Computing coset leaders of binary codes

0.08 MB·English

by  M. Borges-Quintana

Des.CodesCryptogr.manuscriptNo.
(willbeinsertedbytheeditor)
Computing coset leaders of binary codes
M.Borges-Quintana · M.A.Borges-Trenard ·
E.Mart´ınez-Moro
Received:date/Revised:date
0
1
Abstract We present an algorithm for computing the set of all coset leaders of a
0
2 binary code C ⊂Fn2. The method is adapted from some of the techniques related
tothecomputationofGro¨bnerrepresentationsassociatedwithcodes.Thealgorithm
n
provides a Gro¨bner representation of the binary code and the set of coset leaders
a
J CL(C).Itsefficiencystandsofthefactthatitscomplexityislinearonthenumberof
8 elementsofCL(C),whichissmallerthanexhaustivesearchinFn.
2
2
Keywords Binarycodes·Cosetsleaders·Gro¨bnerrepresentations.

]
T
I
1 Introduction
.
s
c
[ Theerror-correctionproblemincodingtheoryaddressesgivenareceivedwordrecov-

eringthecodewordclosesttoitwithrespecttotheHammingdistance.Thisprevious
1
statement is the usual formulationof the Complete Decoding Problem (CDP). The
v
4 t-bounded distance decoding (t-BDD) algorithms determine a codeword (if such a
7 word exists) which is at distance less or equal to t to the received word. If t is the
0 coveringradiusofthecodethentheboundeddistancedecodingproblemisthesame
5 as CDP. In the CDP of a linear code of length n , C ⊂Fn those errorsthat can be
. q
1 correctedarejustthecosetleaders,whicharevectorsofsmallestweightinthecosets
0 Fn/C.Whenthereismorethanoneleaderinacosetthereismorethanonechoice
q
0
fortheerror.Thereforethefollowingproblemnaturallyfollowsknownasthe coset
1
weightsproblem(CWP):
:
v
i PartiallysupportedbySpanishAECIDA/016959/08.
X
r M.Borges-Quintana·M.A.Borges-Trenard
a Dpto.deMatema´tica,FacultyofMathematicsandComputerScience,UniversidaddeOriente,
SantiagodeCuba,Cuba.
E-mail:[email protected],[email protected]
E.Mart´ınez-Moro
Dpto.deMatema´ticaAplicada,UniversidaddeValladolid,Castilla,Spain.
E-mail:[email protected]
2
Input:Abinaryr×nmatrixH,avectors∈Fr andanon-negativeintegert.
2
Problem:Doesabinaryvectore∈Fr ofHammingweightatmostt existsuchthat
2
He=s?
RecallthatH canbeseenastheparitycheckmatrixofabinarycodeandsthesyn-
dromeofareceivedword,thustheknowledgeofewouldsolvethet-BDDproblem.
Unfortunatellythehopeoffindinganefficientt-BDDalgorithmisverybleaksinceit
wasproventhattheCWPisNP-complete[4].Thusthecomputationofallthecoset
leaders is also NP-complete.The study of the set of coset leadersis also related to
the study of the set of minimalcodewords,which havebeen used in the Maximum
LikelihoodDecodingAnalysis[2]andwhicharealsorelatedtotheminimalaccess
structure of secret sharing schemes [13]. Furthermore,the computationof all coset
leadersofacodeallowstoknowmoreaboutitsinternalstructure[7].
Allproblemsmentionedbeforeareconsideredtobehardcomputationalproblems
(see for example [2,4]) even if preprocesing is allowed [10]. However, taking into
accountthenatureoftheproblem,todevelopanalgorithmforcomputingthesetof
all coset leaders of a binary code, in the vector space of 2n vectors, by generating
a number of vectors close to the cardinality of this set may be quite efficient. This
is ourpurposeextendingsome resultsonGro¨bnerrepresentationsforbinarycodes.
