View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 309 (2009) 5011–5016 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On a class of binary linear completely transitive codes with arbitrary covering radiusI J. Rifà a,∗, V.A. Zinoviev b a Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain b Institute for Problems of Information Transmission of the Russian Academy of Sciences, Bol'shoi Karetnyi per. 19, GSP-4, Moscow, 101447, Russia article info a b s t r a c t Article history: An infinite class of new binary linear completely transitive (and so, completely regular) Received 30 October 2007 codes is given. The covering radius of these codes is growing with the length of the code. In Received in revised form 6 November 2008 particular, for any integer ρ ≥ 2, there exist two codes in the constructed class with d D 3, Accepted 5 March 2009 2 ρ 2 ρC1 Available online 29 March 2009 covering radius ρ and lengths 2 and 2 , respectively. The corresponding distance- transitive graphs, which can be defined as coset graphs of these completely transitive codes Keywords: are described. Completely regular code ' 2009 Elsevier B.V. All rights reserved. Completely transitive code Covering radius Distance-regular graph Distance-transitive graph Intersection numbers Outer distance Uniformly packed code 1. Introduction n Let F be the finite field of two elements. A binary .n; N; d/ code C is a subset of F of length n, cardinality N, and minimum n k distance d. When C is a k-dimensional linear subspace of F we will denote it by Tn; k; dU and notice that, in this case, N D 2 . n The support of v D .v1; : : : ; vn/ 2 F is supp.v/ D fj j vj 6D 0g. Say that a vector v covers a vector z if the condition zi 6D 0 implies zi D vi. For a binary .n; N; d/ code C, following Delsarte [7], define the outer distance s D s.C/ of C as the number of nonzero ? D ? ? components ηi , i 1;:::; n of the vector (η0 ; : : : ; ηn / obtained by the MacWilliams transform of the (average) distance distribution η.C/, where η.C/ D (η0; : : : ; ηn/ and 1 ηj D · jf.u; v/ V u; v 2 C; d.u; v/ D jgj: N Hence, if C is a linear Tn; k; dU code then s.C/ is the number of different nonzero weights of the codewords in the dual Tn; n − k; d?U code C?. A linear Tn; k; dU code C can be given by its parity check matrix H of size .n − k/ × n. Code C is the set of all vectors c of t t n length n such that Hc D 0, where c means the transpose vector of c. For any x 2 F we denote by H.x/ the syndrome of vector x, so H.x/ D Hxt. Denote by wt.x/ the (Hamming) weight of the vector x. I This work has been partially supported by the Spanish MEC Grants MTM2006-03250, TSI2006-14005-C02-01 and PCI2006-A7-0616 as well as by the Russian fund of fundamental research (the number of the project, 06 - 01 - 00226). ∗ Corresponding author. E-mail address: [email protected] (J. Rifà). 0012-365X/$ – see front matter ' 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2009.03.004 5012 J. Rifà, V.A. Zinoviev / Discrete Mathematics 309 (2009) 5011–5016 n Given any vector v 2 F , its distance to the code C is d.v; C/ D minx2C fd.v; x/g and the covering radius of the code C is D n f g ρ.C/ maxv2F d.v; C/ : Let D D C C x be a coset of C. The weight wt.D/ of D is the minimum weight of the codewords in D. For a binary code C with covering radius ρ define n C.i/ D fx 2 F V d.x; C/ D ig; i D 1; 2; : : : ; ρ: (1) n n For any vector x 2 F denote by W .x/ the sphere of radius one centered in x, i.e., W .x/ D fy 2 F V d.x; y/ D 1g. We say that two vectors x and y are neighbors if y 2 W .x/. Definition 1 ([8]). A code C is completely regular, if, for all l ≥ 0, every vector x 2 C.l/ has the same number cl of neighbors in C.l − 1/ and the same number bl of neighbors in C.l C 1/. Also, define al D .q − 1/n − bl − cl and notice that c0 D bρ D 0. Define by fb0;:::; bρ−1I c1;:::; cρ g the intersection array of C. For a binary code C let Perm.C/ be its permutation