In this paper infinite families of linear binary nested completely regular
codes are constructed. They have covering radius $\rho$ equal to $3$ or $4$,
and are $1/2^i$-th parts, for $i\in\{1,\ldots,u\}$ of binary (respectively,
extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where
$m=2u$. In the usual way, i.e., as coset graphs, infinite families of embedded
distance-regular coset graphs of diameter $D$ equal to $3$ or $4$ are
constructed. In some cases, the constructed codes are also completely
transitive codes and the corresponding coset graphs are distance-transitive

Borges, J.

Rifà, J.

Zinoviev, V. A.

English

arXiv

Publicació amb motiu de la 14th International Workshop on Algebraic and Combinatorial Coding Theory (September 7-13, 2014, Svetlogorsk, Russia)Infinite families of linear binary nested completely regular codes with covering radius ρ equal to 3 and 4 are constructed. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter D = 3 or 4 are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive

Borges, Joaquim

Rifà i Coma, Josep

Zinoviev, Victor

Diposit Digital de Documents de la UAB

Fourteenth International Workshop on Algebraic and Combinatorial Coding TheorySeptember 7–13, 2014, Svetlogorsk (Kaliningrad region), Russia pp. 85–90On nested completely regular codesand distance regular graphs1J. Borges quim@deic.uab.catJ. Rifa` josep.rifa@autonoma.eduDepartment of Information and Communications Engineering, Autonomous Universityof Barcelona, SpainV. A. Zinoviev zinov@iitp.ruA.A. Kharkevich Institute for Problems of Information Transmission, Moscow, RussiaAbstract. Infinite families of linear binary nested completely regular codes withcovering radius ρ equal to 3 and 4 are constructed. In the usual way, i.e., as cosetgraphs, infinite families of embedded distance-regular coset graphs of diameter D =3 or 4 are constructed. In some cases, the constructed codes are also completelytransitive codes and the corresponding coset graphs are distance-transitive.1 IntroductionLet Fq be the finite field of order q ≥ 2 and C be a binary linear [n, k, d] codeof length n, dimension k and minimum distance d. The automorphism groupAut(C) of C consists of all permutations of the n coordinate positions whichsend C into itself. Aut(C) acts in a natural way over the set of cosets of C:pi(C + v) = C + pi(v) for every v ∈ Fn2 and pi ∈ Aut(C).For any v ∈ Fn2 its distance to the code C is d(v, C) = minx∈C{d(v,x)}and the covering radius of the code C is ρ = maxv∈Fn2{d(v, C)}. Let J ={1, 2, . . . , n} be the set of coordinate positions of vectors from Fn2 . Denote bySupp(x) the support of the vector x = (x1, . . . , xn) ∈ Fn2 , i.e., Supp(x) = {j ∈J : xj 6= 0}. Say that two vectors x,y ∈ Fn2 are neighbors if d(x,y) = 1 andalso say that vector x covers vector y if Supp(y) ⊆ Supp(x).For a given binary code C such that 0 ∈ C and with covering radius ρ defineC(i) = {x ∈ Fn2 : d(x, C) = i}, i = 1, 2, . . . , ρ.Definition 1. A code C with covering radius ρ is completely regular, if for alll ≥ 0 every vector x ∈ C(l) has the same number cl of neighbors in C(l−1) andthe same number bl of neighbors in C(l+1). Also, define al = (q− 1)n− bl− cland note that c0 = bρ = 0.For a completely regular code, define (b0, . . . , bρ−1; c1, . . . , cρ) as the inter-section array of C.1This work has been partially supported by the Spanish MICINN grant TIN2013-40524-P and the Catalan AGAUR grant 2014SGR-691 and by the Russian fund of fundamentalresearches (under the project No. 12-01-00905).86 ACCT 2014Definition 2. [8] A binary linear code C with covering radius ρ is completelytransitive if Aut(C) has ρ+ 1 orbits when acts on the cosets of C.Clearly any completely transitive code is completely regular.Existence and enumeration of completely regular and completely transitivecodes are open hard problems (see [3, 6, 8] and references there). The purposeof this paper is to construct nested infinite