Publicació amb motiu de la 21st Conference on Applications of Computer Algebra (July 20-24, 2015, Kalamata, Greece

Barrolleta, Roland David

Villanueva, M.

Diposit Digital de Documents de la UAB

PD-sets for (nonlinear) Hadamard Z4-linear codesR. D. Barrolleta1, M. Villanueva11 Universitat Autònoma de Barcelona, Spain, {rolanddavid.barrolleta, merce.villanueva}@uab.catThis work was partially supported by the Spanish MEC under Grant TIN2013-40524-P, and by theCatalan AGAUR under Grant 2014SGR-691.Any nonempty subset C of Zn2 is a binary code and a subgroup of Zn2 is calleda binary linear code. Equivalently, any nonempty subset C of Zn4 is a quaternarycode and a subgroup of Zn4 is called a quaternary linear code. Quaternary codescan be seen as binary codes under the usual Gray map Φ : Zn4 → Z2n2 defined asΦ((y1, . . . ,yn)) = (φ(y1), . . . ,φ(yn)), where φ(0) = (0,0), φ(1) = (0,1), φ(2) =(1,1), φ(3) = (1,0), for all y = (y1, . . . ,yn) ∈ Zn4. If C is a quaternary linear code,the binary code C =Φ(C ) is said to be a Z4-linear code.A Z2Z4-additive code C is a subgroup of Zα2 ×Zβ4 . We consider the extensionof the Gray map Φ : Zα2 ×Zβ4 → Zα+2β2 defined as Φ(x,y) = (x,φ(y1), . . . ,φ(yβ )),for all x ∈ Zα2 and y = (y1, . . . ,yβ ) ∈ Zβ4 . This generalization allows us to considerZ2Z4-additive codes also as binary codes. If C is a Z2Z4-additive code, the binarycode C = Φ(C ) is said to be a Z2Z4-linear code. Moreover, since the code Cis isomorphic to an abelian group Zγ2 ×Zδ4 , we say that C (or equivalently thecorresponding Z2Z4-linear code C = Φ(C )) is of type (α,β ;γ,δ ) [3]. Note thatZ2Z4-additive codes can be seen as a generalization of binary (when β = 0) andquaternary (when α = 0) linear codes. The permutation automorphism group ofC and C = Φ(C ), denoted by PAut(C ) and PAut(C), respectively, is the groupgenerated by all permutations that let the set of codewords invariant.A binary Hadamard code of length n has 2n codewords and minimum dis-tance n/2. The Z2Z4-additive codes such that, under the Gray map, give a binaryHadamard code are called Z2Z4-additive Hadamard codes and the correspondingZ2Z4-linear codes are called Hadamard Z2Z4-linear codes, or just Hadamard Z4-linear codes when α = 0. The permutation automorphism group of Z2Z4-additiveHadamard codes with α = 0 was characterized in [9] and the permutation auto-morphism group of Z2Z4-linear Hadamard codes was studied in [6].Let C be a binary code of length n. For a vector v ∈ Zn2 and a set I ⊆ {1, . . . ,n},we denote by vI the restriction of v to the coordinates in I and by CI the set {vI :v ∈C}. Suppose that |C|= 2k. A set I ⊆ {1, . . . ,n} of k coordinate positions is aninformation set for C if |CI|= 2k. If such I exists, C is said to be a systematic code.Permutation decoding is a technique, introduced by MacWilliams [8], whichinvolves finding a subset S of the permutation automorphism group PAut(C) of acode C in order to assist in decoding. Let C be a systematic t-error-correcting code135with information set I. A subset S⊆ PAut(C) is an s-PD-set for the code C if everys-set of coordinate positions is moved out of the information set I by at least oneelement of the set S, where 1≤ s≤ t. If s = t, S is said to be a PD-set.In [4], it is