1 i 2f0 2g

66616111G2=

6161

1200661161061116011111070651006361001601601

14

;168

1i2f0;2g6661611

1611G3=

6161

1160160100001261165100006016016361111661

64

61;1

1i2f0;2g:

NextwedetermineifthemodulespannedbyG1;G2;orG3iscontainedinPm,where7jmbut3m;and5

havesolutionsandtheresultisthatfor7jmbut3m;and5

m;4m;theredonotexist

any4-dimensionalsubmoduleD󰀒Pmsuchthatj󰀟(D)j=9andthentheresultofthelemmafollows.

89;;ifif34jjm

mor5jmbut3

REFERENCES

[1]A.E.Ashikhmin,“GeneralizedHammingweightsforZinProc.IEEEInt.Symp.InformationTheory(Trondheim,4-linearcodes,”

Norway,June1994),p.306.

[2]A.R.Hammons,P.V.Kumar,A.R.Calderbank,N.J.A.Sloane,and

P.Sol´e,“TheZTrans.4-linearityofkerdock,preparata,Goethals,andrelatedcodes,”IEEEInform.Theory,vol.40,pp.301–319,Mar.1994.[3]T.Helleseth,B.Hove,andK.Yang,“Furtherresultsongeneralized

hammingweightsforGoethalsandpreparatacodesoverZ4,”Tech.Rep.R-98-2016,AalborgUniv.,Aalborg,Denmark.

[4]T.HellesethandP.V.Kumar,“ThealgebraicdecodingoftheZcode,”IEEETrans.Inform.Theory,vol.41,pp.2040–2048,4-linear

GoethalsNov.1995.

[5]T.Helleseth,P.V.Kumar,andA.Shanbhag,“Newcodeswiththesame

weightdistributionsastheGoethalscodesandtheDelsarte-Goethalscodes,”Des.,CodesCryptogr.,vol.9,pp.257–266,1996.

[6]P.V.Kumar,T.Helleseth,andA.R.Calderbank,“Anupperbound

forWeilexponentialsumsoverGaloisringsanditsapplications,”IEEETrans.Inform.Theory,vol.41,pp.456–468,Mar.1995.

[7]V.K.Wei,“GeneralizedHammingweightsforlinearcodes,”IEEE

Trans.Inform.Theory,vol.37,pp.1412–1418,Sept.1991.

[8]K.YangandT.Helleseth,“OntheweighthierarchyofPreparatacodes

overZ4,”IEEETrans.Inform.Theory,vol.43,pp.1832–1842,Nov.1997.[9],“OntheweighthierarchyofGoethalscodesoverZTheory,vol.44pp.304–307,Jan.1998.

4,”IEEE

Trans.Inform.NewUpperBoundsonGeneralizedWeights

AlexeiAshikhmin,AlexanderBarg,and

SimonLitsyn,Member,IEEE

Abstract—Wederivenewasymptoticupperboundsonthegeneralizedweightsofabinarylinearcodeofagivensize.Wealsoprovesomeasymptoticresultsonthedistancedistributionofbinarycodes.IndexTerms—Constant-weightcodes,distancedistribution,generalizedweights,linearprogramming.

I.INTRODUCTION

LetCbealinearbinarycodeoflengthnandrateR(C)=

log2jCj=n.ThesupportofasubsetC󰀚Cisdefinedas

suppC=

e2f1;2;111;ng:ce=1

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.4,MAY1999wherethelimitiscalculatedoverallsequencesofcodesCofgrowinglengthnandrateatleastR.Weshallalsobeinterestedinthebehavioroftheinversefunction

Rr(󰀎)=limn!1

supR(C)

wherethelimitiscalculatedoverallsequencesofcodesCofgrowinglengthnandrelativerthweightdr(C)=natleast󰀎.

Letusquotethebestknownboundson󰀎r(R)andRr(󰀎).Below

Hq(x)=xlogq(q01)0xlogqx0(10x)logq(10x)

istheentropyfunctionandHq01

(1)isitsinverse.Thus

Hq:[0;1]![0;1]

and

Hq01

:[0;1]!

:

Further,let󰀕q(R)=Hq01

(10R).Ifq=2,weomitthesubscript

andwritesimplyH(1);H01

(1);󰀕(1).Alowerexistencebound(the“Gilbert–Varshamov”bound)followsfromaresultin[19].Itisthebestknownasymptoticlowerboundon󰀎randRr.

