Minimal linear codes are linear codes such that the support of every codeword does not contain the support of another linearly independent codeword. Such codes have applications in cryptography, e.g. to secret sharing. We pursue here their study and construct asymptotically good families of minimal linear codes. We also push further the study of quasi-minimal and almost-minimal linear codes, re...

A binary linear code C is a Z2-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the Z2[x]-module Z2[x]/(x r −1)×Z2[x]/(x − 1). We determine the structure of Z2-double cyclic codes giving the generator polynomials of these codes. ...

A construction based on Legendre sequences is presented for a doubly-extended binary linear code of length 2p + 2 and dimension p+1. This code has a double circulant structure. For p = 4k +3, we obtain a doubly-even self-dual code. Another construction is given for a class of triply extended rate 1/3 codes of length 3p + 3 and dimension p + 1. For p = 4k + 1, these codes are doubly-even selfort...

Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic c...

The main aim of this paper is to study LCD codes. Linear code with complementary dual(LCD) are those codes which have their intersection with their dual code as t0u. In this paper we will give rather alternative proof of Massey’s theorem[8], which is one of the most important characterization of LCD codes. Let LCDrn, ks3 denote the maximum of possible values of d among rn, k, ds ternary LCD cod...

Klove, T., Support weight distribution of linear codes, Discrete Mathematics 106/107 (1992) 311-316. The main result of the paper is expressions for the support weight distributions of a linear code in terms of the support weight distributions of the dual code.

Two skew cyclic codes can be equivalent for the Hamming metric only if they have the same length, and only the zero code is degenerate. The situation is completely different for the rank metric, where lengths of codes correspond to the number of outgoing links from the source when applying the code on a network. We study rank equivalences between skew cyclic codes of different lengths and, with...

We present a new approach to the theory of cyclic and constacyclic codes and generalize the theory to cover the family of additive (not necessarily linear) cyclic codes. The approach is based on the action of the Galois group (cyclotomic cosets). The conventional representation of cyclic codes as ideals in a factor ring of the polynomial ring is not needed. 1 General facts on linear codes The c...

In this work we study the set of leader codewords of a non-binary linear code. This set has some nice properties related to the monotonicity of the weight compatible order on the generalized support of a vector in Fnq . This allows us to describe a test set, a trial set and the set zero neighbours in terms of the leader codewords.