ForpreviousresultsandapplicationsofGro¨bnerrepresentationsforlinearcodessee
[6,7,5], for a summaryof the whole materialwe refer the readerto [8]. We extend
somesettingsofpreviousworkinordertoobtainthesetofallcosetleaderskeeping
recordoftheadditivestructureinthecosets.Thepresentationofthepaperisdonein
a“Gro¨bnerbases”-freecontext.
Theoutlineofthepaperisasfollows.InSection2wereviewsomeofthestan-
dard facts on binary linear codes and their Gro¨bner representation.Section 3 gives
a concise presentation of the results in this paper. Definition 3 corresponds to the
construction of List, the main object that is used in the algorithm proposed, while
Theorem2 guaranteesthat all coset leaders will belongto List. In this section it is
presentedthealgorithmCLBCforcomputingthesetofallcosetleaders.Attheend
ofthesectionweshowanexampletoillustrateacomputationalapproachappliedtoa
binarylinearcodewith64cosetsand118cosetleaders.InSection4wediscusssome
complexityissues.Finallysomeconclusionaregivenwhichincludefurtherresearch.
2 Preliminaries
LetF bethefinitefieldwith2elementsandFnbetheF -vectorspaceofdimension
2 2 2
n.We willcallthevectorsinFn words.A linearcodeC ofdimensionk andlength
2
n is the image of an injective linear mapping L:Fk →Fn, where k≤n, i.e. C =
2 2
L(Fk). From now on, we will use the term code to mean binary linear code. The
2
elementsinC arecalledcodewords.Forawordy∈Fnthesupportofyissupp(y)=
2
{i∈{1,...,n}|y 6=0} and the Hamming weight of y is given by weight(y) the
i
cardinalofsupp(y).TheHammingdistancebetweentwowordsc ,c isd(c ,c )=
1 2 1 2
weight(c −c )andtheminimumdistancedofacodeistheminimumweightamong
1 2
allthenon-zerocodewords.ItiswellknownthatCDPhasauniquesolutionforthose
3
vectorsinB(C,t)={y∈Fn|∃c∈C s.t.d(c,y)≤ d−1 }where[·] isthegreatest
2 (cid:2) 2 (cid:3)
integerfunction.
Definition1 The words of minimum Hamming weight in the cosets of Fn/C are
2
calledcosetleaders.
CosetscorrespondingtoB(C,t)haveauniqueleaderhoweveringeneraloutside
B(C,t)theremaybealsocosetswithauniqueleader,i.e.thosecosetswithonlyone
leadercouldbemorethan|B(C,t)|.LetCL(C)denotethesetofcosetleadersofthe
codeC andCL(y)thesubsetofcosetleaderscorrespondingtothecosetC+y.Lete
i
i=1,...,nbethei-thvectorofthecanonicalbasisX={e ,...,e }⊂Fn and0the
1 n 2
zero vector of Fn. The followingtheorem givessome relations between cosets [12,
2
Corollary11.7.7].
Theorem1 Letw∈CL(C),suchthatw=w +e forsomew ∈Fnandi∈{1,...,n}.
1 i 1 2
Then,w ∈CL(w ).
1 1
Definition2 A Gro¨bner representation of Fn/C [6,8] is a pair N,f where N is a
2
transversal of Fn/C such that 0∈N and for each n∈N\{0} there exists e, i∈
2 i
1,...,n, such thatn=n′+e and n′ ∈N, and f :N×{e}n →N be the function
i i i=1
thatmapseachpair(n,e)totheelementofNwhichbelongstothecosetofn+e.
i i
3 Computingthesetofcosetleaders
AkeypointofthealgorithmsforcomputingGro¨bnerrepresentationsistheconstruc-
tion of an object we called List which is an ordered set of elements of Fn w.r.t. a
2
linearorder≺definedasfollows:w≺vifweight(w)<weight(v)orweight(w)=
weight(v)andweight(w)≺ weight(v),where≺ isanyadmissibleorderonFn in
1 1 2
thesensegivenin[3,p.167].Wewillcallsuchkindofordersasweightcompatible
orderings.