families of completely regular codeswith covering radius ρ equal to 3 and 4. When m is growing the length of thechain of these nested codes (with constant covering radius) is also growing. Forlength n = 2m−1, where m = 2u, each family is formed by u nested completelyregular codes of length n with the same covering radius ρ = 3. The last codein the nested family, so the code with the smallest cardinality is a 1/2u-th partof a Hamming code of length n. These last codes are known to be completelyregular codes due to Calderbank and Goethals [4]. These nested families ofcompletely regular codes and their extended codes induces infinite families ofembedded distance-regular coset graphs with diameters 3 and 4, which also giveinteresting families of embedded covering graphs. We point out that in somecases such completely regular codes are also completely transitive and hencethe corresponding coset graphs are also distance transitive.2 Preliminary resultsLet Γ be a finite connected simple graph (i.e., undirected, without loops andmultiple edges). Let d(γ, δ) be the distance between two vertices γ and δ (i.e.,the number of edges in the minimal path between γ and δ). The diameterD of Γ is its largest distance. Two vertices γ and δ from Γ are neighbors ifd(γ, δ) = 1. Denote Γi(γ) = {δ ∈ Γ : d(γ, δ) = i}.An automorphism of a graph Γ is a permutation pi of the vertex set of Γsuch that, for all γ, δ ∈ Γ we have d(γ, δ) = 1, if and only if d(piγ, piδ) = 1.Let Γi be the graph with the same vertices of Γ, where an edge (γ, δ) is definedwhen the vertices γ, δ are at distance i in Γ. Clearly, Γ1 = Γ. The graph Γ iscalled primitive if Γ and all Γi (i = 2, . . . ,D) are connected. Otherwise, Γ iscalled imprimitive. A graph is called complete (or a clique) if any two of itsvertices are adjacent.Definition 3. [3] A simple connected graph Γ is called distance-regular if it isregular of valency k, and if for any two vertices γ, δ ∈ Γ at distance i apart,there are precisely ci neighbors of δ in Γi−1(γ) and bi neighbors of δ in Γi+1(γ).Furthermore, this graph is called distance-transitive, if for any pair of verticesγ, δ at distance d(γ, δ) there is an automorphism pi from Aut(Γ) which movethis pair (γ, δ) to any other given pair γ′, δ′ of vertices at the same distanced(γ, δ) = d(γ′, δ′).The sequence (b0, b1, . . . , bD−1; c1, c2, . . . , cD), where D is the diameter of Γ,is called the intersection array of Γ. Clearly b0 = k, bD = c0 = 0, c1 = 1.Borges, Rifa, Zinoviev 87Let C be a linear completely regular code with covering radius ρ and inter-section 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, i.e., there are v ∈ Band v′ ∈ B′ such that d(v,v′) = 1.Lemma 1. [3, 7] Let C be a linear completely regular code with covering radiusρ and intersection array (b0, . . . , bρ−1; c1, . . . cρ) and let ΓC be the coset graph ofC. Then ΓC is distance-regular of diameter D = ρ with the same intersectionarray as the code C. If C is completely transitive, then ΓC is distance-transitive.Given a code C with d = 2e+ 1, denote by C∗ the extended code, i.e., thecode obtained from C by adding an overall parity checking position. Now wegive a lemma, which is an strengthening of a result from [1].Lemma 2. Let C be a completely regular linear code of length n = 2m − 1with minimum distance d = 3, covering radius ρ = 3 and intersection array(n, b1, 1; 1, c2, n). Let the orthogonal code C⊥ have nonzero weights wi, i =1, 2, 3. Then the extended code C∗ is completely regular with covering radiusρ∗ = 4 and intersection array (n+ 1, n, b1, 1; 1, c2, n, n+ 1), if and only if w1 +w3 = 2w2 = n+ 1.3 Completely regular nested codesPresent the elements of F2m as elements in a quadratic extension of F2u . Letβ = αr be a primitive element of F2u and let F2m = F2u [α]. Every elementγ ∈ F2m can be presented as γ = γ1 + γ2α ∈ F2u [α], where γ1, γ2 ∈ F2u . Theparity check matrix of the Hamming code Hm of length n = 2m − 1, which wedenote