We obtain several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power q and any two natural numbers a, b, we construct completely transitive codes over different fields with covering radius ρ = min{a, b} and identical intersection array, specifically, we construct one code over F_{q^r} for each divisor r of a or b. As a corollary, for any prime power q, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters

Rifà i Coma, Josep

Zinoviev, Victor

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This is the accepted version of the article:Rifà i Coma, Josep; Zinoviev, Victor. «About non equivalent completely regularcodes with identical intersection array». Electronic Notes in Discrete Mathemat-ics, Vol. 57 (Març 2017), p. 181-186. DOI 10.1016/j.endm.2017.02.030This version is avaible at https://ddd.uab.cat/record/171006under the terms of the licenseAbout non equivalent completely regular codeswith identical intersection arrayJ. Rifa` 1,2Department of Information and Communications EngineeringUniversitat Auto`noma de BarcelonaSpainV. A. Zinoviev 1,3A.A. Kharkevich Institute for Problems of Information TransmissionMoscow, RussiaAbstractWe obtain several classes of completely regular codes with different parameters, butidentical intersection array. Given a prime power q and any two natural numbersa, b, we construct completely transitive codes over different fields with coveringradius ρ = min{a, b} and identical intersection array, specifically, we construct onecode over Fqr for each divisor r of a or b. As a corollary, for any prime power q, weshow that distance regular bilinear forms graphs can be obtained as coset graphsfrom several completely regular codes with different parameters.Keywords: Completely regular codes, coset graphs, distance regular graphs1 This work has been partially supported by the Spanish MICINN grants TIN2016-77918-Pand MTM2015-69138-REDT, the Catalan AGAUR grant 2014SGR-691 and by the Russianfund of fundamental researches (under the project No. 15-01-08051).2 Email: josep.rifa@uab.cat3 Email: zinov@iitp.ru1 IntroductionLet Fq be a finite field of the order q and F∗q = Fq \ {0}. A q-ary linear codeC of length n is a k-dimensional subspace of Fnq . Given any vector v ∈ Fnq , itsdistance to the code C is d(v, C) = minx∈C{d(v,x)}, the minimum distanceof the code is d = minv∈C{d(v, C\{v})} and the covering radius of the codeC is ρ = maxv∈Fnq{d(v, C)}. Two vectors x and y are neighbors if d(x,y) = 1.We say that C is a [n, k, d; ρ]q-code. Let D = C+x be a coset of C, where +means the component-wise addition in Fq. For a given q-ary code C we defineC(i) = {x ∈ Fnq : d(x, C) = i}, i = 1, 2, ..., ρ.Definition 1.1 [6] A q-ary code C is completely regular, if for all l ≥ 0 everyvector x ∈ C(l) has the same number cl of neighbors in C(l−1) and the samenumber bl of neighbors in C(l + 1). Define al = (q − 1)n − bl − cl and setc0 = bρ = 0. Denote by (b0, . . . , bρ−1; c1, . . . , cρ) the intersection array of C.Let M be a monomial matrix, i.e. a matrix with exactly one nonzero entryin each row and column. If q is a power of a prime number, then Aut(C) consistof all monomial (n× n)-matrices M over Fq such that cM ∈ C for all c ∈ Cand also contains any field automorphism of Fq which preserves C. The groupAut(C) acts on the set of cosets of C in the following way: for all σ ∈ Aut(C)and for every vector v ∈ Fnq we have (v + C)σ = vσ + C.Definition 1.2 [4,10] A linear code C over Fq with covering radius ρ is com-pletely transitive if Aut(C) has ρ+ 1 orbits when acts on the cosets of C.Since two cosets in the same orbit should have the same weight distribution,it is clear, that any completely transitive code is completely regular.Completely regular and completely transitive codes are classical subjectsin algebraic coding theory, which are closely connected with graph theory,combinatorial designs and algebraic combinatorics. Existence, constructionand enumeration of all such codes are open hard problems (see [1,3,5,6] andreferences there).In a recent paper [8] we described an explicit construction, based on theKronecker product of parity check matrices, which provides, for any naturalnumber ρ and for any prime power q, an infinite family of q-ary linear com-pletely regular codes with covering radius ρ. In [9] we presented another classof q-ary linear completely regular codes with the same property, based onlifting of perfect codes. Here, we extend the Kronecker product constructionto the case when component