Theorem1[19]:For0󰀔R󰀔1;0󰀔󰀎󰀔

2

󰀎r(R)󰀕󰀎gv

r(R)=󰀕2

(󰀎):(2)

Otherproofsofthistheoremaregivenin[7],[8],and[17].

Thefollowingbound(the“Elias,”or“Bassalygo–Elias,”bound)wasprovedin[5]forr=2andin[6]forthegeneralcase.Theorem2[5],[6]:For0󰀔R󰀔1;0󰀔󰀎󰀔

2

󰀎r(R)󰀔󰀎elr(R)=10󰀕(R)r+10(10󰀕(R))r+1

(3)Rr(󰀎)󰀔Rel

r(󰀎)=10H(󰀚)

(4)

where󰀚isthesmallestrootof

xr+1+(10x)r+1=10󰀎:

Thisboundcanbeimprovedinacertainrangeofrates;see[18].

Inparticular,thefollowingtheoremholdstrue.Theorem3[18]:For0󰀔R󰀔1󰀎r(R)󰀔

1󰀔min

s󰀔r01

10󰀎up

s(R)

R

10󰀎ups(R)(5)

where󰀎up

󰀎s(R󰀎)and󰀎upr0s(R)areanyasymptoticupperboundsons(R)andr0s(R),respectively.

Thentakingr=2,onecanputs=1andusetheMcEliece–Rodemich–Rumsey–Welch(MRRW)bound[14],whichwequoteinthefollowingtheorem.

Theorem4[14]:For0󰀔R󰀔1;0󰀔󰀎󰀔1=2

󰀎1(R)󰀔󰀎lp

1

(R)=min2

󰀋(10󰀋)0󰀌(10󰀌)

1+2

1210(

10󰀎lp

1(R)

R

10󰀎lp1(R)

1259

nR(󰀎;󰀋)=limn!1

supR(C):

WestartwiththelinearprogrammingboundofMcElieceetal.Theorem5[14]:For0󰀔R󰀔H(󰀋);0󰀔󰀎󰀔2󰀋(10󰀋)R;󰀋)󰀔󰀎lp(R;󰀋)=2

󰀋(10󰀋)0H01(R)(10H01(R))1+2

1210(

C󰀎(1260IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.4,MAY1999

thatfurnishestheminimumtotheright-handsideof(6)(notethat󰀌in(6)isadummyvariablewhosevalueisdetermineduniquelygiven󰀋andR).Proposition6:

󰀋1;m(󰀎)=󰀋2;m

󰀋1;m

=󰀋2;m(R):

Proof:Directsubstitution.

H(󰀋0󰀘)0󰀋H0(10󰀋)H

+Rup(󰀎;󰀗)

0;2

󰀋(10󰀋)0󰀌(10󰀌)1+2

(15)

suchthat

1

log2A󰀘n󰀕R01+H(󰀌)+2H(󰀋)02q(󰀋;󰀌;󰀘=2)n0󰀘0(10󰀘)H

where󰀋and󰀌arearbitrarynumberssatisfying

0󰀔󰀌󰀔󰀋󰀔1=2;

H(󰀋)0H(󰀌)󰀕10R

(16)

10

p

102󰀎1

;

22andq(󰀋;󰀌;󰀍)isgivenby(17)atthebottomofthispage.

Choosing󰀋and󰀌sothat󰀘berestrictedtotheshortestpossibleinterval,onecanprove[13]thatthereexistsanexponentiallylargecomponentoftheweightdistributionfor󰀘2[0;󰀎lp(R)].Moreover,comparingthiscomponenttothe“binomial”weightdistribution

^i=A

jCj2ni.e.,theweightdistributionofatypicallongcode,onecancheckthat

thistheoremimpliesthefollowingcorollary.

Corollary12[13]:Foreverycodeofsufficientlylargelengthn

lp

andrateR,andany󰀑󰀕󰀎1(R),thereexistsa󰀘

󰀘2(0;󰀑]

suchthat

1

log2A󰀘n󰀕max(0;H(󰀘)+R01)nCorollary10:If󰀋2;m(R)󰀔󰀋󰀔1=2,then

lp

󰀎(R;󰀋)󰀔󰀎sm(R;󰀋)=󰀎1(1+R0H(󰀋)):

wheretheequalityisachievedonlyif󰀘=󰀑.

