y Fs(12)1292 236 y Fn(\000)11 b Fq(C)1374 223 y Fs(12)1421 236 y Fv(+)f Fq(D)1505 223 y Fs(12)1816 205 y Fv(\(149\))60 353 y Fu(sw)o(e)16 b(of)i Fq(S)s Fu(,)e Fn(\006)p Fv(1)338 337 y Fr(n)374 353 y Fn(2)d Fq(S)s Fu(:)45 b Fv(same)15 b(as)g(\(147\).)60 501 y Fu(cw)o(e)i(of)g Fq(W)269 485 y Fs(\(1\))327 501 y Fn(\000)10 b Fq(W)421 485 y Fs(\(3\))469 501 y Fu(:)523 632 y Fq(G)i Fv(=)h Fn(h)p Fq(M)681 639 y Fs(4)701 632 y Fq(;)8 b(\015)746 639 y Fs(4)777 632 y Fv(=)13 b(diag)q Fn(f)p Fv(1)p Fq(;)8 b(\021)r(;)g Fn(\000)p Fv(1)p Fq(;)f(\021)q Fn(gi)12 b Fq(;)52 b Fv(order)15 b(768)g Fq(;)60 762 y Fv(with)h(c)o(haracter)e Fq(\037)p Fv(\()p Fq(M)452 769 y Fs(4)472 762 y Fv(\))e(=)h Fq(i)566 746 y Fr(n)589 762 y Fv(,)i Fq(\037)p Fv(\()p Fq(\015)687 769 y Fs(4)707 762 y Fv(\))d(=)h Fq(\021)810 746 y Fr(n)620 892 y Fv(\010)f(=)966 862 y(1)d(+)i Fq(\025)1071 845 y Fs(18)p 718 882 638 2 v 718 924 a Fv(\(1)f Fn(\000)g Fq(\025)p Fv(\)\(1)f Fn(\000)h Fq(\025)981 910 y Fs(4)1000 924 y Fv(\)\(1)g Fn(\000)g Fq(\025)1141 910 y Fs(8)1160 924 y Fv(\)\(1)f Fn(\000)i Fq(\025)1301 910 y Fs(12)1338 924 y Fv(\))510 1053 y Fq(R)h Fv(:)620 1023 y(1)p Fq(;)c(A)698 1006 y Fs(2)717 1023 y Fq(B)753 1006 y Fs(2)773 1023 y Fq(C)809 1006 y Fs(2)829 1023 y Fv(\()p Fq(A)881 1006 y Fs(4)910 1023 y Fv(+)j Fq(B)992 1006 y Fs(4)1012 1023 y Fv(\)\()p Fq(A)1082 1006 y Fs(4)1111 1023 y Fv(+)g Fq(C)1193 1006 y Fs(4)1212 1023 y Fv(\)\()p Fq(B)1284 1006 y Fs(4)1314 1023 y Fn(\000)g Fq(C)1396 1006 y Fs(4)1415 1023 y Fv(\))p 587 1043 879 2 v 587 1085 a Fq(D)q(;)22 b Fv(symmetric)16 b(p)q(olynomials)h(in)f Fq(A)1223 1071 y Fs(4)1242 1085 y Fq(;)8 b Fn(\000)p Fq(B)1334 1071 y Fs(4)1354 1085 y Fq(;)g Fn(\000)p Fq(C)1446 1071 y Fs(4)1816 1053 y Fv(\(150\))60 1202 y Fu(sw)o(e)16 b(of)i Fq(W)267 1185 y Fs(\(1\))324 1202 y Fn(\000)11 b Fq(W)419 1185 y Fs(\(3\))466 1202 y Fu(:)690 1286 y Fv(\010)h(=)966 1256 y(1)d(+)i Fq(\025)1071 1239 y Fs(18)p 788 1276 498 2 v 788 1318 a Fv(\(1)f Fn(\000)g Fq(\025)911 1304 y Fs(4)930 1318 y Fv(\)\(1)f Fn(\000)i Fq(\025)1071 1304 y Fs(8)1090 1318 y Fv(\)\(1)e Fn(\000)i Fq(\025)1231 1304 y Fs(12)1268 1318 y Fv(\))768 1398 y Fq(R)i Fv(:)f(omit)j Fq(D)h Fv(from)e(\(150\))603 b(\(151\))60 1510 y(\(This)15 b(ring)h(has)f(also)g(b)q(een)i(studied)f(in)g([88)o(].\))60 1658 y Fu(cw)o(e)g(of)g Fq(W)267 1641 y Fs(\(1\))322 1658 y Fn(\000)8 b Fq(W)414 1641 y Fs(\(3\))478 1658 y Fu(with)17 b Fv(1)619 1641 y Fr(n)655 1658 y Fn(2)12 b Fq(C)s Fu(:)45 b Fq(G)13 b Fv(=)f Fn(h)p Fq(M)951 1665 y Fs(4)971 1658 y Fq(;)c(\014)1018 1665 y Fs(4)1037 1658 y Fq(;)g(\015)1082 1665 y Fs(4)1100 1658 y Fn(i)p Fv(,)14 b(order)f(6144,)g(with)i(c)o(haracter)e Fq(\037)p Fv(\()p Fq(M)1770 1665 y Fs(4)1790 1658 y Fv(\))f(=)h Fq(i)1884 1641 y Fr(n)1907 1658 y Fv(,)60 1743 y Fq(\037)p Fv(\()p Fq(\014)132 1750 y Fs(4)152 1743 y Fv(\))f(=)h(1,)h Fq(\037)p Fv(\()p Fq(\015)350 1750 y Fs(4)370 1743 y Fv(\))e(=)h Fq(\021)473 1726 y Fr(n)496 1743 y Fv(;)h(k)o(er\()p Fq(\037)p Fv(\))h(has)g(order)g(3072)601 1873 y(\010)d(=)966 1842 y(1)d(+)i Fq(\025)1071 1826 y Fs(32)p 699 1862 675 2 v 699 1904 a Fv(\(1)f Fn(\000)g Fq(\025)822 1891 y Fs(4)842 1904 y Fv(\)\(1)f Fn(\000)h Fq(\025)982 1891 y Fs(8)1001 1904 