Definition3 (ConstructionofList).LetListbetheorderedstructuregivenbythe
followingaxioms
1. 0∈List.
2. Ifv∈List,letN(v)=min {w|w∈List∩(C +v)}.
≺
3. Ifv∈Listissuchthatweight(v)=weight(N(v)),then{v+e :i∈/supp(v),i∈
i
{1,...,n}}⊂List.
Remark1 ThesetNisthesubsetofListsuchthatN(v)istheleastelementinList∩
(C +v),i.e.N(v)∈C+vandN(v)=min {w|w∈List∩(C +v)}.
≺
Remark2 WestartListwith0andwewillseethat,whenevercondition3.holdsfor
v, the vector v will be a coset leader of C +v. Then, we insert v+e to List, for
i
i=1,...,nandi∈/supp(v)(seeTheorem1).Itisnotnecessarytointroducev+e,
i
for i∈supp(v), because in this case v+e ≺v and then v+e has been already
i i
consideredinListsinceitisanorderedstructure.
4
NexttheoremstatesthatListinDefinition3includesthesetofcosetleaders.
Theorem2 Letw∈CL(C),thenw∈List.
Proof WewillproceedbyNoetherianInductiononthewordsofFnwiththeordering
2
≺, since≺hasthepropertyofbeingawell-foundedorderingorNoetherianordering
(i.e., any descendingchain of wordsis finite) [3, Theorem4.62, p. 168].The word
0∈CL(C) and by definition it belongs to List, let w∈CL(C)\{0}. Assume the
property valid for any word less than w with respect to ≺, i.e. u∈List provided
u∈CL(C) and u≺ w. Taking i∈supp(w), we write w =u+e where u ∈Fn,
i 2
then by Theorem 1, u∈CL(C). In addition, supp(u)=supp(w)−1, thus u≺w.
Therefore,byapplyingtheinductionprinciplewehaveu∈List.Ifuisacosetleader
belongingtoListitisclearthatweight(u)=weight(N(u)).Asaconsequence,by3.
inDefinition3,w=u+e ∈List. ⊓⊔
i
Algorithm1 (CLBC)
Input: AparitycheckmatrixofabinarycodeC.
Output: CL(C),f :ThesetofallcosetleadersandthefunctionMatphi.
1: List←[0],N←0/,r←0,CL(C)←[]
2: whileList6=0/ do
3: t ←NextTerm[List],s←t H
4: j←Member[s,{s1,...,sr}]
5: if j6=falsethen
6: forksuchthatt =t ′+ekwitht ′∈Ndof (t ′,ek)←t j
7: ifweight(t )=weight(t j)then
8: CL(C)[t j]←CL(C)[t j]∪{t }
9: List←InsertNext[t ,List]
10: endif
11: else
12: r←r+1,sr←s,tr←t ,N←N∪{tr}
13: CL(C)[tr]←CL(C)[tr]∪{t }
14: List←InsertNext[tr,List]
15: forksuchthattr=t ′+ekwitht ′∈Ndo
16: f (t ′,ek)←tr
17: f (tr,ek)←t ′
18: endfor
19: endif
20: endwhile
21: return CL(C),f
Where
1. InsertNext[t ,List]Insertsallthesumst +e inList,wherek∈/supp(t ),andkeeps
k
Listinincreasingorderw.r.t.theordering≺.
2. NextTerm[List]returnsthefirstelementfromListanddeletesitfromthisset.
3. Member[obj,G]returnstheposition jofobjinGifobj∈Gandfalseotherwise.
5
Theorem3 CLBCcomputesthesetofcosetleadersofagivenbinarycodeandits
correspondingMatphi.
Proof Notethatwhenanelementt istakenoutfromListbyNextTermtheelements
to be inserted in List by InsertNext are of the form t +e , where k∈/ supp(t ). As
k
aconsequence,t +e ≻t . ThenallelementsgeneratedbyCLBCinList,aftert is
k
takenout,aregreaterthant .Therefore,whent isthefirstelementinListinStep3,
allelementsofListthatshallbeanalyzedbyCLBCaregreaterthant .