by Hm, can also be written as the binary matrix of size (2u×n), wherethe columns are binary presentations of [γi, γj ] with γi, γj ∈ {0, β1, . . . , βq−1}.Let Em be the binary representation of the matrix [α0r, αr, . . . , α(n−1)r ].Take the matrix Pm as the vertical join of Hm and Em. It is well known [4]that the code C(u) with parity check matrix Pm is a cyclic binary completelyregular code with covering radius ρ = 3, minimum distance d = 3 and dimensionn− (m+u). The generator polynomial of C(u) is g(x) = mα(x)mαr (x) ∈ F2[x],where mαi(x) means the minimal polynomial associated to αi.Denote by ei the vector with only one nonzero coordinate of value 1 inith position. Binary vectors v ∈ Fn2 can be written as v =∑i∈Ivei, whereIv = Supp(v). The elements in F2m can also be seen as elements in F2u [α].The positions of vectors in Fn2 can be enumerated by using the nonzero ele-ments in F2m, or as elements in F2u [α] by substituting any αi ∈ F2m with thecorresponding αi = γi1 + γi2α ∈ F2u [α], where γi1, γi2 ∈ F2u .88 ACCT 2014For any v =∑i∈Ivei ∈ Fn2 , denote S(v) =∑i∈Ivγi1γi2 ∈ F2u . The nextlemma gives a new description for the code C(u).Lemma 3. The code C(u) consists of elements v ∈ Fn2 , with syndromes HmvT =0 and S(v) = 0.The code C(u) is a binary [n = 2m − 1, k = n − m − u] code and it isa subcode of the [2m − 1, n − m] Hamming code Hm. The number of cosetsC(u)+v, of weight three, is 2u−1. Indeed, their syndromes S(v) are the nonzeroelements of F2u . For i ∈ {0, . . . , u}, taking u−i cosets C(u)+v1, . . . , C(u)+vu−iwith independent syndromes S(v1), . . . , S(vu−i) (independent, means that theyare independent binary vectors in Fu2) we can generate a linear binary codeC(i) = 〈C(u),v1, . . . ,vu−i〉.The dimension of the code C(u) is dim(C(u)) = n −m− u and, in general,dim(C(i)) = u− i+dim(C(u)). Note that the maximum number of independentsyndromes we can take is u, so the biggest code we can obtain is of dimensionu + dim(C(u)) = n − m, which is the Hamming code C(0) = Hm. All theconstructed codes contains C(u) and they are contained in the Hamming codeC(0).Theorem 1. Let i ∈ {0, 1, u} for m = 2u ≥ 8 and i ∈ {0, 1, 2, 3} for m = 6.The codes C(i) and C(i)∗ are completely transitive.We conjecture that codes C(i) and C(i)∗ are completely transitive if and onlyif i = 0, i = 1, i = u or 2i ≤ u+ 1, for i ∈ {2, . . . , u− 1}.Theorem 2. Let i ∈ {0, . . . , u} and m = 2u. The codes C(i) and C(i)∗ arecompletely regular with intersection arrays (2m−1, 2m−2m−i, 1; 1, 2m−i, 2m−1)and (2m, 2m − 1, 2m − 2m−i, 1; 1, 2m−i, 2m − 1, 2m), respectively.4 Nested antipodal distance regular graphsof diameter 3 and 4A graph Γ with diameter D ≥ 3 is called antipodal if all vertices at distanceD from a given vertex are at distance D from each other [3], i.e., graph ΓDis a disjoint union of cliques. Such a graph is imprimitive by definition. Inthis case, the folded graph, or antipodal quotient of Γ is defined as the graphΓ¯, whose vertices are the maximal cliques (which are called fibres) of ΓD, withtwo adjacent if and only if there is an edge between them in Γ. If, in addition,each vertex γ has the same valency as its image under folding, then Γ is calledan antipodal covering graph of Γ¯. If, moreover, all fibres of ΓD have the samesize r, then Γ is also called an antipodal r-cover of Γ¯.Borges, Rifa, Zinoviev 89Denote by Γ(i) (respectively, Γ(i)∗) the coset graph, obtained from the codeC(i) (respectively C(i)∗). Since all cosets of weight 3 (respectively, of weight 4)of the Hamming code Hm (respectively, of the extended Hamming code H∗m)belong to this code, we conclude that all graphs Γ(i) (respectively, Γ(i)∗) areantipodal.Lemma 4. [5] Let Γ be an antipodal distance-regular graph of diameter three.Then Γ is a r-fold covering graph of Kn, for some r and n and recall that c2is the number of common neighbors of two vertices in Γ at distance two. Thenthe intersection array of Γ is (n− 1, (r − 1)c2, 1; 1, c2, n − 1).As a direct result of Theorem 2 and taking into account [5, 7] we obtain thefollowing new distance-regular and distance-transitive coset graphs.Theorem 3. For any m = 2u ≥ 4, there exist a family of embedded an-tipodal distance-regular coset graphs Γ(i) with 22u+i vertices and diameter 3,for i = 1, . . . , u. Graph Γ(0) has diameter 1, i.e., it is a complete graphKn,n = 2m − 1. Specifically:(i) Γ(i), i = 1, . . . , u has intersection array (2m−1, 2m−2m−i, 1; 1, 2m−i , 2m−1).