y Fv(\)\(1)g Fn(\000)g Fq(\025)1142 1891 y Fs(12)1179 1904 y Fv(\)\(1)f Fn(\000)i Fq(\025)1320 1891 y Fs(16)1356 1904 y Fv(\))146 2034 y Fq(R)i Fv(:)224 2003 y(1)p Fq(;)22 b(A)316 1987 y Fs(2)336 2003 y Fq(B)372 1987 y Fs(2)392 2003 y Fq(C)428 1987 y Fs(2)448 2003 y Fq(D)487 1987 y Fs(2)507 2003 y Fv(\()p Fq(A)559 1987 y Fs(4)588 2003 y Fv(+)11 b Fq(B)670 1987 y Fs(4)690 2003 y Fv(\)\()p Fq(A)760 1987 y Fs(4)789 2003 y Fv(+)g Fq(C)871 1987 y Fs(4)890 2003 y Fv(\)\()p Fq(A)960 1987 y Fs(4)990 2003 y Fn(\000)f Fq(D)1074 1987 y Fs(4)1094 2003 y Fv(\)\()p Fq(B)1166 1987 y Fs(4)1196 2003 y Fn(\000)g Fq(C)1277 1987 y Fs(4)1296 2003 y Fv(\)\()p Fq(B)1368 1987 y Fs(4)1398 2003 y Fv(+)h Fq(D)1483 1987 y Fs(4)1502 2003 y Fv(\)\()p Fq(C)1574 1987 y Fs(4)1603 2003 y Fv(+)g Fq(D)1688 1987 y Fs(4)1707 2003 y Fv(\))p 224 2023 1501 2 v 533 2066 a(symmetric)16 b(p)q(olynomials)g(in)g Fq(A)1094 2050 y Fs(4)1114 2066 y Fq(;)8 b Fn(\000)p Fq(B)1206 2050 y Fs(4)1226 2066 y Fq(;)g Fn(\000)p Fq(C)1318 2050 y Fs(4)1337 2066 y Fq(;)g(D)1397 2050 y Fs(4)1816 2034 y Fv(\(152\))60 2182 y Fu(sw)o(e)16 b(of)i Fq(W)267 2166 y Fs(\(1\))324 2182 y Fn(\000)11 b Fq(W)419 2166 y Fs(\(3\))484 2182 y Fu(with)18 b Fv(1)626 2166 y Fr(n)662 2182 y Fn(2)12 b Fq(C)s Fu(:)690 2313 y Fv(\010)g(=)1025 2282 y(1)p 788 2302 498 2 v 788 2344 a(\(1)e Fn(\000)g Fq(\025)911 2331 y Fs(4)930 2344 y Fv(\)\(1)f Fn(\000)i Fq(\025)1071 2331 y Fs(8)1090 2344 y Fv(\)\(1)e Fn(\000)i Fq(\025)1231 2331 y Fs(12)1268 2344 y Fv(\))511 2465 y Fq(R)i Fv(:)1015 2434 y(1)p 589 2455 875 2 v 589 2497 a(symmetric)i(p)q(olynomials)i(in) f Fq(A)1150 2481 y Fs(4)1170 2497 y Fq(;)32 b Fn(\000)11 b Fq(B)1297 2481 y Fs(4)1317 2497 y Fq(;)33 b Fn(\000)10 b Fq(C)1444 2481 y Fs(4)967 2853 y Fv(64)p eop %%Page: 65 69 65 68 bop 60 74 a Fo(7.11.)41 b(F)-5 b(amily)17 b Fg(4)425 56 y Fp(Z)425 87 y Fs(I)q(I)454 74 y Fo(:)24 b(T)n(yp)r(e)18 b(I)r(I)g(self-dual)g(co)r(des)g(o)n(v)n(er)h Ft(Z)1227 81 y Fs(4)60 188 y Fu(cw)o(e)d(of)h Fq(C)s Fu(,)g Fv(1)310 172 y Fr(n)346 188 y Fn(2)c Fq(C)48 b Fv([51)o(],)14 b([27)o(])g(In)i(view)f(of)f(the)h(remarks)f(follo)o(wing)i(Theorem)e (8,)h(this)g(is)g(not)f(a)h(sev)o(ere)60 273 y(restriction.)682 358 y Fq(G)e Fv(=)g Fn(h)p Fq(M)841 365 y Fs(4)860 358 y Fq(;)8 b(\014)907 365 y Fs(4)926 358 y Fq(;)g(\015)971 365 y Fs(4)989 358 y Fn(i)15 b Fq(;)53 b Fv(order)15 b(6144)671 467 y(\010)d(=)859 436 y(\(1)e(+)g Fq(\025)982 419 y Fs(16)1019 436 y Fv(\)\(1)f(+)i Fq(\025)1160 419 y Fs(32)1197 436 y Fv(\))p 769 456 535 2 v 769 498 a(\(1)f Fn(\000)g Fq(\025)892 485 y Fs(8)912 498 y Fv(\))930 485 y Fs(2)949 498 y Fv(\(1)g Fn(\000)g Fq(\025)1072 485 y Fs(16)1109 498 y Fv(\)\(1)f Fn(\000)h Fq(\025)1249 485 y Fs(24)1286 498 y Fv(\))367 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Fv(,)h(4)1803 1728 y Fp(Z)1824 1745 y Fv(\).)60 1892 y Fu(Remark.)44 b Fv(F)l(or)19 b(a)g(co)q(de)h Fq(C)i Fv(from)c(family)i Fq(q)869 1875 y Fs(H)917 1892 y Fv(\(linear)g(o)o(v)o(er)e Fq(F)1195 1899 y Fr(q)1214 1892 y Fv(,)i Fq(q)h Fv(a)e(square,)h(with)g (Hermitian)g(inner)60 1976 y(pro)q(duct\),)15 b(it)g(can)h(b)q(e)g(sho) o(wn)e(that)h(the)g(p)q(olynomial)499 2098 y Fq(S)s Fv(\()p Fq(x;)8 b(y)r Fv(\))i(=)j Fq(q)717 