LetusprovethattheproceduregeneratesListaccordingtoDefinition3.First,by
Step1,0∈List.Lett =NextTerm[List]inStep3,inthisstepthesyndromesoft is
computed.ThuswehavetwocasesregardingtheresultofStep4,namely
1. Assume j=“false”, thus Steps 5 and 11 guaranty us to find N(t ) as the least
elementin List havingthe same syndromeas t . When we are in this case (t =
N(t )),inStep14itisperformed(ii)ofDefinition3.
2. Ontheotherhand,assume j6=“false”(anelementN(t )=t havebeenalready
j
computed),if we have weight(t )=weight(t ), in Step 9 it is performed(ii) of
j
Definition3.
ThereforeCLBCconstructstheobjectListfollowingDefinition3.ByTheorem2,the
setof cosetleadersis a subsetof List. ThenSteps8 and13assure thecomputation
ofthisset.TheprocedurecomputestheMatphistructureasadirectconsequenceof
Steps6,16,17.ThereasonforincludingStep17isthatinthiscase
t +e =t ′≺t
r k r
andthoseelementst +e ,wherek∈supp(t ),arenotinsertedinListsincet ′ has
r k r
been already considered when t is computed as a new element of N. In addition,
r
since ≺ is admissible, all subwordsof t are also least elementsof their cosets ac-
r
cordingto≺,sot ′=N(t ′)(notet ′ isacosetleaderandbyTheorem2itbelongsto
List).
WehaveprovedthatCLBCguaranteestheoutputsweareclaiming.Termination
isaconsequenceofthefactthatthecardinalofthesetofelementsbelongingtoList
is least than n|CL(C)|. Then, to a certain extent(when the set of coset leadershas
beencomputed)nomoreelementsareinsertedinListinSteps9and14.Therefore,
thelistgetempty,andbyStep2,CLBCterminates. ⊓⊔
Remark3
1. Weintroduceinthispapersteps7,8and9.Inourpredecessoralgorithmswhere
itwasnotnecessarytointroducethem[6,7,8].Thesesteps,aswellasamodified
definitionoflist,areforcedbythefactthatallcosetleadersareincorporated.
2. InStepsfrom14to 17,thepairs(t ,e ), wherek∈supp(t ), arenotnecessary
r k r
tocomputethecosetleadersbutforcomputingthestructureMatphiandcouldbe
removedifweareonlyinterestedinthecosetleaders.
Example1 Considerthe[10,4,1]codewithparitycheckmatrixH
6
10 00 100 00 0
 
10 11 010 00 0
11 01 001 00 0
H= .
11 10 000 10 0
 
11 11 000 01 0
 
11 11 000 00 1
We use GAP 4.12 [11] and the GAP’s package GUAVA 3.10 for Coding The-
ory. We have built in this framework a collection of programs we call GBLA LC
“Gro¨bnerBasesbyLinearAlgebraandLinearCodes”[9].Inparticular,wehaverun
thefunction“CLBC” ofGBLA LC(CosetLeadersofBinaryCodes),itgivesa list
ofthreeobjectsasanoutput,thefirstoneisthesetofcosetleaders,thesecondone
thefunctionMatphi,andthethirdonetheerrorcorrectingcapabilityofthecode.The
completesetofcosetleadersisthelistbelowwith64componentsandeachcompo-