LetusanalyzesomeconsequencesofTheorem9.LetR0bethesolutionof(14)

orR0=0:421111:

lp

󰀋2;m(R)=󰀎1(R)

Proof:FollowsfromProposition6.

q(󰀋;󰀌;󰀍)=H(󰀌)+

󰀋(10󰀋)0y(102y)0󰀌(10󰀌)

2(󰀋0y)(10󰀋0y)+

dy:

(17)

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.4,MAY19991261

TABLEI

Theorem13:SupposeacodeofrateR0󰀔R󰀔1,andsufficientlylargelengthnmeetsrelativedistance󰀎lp

thelinearprogrammingbound(6),1

(R).Thenfor󰀎lp1(R)󰀔󰀘󰀔10󰀎lpi.e.,isof

1(R)1

nlog2A󰀘n󰀔H(󰀘)+R01:(18)

Moreover,everysubintervaloftheinterval[󰀎lp

(const)1n+󰀏

of1(R);1=2]oflength

;󰀏>0,containsacomponenttheweightdistribu-tionsuchthat

1

nlog2A󰀘n=H(󰀘)+R01:(19)

Proof:ByTheorem9,if󰀘>󰀋1;m(󰀎lp

1(R))=󰀋2;m(R)R(󰀎lp1

(R);󰀘)󰀔Rlp

1+H(󰀘)01=R+H(󰀘)01:(20)

TheinequalityonRfollowsbysolving󰀋2;m(R)󰀔󰀎lp

(R)withrespecttoR.Thesecondclaimforevery󰀑󰀕󰀎lp

1

Theorem13(technicaldetailsare1(R)followsfromCorollary12andomitted).

max

0<󰀘󰀔2

p

(21)

where󰀎up

(R;󰀘)isanyupperboundon󰀎(R;󰀘)

R0=R01+H(󰀌)+2H(󰀋)02q(󰀋;󰀌;󰀘=2)

0󰀘0(10󰀘)H

(22)

andq(󰀋;󰀌;󰀍)isdefinedin(17).

Proof:ByTheorem11,forany󰀋;󰀌satisfying(16)and󰀘asin(15),thereexistsaconstant-weightcodeD(󰀘)ofsizeA󰀘n󰀕R0andweight󰀘n:ThenD(󰀘)containsapairofcodewordsx;yatadistanceatmost󰀎up(R0;󰀘)napart.Thesizeofthesupportof(x;y;x+y)equals

jsupp(x;y)j=2󰀘n0jx\\yj=󰀘n+

dist(x;y)

2:Thiscompletestheproof.

󰀌(10󰀌):

Thenby(6)wehave󰀘=󰀎lp

1(R).Further,forthischoiceof󰀋;󰀌;and󰀘;(22)takesontheform

R0=H

0R+1

(see[13];thisisprovedsimilarlytothewayCorollary12isderivedfromTheorem11).Nowwecanrewrite(21)using(14)asfollows:

󰀎2(R)󰀔󰀎lp

1(R)+12󰀎sm

=󰀎lp1(R)+1lplp

2󰀎1󰀎1(R)

=3lp

2󰀎1(R):󰀎lp

1(R)+1󰀎lp

󰀎lp321(R)

;0󰀔R1

(R);R0󰀔R󰀔1

(24)

where󰀎lp

1(R)and󰀎lp(R;󰀋)aregivenby(6)and(10),respectively.Bound(24)isbetterthanthepreviouslyknownbounds.For

reference󰀎el

purposeswegive,inTableI,󰀎3

2(24),the2(9),theVl˘adu¸tbound󰀎vlEliasbound

2(8),andthelowerGilbert–Varshamovbound󰀎gv

Tovisualize2(2).

theseresults,inFig.1weplotbounds(24),(9),(8),and(2).

SincetheboundinProposition16isbetterthantheboundspreviouslyknown,itsapplicationin(5)improvesknownboundsongeneralizedweightoforderrgreaterthan2.Letusillustratethisforr=3.Computingtheboundsin(5)showsthattheminimumisattainedfors=1.Thisgivesthefollowingbound:󰀎3(R)󰀔󰀎33(R)=󰀎1(R)+(10󰀎1(R))󰀎3

2

(25)

where󰀎1(R)isgivenin(6)and󰀎32(R)in(24).Bound󰀎33(R)isbetterthantheEliasbound(3)for0V.NEWBOUNDSONGENERALIZEDWEIGHTS:II

InthissectionwepresentanotherwayofusingTheorem11toderiveboundsongeneralizedweights.Webeginwithtwoauxiliarylemmas.