shown how to find s-PD-sets of size s+ 1 that satisfy the Gordon-Schönheim bound for partial permutation decoding for the binary simplex code Smof length 2m− 1, for all m ≥ 4 and 1 < s ≤ ⌊2m−m−1m ⌋. In [1], similar results areestablish for the binary linear Hadamard code Hm (extended code of Sm) of length2m , for all m≥ 4 and 1 < s≤ ⌊2m−m−11+m ⌋, following the techniques described in [4].The paper is organized as follows. In Section 1, we show that the Gordon-Schönheim bound can be adapted to systematic codes, not necessarily linear. More-over, we apply the bound of the minimum size of s-PD-sets for binary Hadamardcodes obtained in [1] to Hadamard Z2Z4-linear codes, which are systematic [2] butnot linear in general. In Section 2, we provide a criterion to obtain s-PD-sets of sizes+ 1 for Z4-linear codes. Finally, in Section 3, we recall a recursive constructionto obtain all Z2Z4-additive codes with α = 0 [7] and we give a recursive method toobtain s-PD-sets for the corresponding Hadamard Z4-linear codes.1 Minimum size of s-PD-setsThere is a well-known bound on the minimum size of PD-sets for linear codesbased on the length, dimension and minimum distance of such codes that can beadapted for systematic codes (not necessarily linear) easily:Proposition 1. Let C be a systematic t-error correcting code of length n, size |C|=2k and minimum distance d. Let r = n− k be the redundancy of C. If S is a PD-setfor C, then|S| ≥⌈nr⌈n−1r−1⌈. . .⌈n− t+1r− t+1⌉. . .⌉⌉⌉. (1)The above inequality (1) is often called the Gordon-Schönheim bound. Thisresult is quoted and proved for linear codes in [5]. We can follow the same proofsince the linearity of the code C is only used to guarantee that C is systematic.In [2], it is shown that Z2Z4-linear codes are systematic. Moreover, a systematicencoding is given for these codes.The Gordon-Schönheim bound can be adapted to s-PD-sets for all s up to theerror correcting capability of the code. Note that the error-correcting capability ofany Hadamard Z2Z4-linear code of length n = 2m is tm = b(d−1)/2c= 2m−2−1.Therefore, the right side of the bound given by (1), for Hadamard Z2Z4-linearcodes of length 2m and for all 1≤ s≤ tm, becomesgm(s) =⌈2m2m−m−1⌈2m−12m−m−2⌈. . .⌈2m− s+12m−m− s⌉⌉. . .⌉⌉. (2)136For any m≥ 4 and 1≤ s≤ tm, we have that gm(s)≥ s+1. The smaller the size ofthe PD-set is, the more efficient permutation decoding becomes. Because of this,we will focus on the case when gm(s) = s+1.2 s-PD-sets of size s+1 for Z4-linear codesLet C be a Z2Z4-additive code of type (0,β ;γ,δ ) and let C = Φ(C ) be the corre-sponding Z4-linear code. Let Φ : PAut(C )→ PAut(C) be the map defined asΦ(τ)(i) ={2τ(i/2), if i is even,2τ( i+12 )−1 if i is odd,for all τ ∈ Sym(β ) and i ∈ {1, . . . ,2β}. The map Φ is a group monomorphism.Given a subset S of PAut(C )⊆ Sym(β ), we define the set S = Φ(S ) = {Φ(τ) :τ ∈S }, which is a subset of PAut(C)⊆ Sym(2β ).A set I = {i1, . . . , iγ+δ} ⊆ {1, . . . ,β} of γ + δ coordinate positions is saidto be a quaternary information set for the code C if the set Φ(I ), defined asΦ(I ) = {2i1−1,2i1, . . . ,2iδ −1,2iδ ,2iδ+1−1, . . . ,2iδ+γ −1}, is an informationset for C =Φ(C ) for some ordering of elements of I .Let S be an s-PD-set of size s+1. The set S is a nested