(ii) Γ(i) is a subgraph of Γ(i+1) for all i = 0, 1, . . . , u− 1.(iii) Γ(i) covers Γ(j) with parameters (2m−1, 2i−j , 22u−i+j), for j ∈ {0, ..., i−1}.(iv) Γ(i) is distance-transitive for i ∈ {0, 1, u} when m ≥ 8 and for i ∈{0, 1, 2, 3} when m = 6.Theorem 4. For any m = 2u ≥ 4 and i = 0, 1, . . . , u there exist a family ofembedded antipodal distance-regular coset graphs Γ(i)∗ with 2m+i+1 vertices anddiameter 4. Specifically:(i) Γ(i)∗ has intersection array (2m, 2m − 1, 2m − 2m−i, 1; 1, 2m−i, 2m − 1, 2m).(ii) Γ(i)∗ is a subgraph of Γ(i+1)∗ for all i = 0, 1, . . . , u− 1.(iii) Γ(i)∗ covers Γ(j)∗, where j = 0, 1, . . . , i − 1 with the size of the fibreri,j = 2i−j .(iv) Γ(i)∗ is distance-transitive for i = 0, 1, u when m ≥ 8 and i = 0, 1, 2, 3when m = 6.We conjecture that the graphs Γ(i) and Γ(i)∗ are distance-transitive for i ∈{2, . . . , u− 1} and 2i ≤ u+ 1.The first graphs Γ(1) and Γ(1)∗ are well known distance-transitive graphs(see [1, 2] and references there). Graphs Γ(u) and Γ(u)∗ are also known. Thecorresponding codes C(u) and C(u)∗ have been presented in a very symmetricform by Calderbank and Goethals [4]. They proved that these codes form as-sociation schemes, which immediately implies the existence of the correspond-ing distance-regular graphs Γ(u) and Γ(u)∗ [3, Ch. 11]. All graphs Γ(i) fori = 0, 1, . . . , u have been constructed by Godsil and Hensel using the QuotientConstruction [5]. But it was not mentioned in all references above that some90 ACCT 2014of these graphs are completely transitive. Besides, except for the graphs Γ(u),it was not stated that these graphs can be constructed as coset graphs. Wecould not found the graphs Γ(i)∗ for i = 2, . . . , u − 1 in the above mentionedliterature.References[1] J. Borges, J. Rifa, V.A. Zinoviev, “New families of completely regular codesand their corresponding distance regular coset graphs”, Designs, Codes andCryptography, (2014), vol.70, pp:139-148. DOI 10.1007/s10623-012-9713-3.[2] J. Borges, J. Rifa, V.A. Zinoviev, “New families of completely transitivecodes”, Discrete Mathematics, 2014, to appear.[3] A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs,Springer, Berlin, 1989.[4] A.M. Calderbank, J.-M. Goethals, Three-weights codes and associationschemes, Philips J. Res., vol. 39, 143-152, 1984.[5] C.D. Godsil, A.D. Hensel, “Distance regular covers of the complete graph”,J. Comb. Theory, Ser. B, 1992, vol. 56, 205 - 238.[6] A. Neumaier, “Completely regular codes,” Discrete Maths., vol. 106/107,pp. 335-360, 1992.[7] J. Rifa`, J. Pujol, “Completely transitive codes and distance transitivegraphs,” Proc, 9th International Conference, AAECC-9, no. 539 LNCS,360-367, Springer-Verlag, 1991.[8] P. Sole´, “Completely Regular Codes and Completely Transitive Codes,”Discrete Maths., vol. 81, pp. 193-201, 1990.

On nested completely regular codes and distance regular graphs

In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius ρ equal to 3 or 4, and are 1/2 i th parts, for i ∈ {1, ... , u} of binary (respectively, extended binary) Hamming codes of length n = 2 m - 1 (respectively, 2 m ), where m = 2u. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter D equal to 3 or 4 are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive. This gives antipodal covers of some distance-regular and distance-transitive graphs

Families of nested completely regular codes and distance-regular graphs

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