codes have different alphabets and connect theresulting completely regular codes with the codes obtained by lifting q-aryperfect codes. This gives several different infinite classes of completely regularcodes with different parameters and with identical intersection arrays.Definition 1.3 For two matrices A = [ar,s] and B = [bi,j ] over Fq define a newmatrix H which is the Kronecker product H = A ⊗ B, where H is obtainedby changing any element ar,s in A by the matrix ar,sB.Definition 1.4 Let C be the [n, k, d]q code with parity check matrix H where1 ≤ k ≤ n− 1 and d ≥ 3. Denote by Cr the [n, k, d]qr code over Fqr with thesame parity check matrix H . Say that code Cr is obtained by lifting C to Fqr .2 Extending the Kronecker product constructionRecall that by C(H) we denote the code defined by the parity check matrix H ,by Hqm we denote the parity check matrix of the q-ary Hamming [n, n−m, 3]qcode C = C(Hqm) of length n = (qm − 1)/(q − 1), and by Cr(Hqm) we denotethe code (with the same length n) obtained by lifting C(Hqm) to the field Fqr .Considering the Kronecker construction obtained in [8] we could see thatthe alphabets of both matrices A = [ai,j] and B should be compatible to eachother in the sense that the multiplication ai,jB can be carried out. To havethis compatibility it is enough that, say, the matrix A is over Fqu and B isover Fq. First, we consider the covering radius of the resulting codes.Lemma 2.1 Let C(Hquma) and C(Hqmb) be two Hamming codes with parameters[na, na−ma, 3]qu and [nb, nb−mb, 3]q, respectively, where na = (quma−1)/(qu−1), nb = (qmb−1)/(q−1), q is a prime power, ma, mb ≥ 2, and u ≥ 1. Thenthe code C with parity check matrix H = Hquma⊗Hqmb, the Kronecker productof Hqumaand Hqmb, has covering radius ρ = min{uma, mb}.The following statement generalizes the results of [8,9].Theorem 2.2 Let C(Hquma) and C(Hqmb) be two Hamming codes with param-eters [na, na −ma, 3]qu and [nb, nb −mb, 3]q, respectively, where na = (quma −1)/(qu − 1), nb = (qmb − 1)/(q − 1), q is a prime power, ma, mb ≥ 2, andu ≥ 1. Let C be the code with parity check matrix H = Hquma⊗Hqmb. Then(i) The code C is a completely transitive, and completely regular, [n, k, d; ρ]qucode with parameters:n = na nb, k = n−mamb, d = 3, ρ = min{uma, mb}.(ii) The intersection numbers of the code C are:bℓ =(quma−qℓ)(qmb−qℓ)(q−1), (0 ≤ ℓ ≤ ρ−1); and cℓ = qℓ−1 qℓ−1q−1, (1 ≤ ℓ ≤ ρ).(iii) The lifted code Cmb(Hquma) is a completely regular code with the sameintersection array as C.Remark 2.3 We have to remark here that in the statement (iii) we cannot choose the code Cmb(Hquma) (instead of Cmb(Hquma)), which seems to benatural. We emphasize that the codes Cmb(Hquma) and Cmb(Hquma) are not onlydifferent completely regular codes, but they induce different distance-regulargraphs with different intersection arrays. So, the code Cmb(Hquma) suits tothe codes from (i) in the sense that it has the same intersection array. Forexample, the code C2(H223 ) induces a distance-regular graph with intersectionarray (315, 240; 1, 20) and the code C2(H26 ) gives a distance-regular graph withintersection array (189, 124; 1, 6).Remark 2.4 Theorem 2.2 above can not be extended to the more general casewhen the alphabets Fqa and Fqb of component codes CA and CB, respectively,neither Fqa is a subfield of Fqb or vice versa Fqb is a subfield of Fqa . We illustrateit by considering the smallest nontrivial example. Take two Hamming codes,the [5, 3, 3] code CA over F22 with parity check matrix H222 , and the [9, 7, 3]code CB over F23 with parity check matrix H232 . Then the resulting [45, 41, 3]code C = C(H222 ⊗ H232 ) over F26 is not even uniformly packed in the widesense, since it has the covering radius ρ = 3 and the outer distance s = 7,which can be checked by considering the parity check matrix of C.3 CR-codes with the same intersection arrayIn [9, Theo. 2.11] it is proved that lifting a q-ary Hamming code C(Hqm)to Fqs we obtain a completely regular code Cs(Hqm) which is not necessarilyisomorphic to the code Cm(Hqs ). However, both codes Cs(Hqm) and Cm(Hqs )have the same intersection array. As we saw above, the code obtained bythe Kronecker product construction, or our extension for the case when thecomponent codes have different alphabets, can have the same intersectionarray. The next statement is one of the main results of our paper.Theorem 3.1 Let q be any prime power and let a, b, u be any natural numbers.Then:(i) There exist the following completely regular [n, k, d; ρ]-codes with differentparameters, where d = 3 and ρ = min{ua, b}:(a) Cua(Hqb ) over Fqua with n =qb−1q−1, k = n− b;(b) Cb(Hqua) over Fqb with n =qua−1q−1, k = n− ua;(c) C(Hqb ⊗Hqua) over Fq with n =qua−1q−1× qb−1q−1, k = n− bua;(d) C(Hqb ⊗Hqau ) over Fqa with n =qb−1q−1× qua−1qa−1, k = n− bu;(e) C(Hqb ⊗Hqua ) over Fqu with n =qb−1q−1× qua−1qu−1, k = n− ba;(ii) All above codes have the same intersection numbersbℓ =(qb−qℓ)(qua−qℓ)(q−1), (0 ≤ ℓ ≤ ρ− 1); and cℓ = qℓ−1 qℓ−1q−1, (1 ≤ ℓ ≤ ρ).