1262IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.4,MAY1999

Fig.1.BoundsonthesecondweightofacodeofrateR.

Lemma17:LetWbeaconstant-weightcodeoflengthnandweightw.If0󰀔w1󰀔w,and

jWj

min(`;w)

󰀕M;1󰀔M󰀔jWj

thenthereexist`positionssuchthatatleastMcodewordsofWhaveweightatleastw1onthem.

Proof:Wewillfindtheaverageweightofsubwordsofthecodeoverallpossiblechoicesof`positions.LetL󰀚f1;2;111;ng;jLj=`󰀟.Forc)=acodeword1ifthetotalc2numberW,defineof1’stheinindicatorthepositions󰀟L(cL)byofclettingL(isatleastw1.Wehave

min(`;w)

X=

󰀟(c)=jWj:

jLj=`

HencethereexistsachoiceofLsuchthatthenumberof“good”

codewordsisatleastX=(n`

),givingtheclaim.Mc

01

differentnontriviallinearcombinations).Denotethissubsetofcode-wordsbyW:

Thesecondauxiliaryresultthatweuseisastandarddouble(“row–column”)countingargumentfortheaveragesupportofrgivenvectorsofacodeD(herewedonotassumelinearity).InthematrixwithcodewordsofDasrowsletuibethenumberofonesintheithcolumn.

Lemma18:LetDbeacodeoflengthnandsizeD.Then01

n

minsupp(c1;111;cr)󰀔

0

r

):Soweget

n

minsupp(c1;111;cr)󰀔0

NowletususethelasttwolemmastogetherwithTheorem11toderivethemainresultofthissection.Beloww=󰀘n;w`=󰀗n:

1=!n;Theorem19:LetCbeabinarycodeofrateR.LetR0=R0(󰀋;󰀌;󰀘)=R01+H(󰀌)+2H(󰀋)

02q(󰀋;󰀌;󰀘=2)0󰀘0(10󰀘)H

whereq(󰀋;󰀌;󰀍)isdefinedin(17).Then

󰀎r(R)󰀔󰀎33

r(R)=min󰀋;󰀌

max󰀘

0min

󰀔!󰀔󰀘

r

10

(26)

where󰀋;󰀌;󰀘satisfy(15)and(16),and󰀗=󰀗(R0;󰀘;!)isasolutionoftheequation

R0+󰀘H

+(10󰀘)H

0H(󰀗)=0

suchthat!󰀔󰀗.

Proof:Theproofconsistsofthreesteps:takeaconstant-weightcodeWgivenbyTheorem11,applyLemma17onit,andapplyLemma18onthecodeobtainedbyarestrictionofWton0`coordinates.

WeneedtopresentrlinearlyindependentcodewordsofCwithsupportgivenbytheright-handsideof(26).Theorem11guaranteesthatthereisasubsetofvectorsofCatadistancewfromacertaincodeworda2C,or,uponshiftingtheentirespacebya,acodeWofconstantweightw.Further,thereisasubsetofWofsizeMthatsatisfiestheweightrestrictionsofLemma17.Bytheremarkafterthislemma,thereisasubsetW󰀚Woflogsatisfytheserestrictions,2Mlinearlyindependentcodewordsthatalsoi.e.,areofweightw1onacertainsubsetof`coordinates,1󰀔`󰀔n.ConsidertherestrictionofWtotheremainingn0`coordinates,denotethiscodebyD,andapplyLemma18onit.Wehave

(c

min

)2D

supp(c1;111;cr)

01n0`

󰀔

0

;111;c

c[Dr0(D0ui)r]

i=1

=(n0`)

(D0ui)r

i=1

n[-convexfunctionsi=1

0`uiof=uD(w0w1):Summationtermsin(27)arei.Finally,theright-handsideof(27)issymmetricinui.InvokingLagrangemultipliersnowimpliesthatitattainsmaximumforui=D(w0w1)=(n0`),andthatthe

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.4,MAY19991263

TABLEII

Fig.2.BoundsonthethirdweightofacodeofrateR.

maximumisstrict.Henceweget

(c

min

)2D

supp(c1;111;cr)󰀔(n0`)10

w0w1n0`:

Thuswehaveprovedtheexistenceofrlinearlyindependentcode-wordsofCwithsupportatmost

`+(n0`)

10

w0w1n0`:

Thiscompletestheproof.