2080 y Fl(\000)p Fr(n=)p Fs(2)803 2098 y Fq(W)6 b Fv(\(\()888 2068 y Fn(p)p 926 2068 22 2 v 30 x Fq(q)12 b Fn(\000)e Fv(1\))p Fq(x)g Fv(+)g(\()1143 2068 y Fn(p)p 1181 2068 V 30 x Fq(q)i Fv(+)e(1\))p Fq(y)r(;)22 b(y)12 b Fn(\000)f Fq(x)p Fv(\))60 2221 y(has)f(nonnegativ)o(e)h(\(but)f(not)g(necessarily)i(in)o (tegral\))f(co)q(e\016cien)o(ts;)h(note)f(that)e(this)i(agrees)f(with)h (the)g(shado)o(w)60 2305 y(en)o(umerator)h(for)f Fq(q)k Fv(=)e(4.)18 b(This)13 b(can)g(b)q(e)g(used)g(to)f(strengthen)g(the)g (LP)h(b)q(ound)g(in)h(those)e(cases.)19 b(The)12 b(kno)o(wn)60 2390 y(pro)q(of)i(that)g(this)g(is)h(nonnegativ)o(e)g(in)o(v)o(olv)o (es)g(constructing)f(a)g(quan)o(tum)g(co)q(de)h Fq(Q)f Fv(from)g Fq(C)j Fv(\([248)o(]\);)c Fq(S)s Fv(\()p Fq(x;)8 b(y)r Fv(\))60 2475 y(is)19 b(then)g(the)g(shado)o(w)f(en)o(umerator)g (of)g Fq(Q)g Fv(\([251)o(],)g(pro)o(v)o(ed)h(nonnegativ)o(e)f(in)i ([252)o(]\).)29 b(There)19 b(is)g(surely)g(a)60 2559 y(more)c(direct)h(pro)q(of.)131 2644 y(One)e(w)o(a)o(y)e(to)h(apply)h (the)f(linear)h(programming)f(b)q(ound)h(is)g(to)f(ignore)g(the)h (constrain)o(t)f(that)f(the)i(co)q(e\016-)60 2729 y(cien)o(ts)f(of)g Fq(W)6 b Fv(\()p Fq(x;)i(y)r Fv(\))j(b)q(e)i(nonnegativ)o(e,)h(and)f (simply)h(ask)e(that)g(the)h(lo)o(w)f(order)h(co)q(e\016cien)o(ts)g(b)q (e)h(as)e(sp)q(eci\014ed.)967 2853 y(69)p eop %%Page: 70 74 70 73 bop 60 74 a Fv(This)16 b(giv)o(es)g(a)g(surprisingly)h(go)q(o)q (d)f(b)q(ound)g(for)f(T)o(yp)q(e)h(I)q(I)h(binary)g(co)q(des.)k(Recall) d(from)d(Theorem)h(13)f(that)60 159 y(for)g Fq(C)i Fv(of)e(T)o(yp)q(e)h (I)q(I,)g Fq(W)6 b Fv(\()p Fq(x;)i(y)r Fv(\))13 b(lies)k(in)f(the)f (ring)586 289 y Fq(R)d Fv(=)h Ft(C)6 b Fv([)p Fq(x)747 271 y Fs(8)780 289 y Fv(+)k(14)p Fq(x)897 271 y Fs(4)916 289 y Fq(y)940 271 y Fs(4)970 289 y Fv(+)h Fq(y)1040 271 y Fs(8)1059 289 y Fq(;)d(x)1106 271 y Fs(4)1125 289 y Fq(y)1149 271 y Fs(4)1169 289 y Fv(\()p Fq(x)1213 271 y Fs(4)1242 289 y Fn(\000)j Fq(y)1312 271 y Fs(4)1331 289 y Fv(\))1349 271 y Fs(4)1369 289 y Fv(])p Fq(;)60 420 y Fv(and)18 b(if)g Fq(C)i Fv(has)e(length)g Fq(n)p Fv(,)g Fq(W)6 b Fv(\()p Fq(x;)i(y)r Fv(\))17 b(has)g(degree)h Fq(n)p Fv(.)28 b(The)18 b(subspace)g(of)f Fq(R)h Fv(of)f(degree)h Fq(n)g Fv(has)f(dimension)60 504 y Fq(D)d Fv(=)e([)184 487 y Fr(n)p 177 494 36 2 v 177 520 a Fs(24)217 504 y Fv(])7 b(+)g(1.)20 b(This)14 b(lets)h(us)e(set)h(the)g(\014rst)f Fq(D)i Fv(co)q(e\016cien)o(ts)g(of)e Fq(W)6 b Fv(\()p Fq(x;)i(y)r Fv(\))k(arbitrarily;)j(in)g(particular,)f(there)60 589 y(exists)19 b(a)f(unique)i(elemen)o(t)g Fq(W)602 573 y Fl(\003)622 589 y Fv(\()p Fq(x;)8 b(y)r Fv(\))17 b(of)h Fq(R)g Fv(suc)o(h)h(that)f Fq(W)1111 573 y Fl(\003)1131 589 y Fv(\(1)p Fq(;)8 b(y)r Fv(\))16 b(=)j(1)12 b(+)h Fq(O)q Fv(\()p Fq(y)1466 573 y Fs(4)p Fr(D)1515 589 y Fv(\).)30 b(This)19 b(is)g(kno)o(wn)f(as)60 674 y(the)f Fw(extr)n(emal)f Fv(en)o(umerator,)g(since)i Fq(W)741 657 y Fl(\003)777 674 y Fv(has)f(the)f(largest)h(minimal)h(distance)f (of)g(an)o(y)f(T)o(yp)q(e)h(I)q(I)g(self-dual)60 759 y(en)o(umerator.)i(It)14 b(follo)o(ws)g(immediately)i(that)e(the)g (minimal)i(distance)f(of)f(an)o(y)f(T)o(yp)q(e)i(I)q(I)g(co)q(de)g(of)f (length)h Fq(n)60 843 y Fv(is)h(b)q(ounded)g(ab)q(o)o(v)o(e)f(b)o(y)g (the)h(minimal)g(distance)g(of)f Fq(W)1016 827 y Fl(\003)1036 843 y Fv(.)60 978 y Fu(Theorem)i(26.)22 b Fv([195)o(])e Fw(The)f(\014rst)h(nonzer)n(o)e(c)n(o)n(e\016cient)h(of)h Fq(W)1176 961 y Fl(\003)1196 978 y Fv(\(1)p Fq(;)8 b(y)r Fv(\))18 b Fw(o)n(c)n(curs)i(pr)n(e)n(cisely)e(at)i(de)n(gr)n(e)n(e)f Fv(4)p Fq(D)q Fw(;)60 1062 y(in)g(p)n(articular,)h(the)g(minimal)e (distanc)n(e)h(of)g(a)g(T)m(yp)n(e)f(II)h(self-dual)g(binary)g(c)n(o)n (de)f(of)i(length)e Fq(n)i Fw(is)e(at)i(most)60 1147 y Fv(4[)p Fq(n=)p Fv(24])9 b(+)h(4)p Fw(.)131 1282 y Fv(In)20 b(fact)g(it)g(is)g(p)q(ossible)i(to)d(use)i(the)f(B)q(\177)-24 b(urmann-Lagrange)20 b(theorem)f(\(Theorem)h(32\))f(to)g(deriv)o(e)i (an)60 1366 y(explicit)k(form)o(ula)d(for)g(the)h(n)o(um)o(b)q(er)g(of) f(w)o(ords)g(of)g(w)o(eigh)o(t)h(4)p Fq(D)g Fv(in)h(the)e(extremal)h (en)o(umerator.)42 b(Let)60 1451 y Fq(\026)13 b Fv(=)g([)p Fq(n=)p Fv(24],)g(so)i(that)g Fq(D)e Fv(=)g Fq(\026)e Fv(+)f(1.)20 b(Then)15 b(w)o(e)g(ha)o(v)o(e)60 1573 y Fu(Theorem)i(27.)22 b Fv(\(Mallo)o(ws)j(and)f(Sloane)i([195)n(].\))46 b Fq(A)1031 1556 y Fl(\003)1031 1585 y Fs(4)p Fr(\026)p Fs(+4)1117 1573 y Fw(,)27 b(the)e(numb)n(er)g(of)g(c)n(o)n(dewor)n(ds)f (of)h(minimal)60 1658 y(nonzer)n(o)15 b(weight)i Fv(4)p Fq(D)c Fv(=)g(4)p Fq(\026)d Fv(+)h(4)16 b Fw(in)f(the)i(extr)n(emal)f (weight)h(enumer)n(ator,)f(is)g(given)g(by:)567 1716 y Fm( )600 1757 y Fq(n)602 1819 y Fv(5)627 1716 y Fm(! )(693)1757 y fv(5)p fq(026)10 b fn(000)h fv(2)704 1819 y fq(026)g fn(000)f fv(1)821 y y fM(!)854 1741 y(888)1716 y()921 1757 y fv(4)p fq(026)g fv(+)h(4)974 1819 y y(5)5 y fM(!)(\026)J((403)b)Fv(\(162)),436,1923,(y)y fq(n)p fv(()p fq(n),b fn(α)g fv(ω)(p)fq(n)g fn(α)gfv(α)(p)fq(n)g fn(α)g fv(ω)y y((α)p fq(α)p fv(\))1082 1788 Y FQ(;)54 B Fw(IF)46 B fq(n)13 b Fv(=)g(24)p fqp 919 1943 217 217 V 919 1985 1985 fq(\026)p Fv(!)(4)p fq(026)g fv(+)g(4)!(54)B(Fw)(46)B Fq(n)13 b Fv(=)p Fq(\026)d Fv(+)G(8)16 b Fq();267πFv(\(163))566 y(α)pαv Vα(α)y fq(n)p fv(\)p fq(n)b b fn(α)f fv(α)y((α)p fq(\)g fv(+)h(ω))1140 1954 Y FQp 767 2082 217 217 V 767 767 2123 Fq(\026)p Fv(!)(4)p Fq(026)E Fv(+)I(4)!988 2092 y Fq(;)53 b Fw(if)47 b Fq(n)13 b Fv(=)g(24)p Fq(\026)d Fv(+)g(16)16 b Fq(;)396 b Fv(\(164\))60 2204 y Fw(and)16 b(is)g(never)g(zer)n(o.)131 2326 y Fv(F)l(or)e(the)i(pro)q(of,)f(see)h ([195)o(])f(or)g([189)o(],)f(Chapter)i(19.)k(There)c(is)g(a)f(similar)i (form)o(ula)e(for)g(T)o(yp)q(e)g(I)h(binary)60 2410 y(co)q(des)g(|)f (see)h([189)o(],)e(Chapter)h(19,)f(Problem)i(\(12\).)131 2495 y(Results)g(similar)g(to)f(Theorem)g(26)f(hold)i(for)f(other)g (families:)60 2630 y Fu(Theorem)i(28.)22 b Fw(The)c(minimal)f(distanc)n (e)f(of)i(a)g(T)m(yp)n(e)e(I)h(binary)h(self-dual)f(c)n(o)n(de)g(is)g (at)h(most)f Fv(2[)p Fq(n=)p Fv(8])10 b(+)h(2)p Fw(.)60 2714 y(The)17 b(minimal)f(distanc)n(e)g(of)h(a)h(T)m(yp)n(e)d(II)h (binary)h(self-dual)g(c)n(o)n(de)f(is)h(at)g(most)g Fv(4[)p Fq(n=)p Fv(24])9 b(+)i(4)p Fw(.)23 b(The)16 b(minimal)967 2853 y Fv(70)p eop %%Page: 71 75 71 74 bop 60 74 a Fw(distanc)n(e)17 b(of)h(a)g(self-dual)g(c)n(o)n(de)g (fr)n(om)g(family)g Fv(3)f Fw(is)h(at)g(most)g Fv(3[)p Fq(n=)p Fv(12])10 b(+)i(3)p Fw(.)25 b(The)18 b(minimal)g(distanc)n(e)f (of)h(a)60 159 y(self-dual)c(c)n(o)n(de)f(fr)n(om)i(family)e Fv(4)599 143 y Fs(H)642 159 y Fw(is)g(at)h(most)h Fv(2[)p 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y(for)f(an)o(y)g(constan)o(t)f Fq(\013)p Fv(,)i(the)f(minimal)i(distance)f(can)g(b)q(e)g(within)g Fq(\013)g Fv(of)e(the)i(b)q(ound)g(only)g(\014nitely)h(often.)60 1728 y(F)l(or)c(T)o(yp)q(e)h(I)q(I)h(binary)f(co)q(des,)g(for)f (instance,)i(it)f(w)o(as)e(sho)o(wn)i(in)g([192)o(])f(that)g(the)h Fq(A)1524 1711 y Fl(\003)1524 1739 y Fs(4)p Fr(n)p Fs(+8)1628 1728 y Fv(term)f(\014rst)g(go)q(es)60 1813 y(negativ)o(e)e(when)g Fq(n)g Fv(is)g(around)g(3720.)j(Ma)c(and)h(Zh)o(u)g([183)o(])f(and)h (Zhang)f([337)o(])g(ha)o(v)o(e)g(recen)o(tly)i(determined)60 1897 y(precisely)f(when)e(the)g Fq(A)470 1881 y Fl(\003)470 1909 y Fs(4)p Fr(n)p Fs(+8)570 1897 y Fv(term)f(\014rst)h(go)q(es)f (negativ)o(e,)i(and)f(ha)o(v)o(e)f(obtained)i(similar)g(results)f(for)g (sev)o(eral)60 1982 y(other)i(families.)21 b(The)16 b(follo)o(wing)f (result)h(incorp)q(orates)f(the)h(w)o(ork)e(of)h(sev)o(eral)g(authors.) 60 2099 y Fu(Theorem)i(29.)22 b Fv([337)o(])c Fw(L)n(et)e Fq(C)21 b 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Fw(go)n(es)f(ne)n(gative)f(for)i(T)m(yp)n(e)e(II)g(c)n(o)n(des)h (is)g(at)g Fv(24)10 b Fn(\002)g Fv(154)i(=)h(3696)p Fw(.)967 2853 y Fv(71)p eop %%Page: 72 76 72 75 bop 131 74 a Fv(Of)13 b(course)g(other)f(co)q(e\016cien)o(ts)i (in)g(the)f(extremal)g(w)o(eigh)o(t)g(en)o(umerator)f(ma)o(y)g(go)h (negativ)o(e)g(b)q(efore)g(this.)60 159 y(In)19 b(the)f(case)h(of)f (ternary)g(self-dual)i(co)q(des,)f(for)f(example,)h(family)g(3,)g(the)f (extremal)h(Hamming)f(w)o(eigh)o(t)60 244 y(en)o(umerator)c(con)o (tains)i(a)f(negativ)o(e)g(co)q(e\016cien)o(t)h(for)f(lengths)g(72,)g (96,)f(120)g(and)i(all)g Fq(n)d Fn(\025)g Fv(144.)131 328 y(The)i(b)q(est)g(asymptotic)g(b)q(ound)h(presen)o(tly)g(kno)o(wn)f (for)g(T)o(yp)q(e)g(I)q(I)h(co)q(des)g(is)g(the)f(follo)o(wing.)60 449 y Fu(Theorem)i(30.)22 b Fv(\(Krasik)o(o)o(v)15 b(and)h(Litsyn)g ([172)o(].\))k Fw(The)c(minimal)g(distanc)n(e)f Fq(d)h Fw(of)h(a)f(T)m(yp)n(e)g(II)f(binary)h(c)n(o)n(de)60 533 y(of)g(length)g Fq(n)h Fw(satis\014es)625 618 y Fq(d)12 b Fn(\024)h Fv(0)p Fq(:)p Fv(166315)8 b Fq(:)f(:)h(:)t(n)i Fv(+)h Fq(o)p Fv(\()p Fq(n)p Fv(\))p Fq(;)53 b(n)13 b Fn(!)g(1)k Fq(:)60 729 y Fw(The)f(c)n(onstant)f(in)h(this)g(expr)n (ession)f(is)h(the)g(r)n(e)n(al)g(r)n(o)n(ot)g(of)h Fv(8)p Fq(x)1102 712 y Fs(5)1131 729 y Fn(\000)11 b Fv(24)p Fq(x)1249 712 y Fs(4)1278 729 y Fv(+)f(40)p Fq(x)1395 712 y Fs(3)1425 729 y Fn(\000)g Fv(30)p Fq(x)1542 712 y Fs(2)1571 729 y Fv(+)h(10)p Fq(x)e Fn(\000)i Fv(1)p Fw(.)60 849 y Fv(The)k(pro)q(of)g(uses)h(a)f(v)m(arian)o(t)g(of)g(the)g (linear)h(programming)f(b)q(ound.)131 934 y(F)l(or)k(T)o(yp)q(e)h(I)g (binary)g(co)q(des,)h(the)f(b)q(ound)g(of)g(Theorem)f(28)g(is)i(esp)q (ecially)h(w)o(eak.)33 b(W)l(ard)19 b([318)o(])g(has)60 1018 y(sho)o(wn)e(that)g(the)g(minimal)i(distance)f(can)g(b)q(e)g(2[)p Fq(n=)p Fv(8])10 b(+)i(2)17 b(precisely)i(when)f Fq(n)g Fv(is)g(one)f(of)g(2,)g(4,)g(6,)h(8,)f(12,)60 1103 y(14,)f(22)g(or)h (24.)24 b(This)17 b(suggests)f(that)h(the)f(b)q(ound)i(can)f(b)q(e)h 