s-PD-set if there is anordering of the elements of S, S= {σ1, . . . ,σs+1}, such that Si = {σ1, . . . ,σi+1}⊆ Sis an i-PD-set of size i+1, for all i ∈ {1, . . . ,s}.Proposition 2. Let C be a Z2Z4-additive code of type (0,β ;γ,δ ) with quaternaryinformation setI and let s be a positive integer. If τ ∈ PAut(C ) has at least γ+δdisjoint cycles of length s+1 such that there is exactly one quaternary informationposition per cycle of length s+ 1, then S = {Φ(τ i)}s+1i=1 is an s-PD-set of size s+1 for the Z4-linear code C = Φ(C ) with information set Φ(I ). Moreover, anyordering of the elements of S gives a nested r-PD-set for any r ∈ {1, . . . ,s}.Example 3. Let C0,3 be the Z2Z4-additive Hadamard code of type (0,16;0,3) withgenerator matrixG0,3 = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 1 2 3 0 1 2 3 0 1 2 3 0 1 2 30 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 .Let τ = (1,16,11,6)(2,7,12,13)(3,14,9,8)(4,5,10,15) ∈ PAut(C0,3)⊆ Sym(16)[9]. It is straightforward to check that I = {1,2,5} is a quaternary informationset forC0,3. Note that each information position inI is in a different cycle of τ . Letσ = Φ(τ) ∈ PAut(C0,3) ⊆ Sym(32), where C0,3 = Φ(C0,3). Thus, by Proposition1372, S = {σ ,σ2,σ3,σ4} is a 3-PD-set of size 4 for C0,3 with information set I ={1,2,3,4,9,10}. Note that C0,3 is the smallest Hadamard Z4-linear code that is abinary nonlinear code.3 s-PD-sets for Hadamard Z4-linear codesLet 0,1,2 and 3 be the repetition of symbol 0, 1, 2 and 3, respectively. Let Gγ,δ be agenerator matrix of the Z2Z4-additive Hadamard code Cγ,δ of length β = 2m−1 andtype (0,β ;γ,δ ), where m = γ+2δ −1. A generator matrix for the Z2Z4-additiveHadamard code Cγ+1,δ of length β ′ = 2β = 2m and type (0,β ′;γ + 1,δ ) can beconstructed as follows [7]:Gγ+1,δ =(0 2Gγ,δ Gγ,δ). (3)Equivalently, a generator matrix for the Z2Z4-additive Hadamard code Cγ,δ+1of length β ′′ = 4β = 2m+1 and type (0,β ′′;γ,δ +1) can be constructed as [7]:Gγ,δ+1 =(Gγ,δ Gγ,δ Gγ,δ Gγ,δ0 1 2 3). (4)Note that a generator matrix for every code Cγ,δ can be obtained by applying(3) and (4) recursively over the generator matrix G0,1 = (1) of the code C0,1. Fromnow on, we assume that Cγ,δ is obtained by using these constructions.Proposition 4. Let Cγ,δ be a Z2Z4-additive Hadamard code of type (0,β ;γ,δ )with quaternary information set I . The set I ∪{β +1} is a suitable quaternaryinformation set for both codes Cγ+1,δ and Cγ,δ+1 obtained from Cγ,δ by applyingconstructions (3) and (4), respectively.Despite the fact that the quaternary information set is the same for Cγ+1,δ andCγ,δ+1, the information set for the corresponding binary codes Cγ+1,δ and Cγ,δ+1are I′ =Φ(I )∪{2β +1} and I′′ =Φ(I )∪{2β +1,2β +2}, respectively.Given two permutations σ1 ∈ Sym(n1) and σ2 ∈ Sym(n2), we define the per-mutation (σ1|σ2)∈ Sym(n1+n2), where σ1 acts on the coordinates {1, . . . ,n1} andσ2 acts on the coordinates {n1+1, . . . ,n1+n2}. Given σi ∈ Sym(ni), i∈{1, . . . ,4},we define the permutation (σ1|σ2|σ3|σ4) in the same way.Proposition 5. Let S be an s-PD-set of size l for the Hadamard Z4-linear codeCγ,δ of binary length n= 2β and type (0,β ;γ,δ ) with respect to an information setI. Then the set (S|S) = {(σ |σ) : σ ∈ S} is an s-PD-set of size l with respect