nentcorrespondstoacosetwithitscosetsleaders,thesetNiscomposedbythefirst
elementsofeachcomponent.We haveindicatedwitharrowssomeplacesinthelist
below,whichwearegoingtouseduringtheexample.TheelementsinthelistCL(C)
are
[[1],[e ],[e ],[e ],[e ],[e ],[e ],[e ],[e ],[e ],[e ],
1 2 3 4 5 6 7 8 9 10
[e +e ,→e +e ←],[e +e ,e +e ],[e +e ,e +e ],
1 2 5 6 1 3 5 7 1 4 5 8
[e +e ,e +e ,e +e ,e +e ],[e +e ,e +e ],[e +e ,e +e ],
1 5 2 6 3 7 4 8 1 6 2 5 1 7 3 5
[e +e ,→e +e ←],[e +e ],[e +e ],[e +e ,e +e ],
1 8 4 5 1 9 1 10 2 3 6 7
[e +e ,e +e ],[e +e ,e +e ],[e +e ,→e +e ←],[e +e ],[e +e ],
2 4 6 8 2 7 3 6 2 8 4 6 2 9 2 10
[e +e ,e +e ],[e +e ,e +e ],[e +e ],[e +e ],
3 4 7 8 3 8 4 7 3 9 3 10
[e +e ],[e +e ],[e +e ],[e +e ],[e +e ],[e +e ],
4 9 4 10 5 9 5 10 6 9 6 10
[e +e ],[e +e ],[e +e ],[e +e ],[e +e ],
7 9 7 10 8 9 8 10 9 10
[e +e +e ,e +e +e ,e +e +e ,e +e +e ],
1 2 3 1 6 7 2 5 7 3 5 6
→[e +e +e ,e +e +e ,e +e +e ,e +e +e ]←,
1 2 4 1 6 8 2 5 8 4 5 6
[e +e +e ,e +e +e ,e +e +e ,e +e +e ],
1 2 7 1 3 6 2 3 5 5 6 7
[e +e +e ,e +e +e ,e +e +e ,e +e +e ],
1 2 8 1 4 6 2 4 5 5 6 8
[e +e +e ,e +e +e ],[e +e +e ,e +e +e ],
1 2 9 5 6 9 1 2 10 5 6 10
[e +e +e ,e +e +e ,e +e +e ,e +e +e ],
1 3 4 1 7 8 3 5 8 4 5 7
[e +e +e ,e +e +e ,e +e +e ,e +e +e ],
1 3 8 1 4 7 3 4 5 5 7 8
[e +e +e ,e +e +e ],[e +e +e ,e +e +e ],[e +e +e ,e +e +e ],
1 3 9 5 7 9 1 3 10 5 7 10 1 4 9 5 8 9
[e +e +e ,e +e +e ],[e +e +e ,e +e +e ,e +e +e ,e +e +e ],
1 4 10 5 8 10 1 5 9 2 6 9 3 7 9 4 8 9
[e +e +e ,e +e +e ,e +e +e ,e +e +e ],[e +e +e ,e +e +e ],
1 5 10 2 6 10 3 7 10 4 8 10 1 6 9 2 5 9
[e +e +e ,e +e +e ],[e +e +e ,e +e +e ],[e +e +e ,e +e +e ],
1 6 10 2 5 10 1 7 9 3 5 9 1 7 10 3 5 10
[e +e +e ,e +e +e ],[e +e +e ,e +e +e ],[e +e +e ],
1 8 9 4 5 9 1 8 10 4 5 10 1 9 10
[e +e +e ,e +e +e ,e +e +e ,e +e +e ],[e +e +e ]]
2 3 8 2 4 7 3 4 6 6 7 8 5 9 10
Note that y=e +e +e in CL(C) =[e +e +e ,e +e +e ,e +e +
4 5 6 46 1 2 4 1 6 8 2 5
e ,e +e +e ] (pointed by arrows) is a coset leader such that none subword of y
8 4 5 6
is in N (see the previous elements pointed by arrows). This shows the importance
of considering all coset leaders in 3. of Definition 3 and not only the coset leaders
belongingtoNlikeinpreviousworks.
7
Thealgorithmcouldbeadaptedwithoutincrementingthecomplexitytogetmore
informationliketheCoveringradiusandtheNewtonradius.Inthiscasebyanalyzing
thelastelementCL(C) =[e +e +e ]wehaveacosetofhighestweightwhich
64 5 9 10
alsocontainsonlyoneleader;therefore,theCoveringradiusandtheNewtonradius
areequalto3.Moreover,forcomputingtheseparametersthealgorithmdonotneed
torununtiltheveryend.