(iii) All above codes from Kronecker constructions are completely transitive.It is easy to see that the number of different completely transitive (and,therefore, completely regular) codes with different parameters and the sameintersection array is growing with q.4 Coset distance-regular graphsLet C be a linear completely regular code with covering radius ρ and intersec-tion array (b0, . . . , bρ−1; c1, . . . cρ). Let {B} be the set of cosets of C. Definethe graph ΓC , which is called the coset graph of C, taking all different cosetsB = C + x as vertices, with two vertices γ = γ(B) and γ′ = γ(B′) adjacent ifand only if the cosets B and B′ contain neighbor vectors.Lemma 4.1 [1, 7] Let C be a linear completely regular code with coveringradius ρ and intersection array (b0, . . . , bρ−1; c1, . . . cρ) and let ΓC be the cosetgraph of C. Then ΓC is distance-regular of diameter D = ρ with the sameintersection array. If C is completely transitive, then ΓC is distance-transitive.From all different completely transitive codes described in Theorem 3.1, weobtain distance-transitive graphs with classical parameters [1]. These graphshave quab vertices, diameter D = min{ua, b}, and intersection array given by(ii) in Theorem 3.1.Notice that bilinear forms graphs [1, Sec. 9.5] have the same parametersand are distance-transitive too. These graphs are uniquely defined by theirparameters [1, Sec. 9.5]. Therefore, all graphs coming from the completelyregular and completely transitive codes described in Theorem 3.1 are bilinearforms graphs. We did not find in the literature (in particular in [2], where theassociation schemes, formed by bilinear forms, have been introduced) the de-scription of these graphs, as many different coset graphs of completely regularcodes. It is also known that these graphs are not antipodal and do not haveantipodal covers [1, Sec. 9.5].Theorem 4.2 Let C1, C2, . . ., Ck be a family of linear completely transitivecodes constructed by Theorem 2.2 and let ΓC1 , ΓC2 , . . . ,ΓCk be their corre-sponding coset graphs. Then:(i) Any graph ΓCi is a distance-transitive graph, induced by bilinear forms.(ii) If any two codes Ci and Cj have the same intersection array, then thegraphs ΓCi and ΓCj are isomorphic.(iii) If the graph ΓCi has qm vertices, where m is not a prime, then it canbe presented as a coset graph by several different ways, depending on thenumber of factors of m.References[1] A.E. Brouwer, A.M. Cohen, A. Neumaier, “Distance-Regular Graphs”, Ed.Springer, Berlin, 1989.[2] P. Delsarte, Bilinear forms over a finite field with applications to coding theory,Journal of Combinatorial Theory, Series A, 25 (1978), 226-241.[3] N. I. Gillespie, C. E. Praeger, Uniqueness of certain completely regularHadamard codes, Journal of Combinatorial Theory, Series A, 120.7 (2013),1394-1400.[4] M. Giudici, C.E. Praeger, Completely Transitive Codes in Hamming Graphs,Europ. J. Combinatorics 20 (1999), 647-662.[5] J. H. Koolen, W. S. Lee, W. J. Martin, Characterizing completely regularcodes from an algebraic viewpoint, Combinatorics and Graphs, ContemporaryMathematics 31 (2009), 223-242.[6] A. Neumaier, Completely regular codes, Discrete Maths., 106/107 (1992), 335-360.[7] J. Rifa`, J. Pujol, Completely transitive codes and distance-transitive graphs, in:Proc. 9th International Conference, AAECC-9, in: LNCS, 539 (1991), 360-367.[8] J. Rifa`, V.A. Zinoviev, New completely regular q-ary codes, based on Kroneckerproducts, IEEE Transactions on Information Theory, 56.1 (2010), 266-272.[9] J. Rifa`, V.A. Zinoviev, On lifting perfect codes, IEEE Transactions onInformation Theory, 57.9 (2011), 5918-5925.[10] P. Sole´, Completely Regular Codes and Completely Transitive Codes, DiscreteMaths., 81 (1990), 193-201.

About non equivalent completely regular codes with identical intersection array

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About non equivalent completely regular codes with identical intersection array

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