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1419 y Fs(2[)p Fr(n=)p Fs(24]+2)820 1410 y Fv(,)f(w)o(e)g(note)g(that)465 1486 y Fm(X)489 1577 y Fr(i)533 1527 y Fq(c)553 1534 y Fr(i)566 1527 y Fv(\(1)10 b(+)g Fq(y)686 1508 y Fs(2)706 1527 y Fv(\))724 1508 y Fr(n=)p Fs(2)p Fl(\000)p Fs(4)p Fr(i)839 1527 y Fv(\()p Fq(y)881 1508 y Fs(2)901 1527 y Fv(\(1)f Fn(\000)i Fq(y)1021 1508 y Fs(2)1040 1527 y Fv(\))1058 1508 y Fs(2)1078 1527 y Fv(\))1096 1508 y Fr(i)1122 1527 y Fv(=)i(1)d(+)g Fq(O)q Fv(\()p Fq(y)1326 1508 y Fs(4[)p Fr(n=)p Fs(24]+6)1485 1527 y Fv(\))p Fq(:)60 1660 y Fv(Dividing)17 b(b)q(oth)e(sides)h(b)o(y) g(\(1)9 b(+)h Fq(y)644 1644 y Fs(2)664 1660 y Fv(\))682 1644 y Fr(n=)p Fs(2)756 1660 y Fv(and)15 b(substituting)h Fq(y)f Fv(=)1182 1622 y Fn(p)p 1220 1622 37 2 v 38 x Fq(Y)10 b Fv(,)15 b(w)o(e)g(get:)472 1750 y Fm(X)496 1841 y Fr(i)539 1791 y Fq(c)559 1798 y Fr(i)581 1719 y Fm( )618 1760 y Fq(Y)c Fv(\(1)e Fn(\000)i Fq(Y)f Fv(\))805 1743 y Fs(2)p 618 1780 207 2 v 637 1822 a Fv(\(1)f(+)i Fq(Y)f Fv(\))787 1809 y Fs(4)830 1719 y Fm(!)863 1730 y Fr(i)889 1791 y Fv(=)j(\(1)d(+)g Fq(Y)h Fv(\))1088 1772 y Fl(\000)p Fr(n=)p Fs(2)1184 1791 y Fv(+)f Fq(O)q Fv(\()p Fq(Y)1319 1772 y Fs(2[)p Fr(n=)p Fs(24]+3)1478 1791 y Fv(\))p Fq(:)60 1914 y Fv(W)l(e)15 b(can)h(then)f(apply)h(B)q (\177)-24 b(urmann-Lagrange,)15 b(with)493 2031 y Fq(f)5 b Fv(\()p Fq(Y)11 b Fv(\))h(=)h(\(1)c(+)i Fq(Y)f Fv(\))803 2012 y Fr(n=)p Fs(2)862 2031 y Fq(;)52 b(g)r Fv(\()p Fq(Y)10 b Fv(\))i(=)h Fq(Y)e Fv(\(1)e Fn(\000)i Fq(Y)f Fv(\))1270 2012 y Fs(2)1289 2031 y Fv(\(1)g(+)g Fq(Y)h Fv(\))1440 2012 y Fl(\000)p Fs(4)60 2148 y Fv(to)k(obtain)342 2265 y Fq(c)362 2272 y Fr(i)417 2265 y Fv(=)499 2234 y(1)p 499 2254 23 2 v 503 2296 a Fq(i)527 2265 y Fv([co)q(e\013.)k(of)c Fq(Y)751 2248 y Fr(i)p Fl(\000)p Fs(1)826 2265 y Fv(in)h([)911 2247 y Fr(d)p 897 2254 49 2 v 897 2280 a(d)r(Y)950 2265 y Fv(\(1)9 b(+)i Fq(Y)f Fv(\))1100 2248 y Fl(\000)p Fr(n=)p Fs(2)1186 2265 y Fv(])1207 2230 y Fm(\000)1225 2265 y Fv(\(1)g(+)g Fq(Y)h Fv(\))1376 2248 y Fs(4)1395 2265 y Fv(\(1)f Fn(\000)g Fq(Y)h Fv(\))1546 2248 y Fl(\000)p Fs(2)1592 2230 y Fm(\001)1611 2240 y Fr(i)1625 2265 y Fv(])417 2370 y(=)499 2339 y Fn(\000)p Fq(n)p 499 2359 63 2 v 511 2401 a Fv(2)p Fq(i)567 2370 y Fv([co)q(e\013.)19 b(of)c Fq(Y)791 2353 y Fr(i)p Fl(\000)p Fs(1)866 2370 y Fv(in)h(\(1)9 b(+)i Fq(Y)f Fv(\))1069 2353 y Fl(\000)p Fr(n=)p Fs(2)p Fl(\000)p Fs(1+4)p Fr(i)1257 2370 y Fv(\(1)g Fn(\000)g Fq(Y)h Fv(\))1408 2353 y Fl(\000)p Fs(2)p Fr(i)1466 2370 y Fv(])417 2475 y(=)499 2444 y Fn(\000)p Fq(n)p 499 2464 V 511 2506 a Fv(2)p Fq(i)567 2475 y Fv([co)q(e\013.)19 b(of)c Fq(Y)791 2458 y Fr(i)p Fl(\000)p Fs(1)866 2475 y Fv(in)h(\(1)9 b(+)i Fq(Y)f Fv(\))1069 2458 y Fl(\000)p Fr(n=)p Fs(2)p Fl(\000)p Fs(1+6)p Fr(i)1257 2475 y Fv(\(1)g Fn(\000)g Fq(Y)1390 2458 y Fs(2)1409 2475 y Fv(\))1427 2458 y Fl(\000)p Fs(2)p Fr(i)1486 2475 y Fv(])p Fq(:)60 2592 y Fv(In)16 b(particular,)f(for)g Fq(i)d Fv(=)h(2[)p Fq(n=)p Fv(24])c(+)h(2,)110 2709 y Fq(c)130 2718 y Fs(2[)p Fr(n=)p Fs(24]+2)301 2709 y Fv(=)434 2678 y Fn(\000)p Fq(n)p 354 2699 222 2 v 354 2740 a Fv(4[)p Fq(n=)p Fv(24])f(+)h(4)581 2709 y([co)q(e\013.)19 b(of)c Fq(Y)806 2693 y Fs(2[)p