Aswestatebefore,thesetNwouldbeenoughtocomputetheWeightDistribution
of the Coset Leaders WDCL=(a ,...,a ) where a is the numberof cosets with
0 n i
cosetleadersofweighti,i=1,...,nofthecode.Wecanprovidealsoanobjectwhich
givesmoreinformationaboutthestructureofthecode,thatwouldbethenumbersof
cosetleadersineachcoset(#(CL)).
WDCL=[1,10,30,23,0,0,0,0,0,0,0].
#(CL)=[1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,2,4,1,1,1,1,2,2,2,2,2,2,4,2,2,2,4,4,
1,1,1,4,4,4,2,2,2,2,2,4,4,1,1,1,2,2,2,2,4,1,1,1,2,2,2,1,1,1,1].
It is also interesting to note that there are more cosets with one leader (19) out the
cosetscorrespondingtoB(C,t)thancosetscorrespondingtoB(C,t)
|{CL(y)|y∈B(C,t)}|=11.
Therefore,thereare30ofthe64cosetswheretheCDPhasauniquesolution.
4 ComplexityAnalysis
Foradetailedcomplexityanalysisandsomeusefulconsiderationsfromthecompu-
tational point of view we refer the reader to [7]. Section 6 of that paper is devoted
to discuss in details about the computational complexity and space complexity of
thesettingforcomputingGro¨bnerbasisrepresentationforbinarycodesandgeneral
linear codes. The method is also compared with other existent methodsfor similar
purposes.In the case of this paper,the differenceis that we work out the set of all
coset leaders and not only a set of canonical forms. Next theorem shows an upper
boundforthenumberofiterationsthatwillperformCLBC.
Theorem4 CLBC computes the set of coset leaders of a given binary code C of
lengthnafteratmostn|CL(C)|iterations.
Proof Note that the numberof iterations is exactly the size of List. In the proofof
Theorem3wasshownthatthealgorithmfollowsthisdefinitiontoconstructtheobject
List.ItisclearthatthesizeofListisboundedbyn|CL(C)|,notethatwecanwrite
List,asasetasfollows
List={w+e |w∈CL(C)and ∈{1,...,n}}.
i
⊓⊔
8
Remark4 1. BytheproofabovewerequireamemoryspaceofO(n|CL(C)|). We
assumethatforcomputingthesetofcosetleadersitisrequiredatleastO(|CL(C)|);
therefore,CLBCisneartheoptimalcaseofmemoryrequirements.
2. CLBC generates at most n|CL(C)| words from Fn to compute the set of coset
2
leaders.AnalgorithmforcomputingthissetneedstogeneratefromFn atleasta
2
subsetformedbyallcosetleaders,i.e.|CL(C)|;therefore,thealgorithmisnear
theoptimalcaseofcomputationalcomplexity.
5 Conclusion
The Algorithm CLBC is formulated in this paper, which turns out to be quite effi-
cientforcomputingallcosetleadersofabinarycodefrommemoryrequirementsand
computational complexity view. Although, as it is expected, the complexity of the
algorithmisexponential(inthenumberofcheckpositions).
The difference of CLBC with its predecessors rely also in the computation of
allcosetleadersinsteadof a setofrepresentativeleadersforthecosets, however,it
supports the computation of the function Matphi. We remark that the computation
of Matphi is not necessary at all for the main goal of computingthe coset leaders.
We have kept this resource in the algorithm because this structure provides some
computationaladvantages(see [6,7,8]) and, although the algorithm will be clearly
faster without computing Matphi, the nature of the computational complexity and
spacecomplexitywillremainthesame.
Unfortunately,ageneralizationtonon-binarylinearcodesisnottrivialfromthis
work.ThemainreasonseemstobethatthesolutionbasedonHammingweightcom-
patibleorderingswillnotcontinuebeingpossible;theerrorvectororderingwehave
used in the general approach [7] is not a total ordering, although it allowed to set
up the computationalenvironmentin order to computeGro¨bnerrepresentationsfor
linearcodes.
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