stabilizer group, hence the group of all the coordinate permutations θ such that any vector v D .v1; v2; : : : ; vn/ 2 C is transformed in θ.v/ D .vθ.1/; vθ.2/; : : : ; vθ.n// 2 C. For any θ 2 Perm.C/ and any translate D D C C x of C define the action of θ on D as θ.D/ D C C θ.x/. Definition 2 ([11]). Let C be a binary linear code with covering radius ρ. The code C is said to be completely transitive when n the set fC C x V x 2 F g of all different cosets of C is partitioned under the action of Perm.C/ into exactly ρ C 1 orbits. Since two cosets in the same orbit should have the same weight distribution, it is clear, that any completely transitive code is completely regular too. Classification of completely regular codes is a hard open problem of algebraic coding theory, which is extremely important for distance-regular graphs and association schemes (see [6–8] and references therein). Many completely regular codes come from perfect codes [4,10] and it has been conjectured for a long time [8] that if C is a completely regular code and jCj > 2, then we have e ≤ 3. For completely transitive codes, the problem of existence is solved in [3,5] in the sense that for e ≥ 4 such nontrivial codes do not exist. Solé [11] (see also [1]) describes a construction of an infinite family of completely regular codes with growing covering radius. Taking s copies of perfect 1-error correcting codes of length n we obtain a completely regular code of length s · n and with covering radius ρ D s. In the literature, there are no other infinite families of completely transitive codes with growing covering radius. Our purpose in this paper is to construct a class of binary linear completely regular and completely transitive codes for which the covering radius is growing with the length of the code. These codes are defined by their parity check matrices, which have a very simple structure. The paper is organized as follows. In Section 2 we give the main construction of the completely transitive codes, which have special parity check matrices. The corresponding distance-regular graphs, which can be defined as coset graphs of these completely regular codes are shortly described in Section 3. 2. Main construction m For a given natural number m where m ≥ 3 define F` as the set of all binary vectors of length m and weight `. .m;`/ × m m Definition 3. Denote by H the binary matrix of size m ` , whose columns are exactly all vectors from F` (i.e., each m .m;`/ .m;`/ vector from F` occurs once as a column of H ). Now define the binary linear code C , whose parity check matrix is the matrix H.m;`/. Lemma 1. For any natural numbers m and `, 2 ≤ ` ≤ m − 1 the binary linear Tn; k; dU code C D C.m;`/ has parameters: m for even `: n D ; k D n − m C 1; d ≥ 3; ` and m for odd `: n D ; k D n − m; d ≥ 4: ` Proof. The proof is immediate. .m;l/ .m;l/ Let Perm.C / be the permutation stabilizer group of the code C . Denote by Pi;j the m × m permutation matrix which transposes coordinates i and j, so Pi;j is the m × m identity matrix with the ith column and jth column transposed. .m;l/ .m;l/ Note that Pi;jH coincides with the matrix H after swapping the ith and jth rows. J. Rifà, V.A. Zinoviev / Discrete Mathematics 309 (2009) 5011–5016 5013 .m;`/ .m;l/ n Lemma 2. For any two rows ith and jth in H there exists a permutation θi;j 2 Perm.C /, such that for any x 2 F : .m;l/ .m;l/ H (θi;j.x// D .Pi;jH /.x/: n .m;l/ Proof. Let ri; rj 2 F be, respectively, the ith row and jth row vectors in the matrix H . Vectors ri and rj disagree in − − − − − 2 m 1 − 2 m 2 D 2 m 1 m 2 coordinates and share exactly m 2 nonzero coordinates. The column vectors, where `−1 `−2 `−1 `−2 `−2 ri; rj disagree, can be taken in p pairs and, in each pair, the two columns have exactly the same coordinates in all the rows different from the ith and the jth. Let is1, js1 be the two columns of the first pair, and so on, is2, js2; :::; isp, jsp, where − − p D m l m 2 .