Fr(n=)p Fs(24]+1)980 2709 y Fv(in)h(\(1)9 b(+)i Fq(Y)f Fv(\))1183 2693 y Fl(\000)p Fr(n=)p Fs(2+12[)p Fr(n=)p Fs(24]+11)1488 2709 y Fv(\(1)g Fn(\000)g Fq(Y)1621 2693 y Fs(2)1641 2709 y Fv(\))1659 2693 y Fl(\000)p Fs(4[)p Fr(n=)p Fs(24])p Fl(\000)p Fs(4)1844 2709 y Fv(])p Fq(:)967 2853 y Fv(73)p eop %%Page: 74 78 74 77 bop 60 74 a Fv(It)17 b(follo)o(ws)f(that)g Fq(c)381 84 y Fs(2[)p Fr(n=)p Fs(24]+2)554 74 y Fn(\024)f Fv(0,)h(with)h (equalit)o(y)h(only)f(when)g Fq(n)e Fn(\021)g Fv(22)h(\()r(mo)q(d)h (24\),)f(since)i(all)f(co)q(e\016cien)o(ts)60 159 y(of)e(an)o(y)g(p)q (o)o(w)o(er)f(series)i(of)f(the)g(form)g(\(1)9 b(+)i Fq(Y)f Fv(\))840 143 y Fr(a)861 159 y Fv(\(1)f Fn(\000)i Fq(Y)993 143 y Fs(2)1013 159 y Fv(\))1031 143 y Fl(\000)p Fr(b)1090 159 y Fv(are)k(p)q(ositiv)o(e)h(whenev)o(er)g Fq(a)p Fv(,)f Fq(b)d(>)h Fv(0.)131 244 y(Similarly)l(,)18 b(w)o(e)d(\014nd)i(that)e(the)h(co)q(e\016cien)o(ts)h(of)f(the)g (expansion)h(of)e Fq(c)1318 253 y Fs(2[)p Fr(n=)p Fs(24]+2)1493 244 y Fv(in)i(terms)e(of)g(the)i Fq(b)1826 251 y Fr(j)1859 244 y Fv(are)60 328 y(p)q(ositiv)o(e.)33 b(This)19 b(pro)o(v)o(es)g (the)g(b)q(ound,)i(except)f(when)f Fq(n)h Fn(\021)f Fv(22)g(\(mo)q(d)f (24\);)i(the)g(pro)q(of)e(that)h(the)g(b)q(ound)60 413 y(holds)d(in)h(that)e(case)g(and)h(that)f(a)h(co)q(de)g(meeting)g(the)g (b)q(ound)g(is)g(ev)o(en)g(if)g Fq(n)e Fn(\021)g Fv(0)h(\()r(mo)q(d)j (24\))d(is)h(left)g(to)f(the)60 498 y(reader.)p 242 496 16 16 v 131 583 a(This)i(b)q(ound)g(agrees)f(with)h(the)f(full)i (linear)g(programming)d(b)q(ound)j(for)e Fq(n)f Fn(\024)g Fv(200,)g(and,)i(most)e(lik)o(ely)l(,)60 667 y(for)f(m)o(uc)o(h)i (larger)e Fq(n)p Fv(.)20 b(Ho)o(w)o(ev)o(er,)14 b(it)i(is)f(lik)o(ely)i (that)d(again)h(it)h(can)f(only)h(b)q(e)f(attained)g(for)g(\014nitely)h (man)o(y)f Fq(n)p Fv(.)131 752 y(There)g(is)h(also)f(an)g(analogue)g 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Fv(\(4\))13 b(this)60 1322 y(agrees)j(with)g(the)g(historical)h (usage.)23 b(F)l(or)15 b(T)o(yp)q(e)i(I)f(binary)h(co)q(des,)f(ho)o(w)o (ev)o(er,)f(\\extremal")h(has)g(generally)60 1407 y(b)q(een)21 b(used)f(to)f(mean)h(a)f(co)q(de)h(meeting)g(the)g(m)o(uc)o(h)g(w)o (eak)o(er)f(b)q(ound)h(of)g(Theorem)f(28;)i(in)f(the)g(ligh)o(t)g(of)60 1491 y(Theorem)15 b(31,)f(it)i(seems)f(appropriate)g(to)g(c)o(hange)g (the)g(de\014nition.)131 1576 y(Concerning)h(co)q(des)f(o)o(v)o(er)g Ft(Z)621 1583 y Fs(4)638 1576 y Fv(,)g(Bonnecaze,)h(Sol)o(\023)-21 b(e,)14 b(Bac)o(ho)q(c)i(and)f(Mourrain)g([27)o(])g(sho)o(w:)60 1682 y Fu(Theorem)i(34.)22 b Fw(Supp)n(ose)c Fq(C)i Fw(is)e(a)g(T)m(yp) n(e)e(II)h(self-dual)h(c)n(o)n(de)f(over)h Ft(Z)1270 1689 y Fs(4)1304 1682 y Fw(of)g(length)f Fq(n)p Fw(.)25 b(Then)17 b(the)h(minimal)60 1767 y(Euclide)n(an)e(norm)g(of)g Fq(C)j Fw(is)d(at)h(most)872 1852 y Fv(8)903 1792 y Fm(\024)938 1821 y Fq(n)p 929 1841 46 2 v 929 1883 a Fv(24)979 1792 y Fm(\025)1011 1852 y Fv(+)11 b(8)16 b Fq(:)707 b Fv(\(165\))131 1958 y(The)16 b(pro)q(of)h(uses)f Fq(C)k Fv(to)c(de\014ne)h(an)g(ev)o (en)g(unimo)q(dular)h Fq(n)p Fv(-dimensional)h(lattice)e(\003\()p Fq(C)s Fv(\))d(=)h Fn(f)1730 1940 y Fs(1)p 1730 1947 18 2 v 1730 1974 a(2)1753 1958 y Fq(u)g Fn(2)g Ft(R)1872 1942 y Fr(n)1907 1958 y Fv(:)60 2043 y Fq(u)g Fv(\()s(mo)q(d)i(4\))12 b Fn(2)h Fq(C)s Fn(g)p Fv(,)h(and)i(examines)g(its)f(theta)g(series.) 