to theinformation set I′ = I∪{n+1} for the Hadamard Z4-linear code Cγ+1,δ of binarylength 2n and type (0,2β ;γ+1,δ ) constructed from (3) and the Gray map.138Example 6. Let S be the 3-PD-set of size 4 for C0,3 of binary length 32 with respectto the information set I = {1,2,3,4,9,10}, given in Example 3. By Propositions 4and 5, the set (S|S) is a 3-PD-set of size 4 for the Hadamard Z4-linear code C1,3 ofbinary length 64 with respect to the information set I′ = {1,2,3,4,9,10,33}.Proposition 5 can not be generalized directly for Hadamard Z4-linear codesCγ,δ+1 constructed from (4). Note that if S is an s-PD-set for the Hadamard Z4-linear code Cγ,δ , then the set (S|S|S|S) = {(σ |σ |σ |σ) : σ ∈ S} is not in general ans-PD-set for the Hadamard Z4-linear code Cγ,δ+1.Proposition 7. LetS ⊆ PAut(Cγ,δ ) such thatΦ(S ) is an s-PD-set of size l for theHadamard Z4-linear code Cγ,δ of binary length n = 2β and type (0,β ;γ,δ ) withrespect to an information set I. Then the set Φ((S |S |S |S )) = {Φ((τ|τ|τ|τ)) :τ ∈S } is an s-PD-set of size l with respect to the information set I′′ = I ∪{n+1,n+ 2} for the Hadamard Z4-linear code Cγ,δ+1 of binary length 4n and type(0,4β ;γ,δ +1) constructed from (4) and the Gray map.Example 8. Let S = {τ,τ2,τ3,τ4}, where τ is defined as in Example 3. ByProposition 7, the set Φ((S |S |S |S )) is a 3-PD-set of size 4 for the HadamardZ4-linear code C0,4 of binary length 128 with respect to the information set I′ ={1,2,3,4,9,10,33,34}.Propositions 5 and 7 can be applied recursively to acquire s-PD-sets for theinfinite family of Hadamard Z4-linear codes obtained (by using constructions (3)and (4)) from a given Hadamard Z4-linear code where we already have such set.References[1] R. Barrolleta and M. Villanueva, “Partial permutation decoding for binary linear Hadamardcodes," Electronic Notes in Discrete Mathematics, 46 (2014) 35-42.[2] J. J. Bernal, J. Borges, C. Fernández-Córboda, and M. Villanueva, “Permutation decoding ofZ2Z4-linear codes," Des. Codes and Cryptogr., DOI 10.1007/s10623-014-9946-4, 2014.[3] J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifà, and M. Villanueva, “Z2Z4-linear codes:generator matrices and duality," Des. Codes and Cryptogr., vol. 54: 167–179, 2010.[4] W. Fish, J. D. Key, and E. Mwambene, “Partial permutation decoding for simplex codes,"Advances in Mathematics of Comunications, vol. 6(4): 505–516, 2012.[5] W. C. Huffman, Codes and groups, Handbook of coding theory, 1998.[6] D. S. Krotov and M. Villanueva “Classification of the Z2Z4-linear Hadamard codes and theirautomorphism groups," IEEE Trans. Inf. Theory, vol. 61(2): 887–894, 2015.[7] D. S. Krotov, “Z4-linear Hadamard and extended perfect codes," Electronic Notes in DiscreteMathematics, vol. 6 (2001), 107-112.[8] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, 1977.[9] J. Pernas, J. Pujol and M. Villanueva. “Characterization of the automorphism group of quater-nary linear Hadamard Codes," Des. Codes Cryptogr., 70(1-2), 105–115, 2014.139

PD-sets for (nonlinear) Hadamard Z₄-linear codes

https://ddd.uab.cat/pub/caplli/2016/142919/promatstaC_a2016.pdf

PD-sets for (nonlinear) Hadamard Z₄-linear codes

Abstract

Similar works

Full text

Available Versions

Diposit Digital de Documents de la UAB