131 2128 y(As)g(usual,)g(one)h(can)f(deriv)o(e)h(an)f(analogue)g(for)g (T)o(yp)q(e)g(I)h(co)q(des:)60 2234 y Fu(Theorem)h(35.)22 b Fv([255)o(])c Fw(Supp)n(ose)g Fq(C)j Fw(is)d(a)g(T)m(yp)n(e)g(I)f (self-dual)h(c)n(o)n(de)g(over)h Ft(Z)1368 2241 y Fs(4)1403 2234 y Fw(of)g(length)e Fq(n)p Fw(.)27 b(The)18 b(minimal)60 2319 y(Euclide)n(an)e(norm)g(of)g Fq(C)j Fw(is)d(at)h(most)872 2403 y Fv(8)903 2344 y Fm(\024)938 2373 y Fq(n)p 929 2393 46 2 v 929 2435 a Fv(24)979 2344 y Fm(\025)1011 2403 y Fv(+)11 b(8)16 b Fq(;)707 b Fv(\(166\))60 2506 y Fw(exc)n(ept)16 b(when)g Fq(n)d Fn(\021)g Fv(23)j(\()s(mo)q(d)i(24\)) p Fw(,)d(in)h(which)h(c)n(ase)e(the)i(b)n(ound)f(is)860 2617 y Fv(8)891 2558 y Fm(\024)927 2586 y Fq(n)p 917 2607 V 917 2648 a Fv(24)968 2558 y Fm(\025)1000 2617 y Fv(+)10 b(12)16 b Fq(:)696 b Fv(\(167\))60 2729 y Fw(If)16 b(e)n(quality)g(holds)g(in)g(\(167\))g(then)g Fq(C)j Fw(is)d(a)g(shortene)n(d)g(version)g(of)g(a)h(T)m(yp)n(e)e(II)g(c)n(o)n (de)h(of)g(length)g Fq(n)10 b Fv(+)h(1)p Fw(.)967 2853 y Fv(74)p eop %%Page: 75 79 75 78 bop 131 74 a Fv(W)l(e)17 b(sa)o(y)f(that)h(co)q(des)h(meeting)f (either)h(of)f(these)g(b)q(ounds)h(are)f Fw(norm-extr)n(emal)p Fv(.)26 b(F)l(or)17 b(T)o(yp)q(e)g(I)q(I)h(co)q(des)60 159 y(this)e(agrees)e(with)i(the)f(de\014nition)i(giv)o(en)f(in)g([27)o (].)131 244 y(There)c(should)i(b)q(e)f(an)f(analogous)g(concept)h(of)f Fw(L)n(e)n(e-extr)n(emal)p Fv(,)f(but)i(at)f(presen)o(t)g(w)o(e)g(do)h (not)f(kno)o(w)g(what)60 328 y(this)19 b(is.)32 b(Of)19 b(course,)h(the)f(b)q(ounds)h(\(166\))d(and)i(\(167\))f(also)h(apply)h (to)e(Lee)i(w)o(eigh)o(t.)31 b(But)19 b(this)g(is)h(not)e(a)60 413 y(satisfactory)f(b)q(ound,)j(since)g(it)e(is)h(not)f(ev)o(en)h (tigh)o(t)f(at)g(length)h(24,)f(where)h(the)f(highest)h(attainable)g (Lee)60 498 y(w)o(eigh)o(t)c(is)h(12)e(rather)h(than)g(16)g(\(see)g(T)l (able)h(XVI\).)131 583 y(The)f(fact)h(that,)e(from)h(Theorem)h(31,)e (an)i(extremal)g(binary)g(co)q(de)g(of)f(length)i(a)e(m)o(ultiple)i(of) e(24)h(m)o(ust)60 667 y(b)q(e)d(doubly-ev)o(en)h(suggests)e(that)g (these)h(co)q(des)g(are)f(lik)o(ely)j(to)d(b)q(e)h(particularly)h (nice.)20 b(Indeed,)14 b(w)o(e)f(ha)o(v)o(e)f(the)60 752 y(follo)o(wing)18 b(result,)f(whic)o(h)h(is)g(a)e(consequence)j(of) d(the)h(Assm)o(us-Mattson)f(theorem)h(\(see)g([189)n(,)g(Chap.)g(6],)60 837 y(Theorem)e(11.14)f(of)g(Chapter)h(1,)g(Section)h(5)f(of)g(Chapter) g(xx)g(\(T)l(onc)o(hev\)\).)60 961 y Fu(Theorem)i(36.)22 b Fw(L)n(et)17 b Fq(C)j 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y(linear)f(co)q(des,)f (there)g(is)h(also)f(a)f(nonconstructiv)o(e)i(lo)o(w)o(er)e(b)q(ound,)i (analogous)e(to)h(the)g(Gilb)q(ert-V)l(arshamo)o(v)60 2288 y(b)q(ound)i(\(cf.)k(Theorems)15 b(3.1,)f(3.4,)f(3.5)i(of)f (Chapter)h(1\).)131 2373 y(W)l(e)g(\014rst)g(consider)h(the)f(case)g (of)g(self-dual)i(binary)f(co)q(des)f(\(family)h(2\).)60 2498 y Fu(Theorem)h(37.)22 b Fv([298)o(],(H)H fw(n)h fw(b)n(e)g(p)n(o)f(iTe)n(g)(26)b(l)n(et)17 b fq(d)1430 2505 y fR(gv)1506 2498 y fw(b)n(e)g(i)n(最大)e(in)n(GER)60 2582 y(等)f(771)771 y Fm(x)ρy fk(α)p Fi()17 B((190)O