Related papers: On abelian and cyclic group codes

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of $q$-ary linear codes under some certain conditions, where $q$ is a power…

We study codes with a single check element derived from group rings, namely, checkable codes. The notion of a code-checkable group ring is introduced. Necessary and sufficient conditions for a group ring to be code-checkable are given in…

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three zeros. Recently, \cite{Y-X-D12}…

Let $r,s,t$ be three positive integers and $\mathcal{C}$ be a binary linear code of lenght $r+s+t$. We say that $\mathcal{C}$ is a triple cyclic code of lenght $(r,s,t)$ over $\mathbb{Z}_2$ if the set of coordinates can be partitioned into…

For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $\chi(G,h)$ (resp.~$\chi \hat{\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold…

Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of $G$}, denoted by $\Gamma_c(G)$, is a graph with all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\Gamma_c(G)$ are adjacent if and…

A finitely presented, torsion free, abelian-by-cyclic group can always be written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n integer matrix M with det(M) \ne 0. The group Gamma_M is polycyclic if and only if…

Irreducible cyclic codes are an interesting type of codes and have applications in space communications. They have been studied for decades and a lot of progress has been made. The objectives of this paper are to survey and extend earlier…

We give sufficient conditions for self-orthogonality with respect to symplectic, Euclidean and Hermitian inner products of a wide family of quasi-cyclic codes of index two. We provide lower bounds for the symplectic weight and the minimum…

We study affine cartesian codes, which are a Reed-Muller type of evaluation codes, where polynomials are evaluated at the cartesian product of n subsets of a finite field F_q. These codes appeared recently in a work by H. Lopez, C.…

The cyclic codes with parity check polynomial the reciprocal of the characteristic polynomial of the Fibonacci recurrence over a prime finite field are shown to have either one weight or two weights. When these codes are irreducible cyclic…

It is well known that the problem of determining the weight distributions of families of cyclic codes is, in general, notoriously difficult. An even harder problem is to find characterizations of families of cyclic codes in terms of their…

The group isomorphism problem asks whether two given groups are isomorphic or not. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism…

Let G be a finite abelian group and F a field such that char(F) does not divide |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I and J of FG are G-equivalent if there exists an…

Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on…

Let $p$ be a prime number. Irreducible cyclic codes of length $p^2-1$ and dimension $2$ over the integers modulo $p^h$ are shown to have exactly two nonzero Hamming weights. The construction uses the Galois ring of characteristic $p^h$ and…

It is well known that the discrete analogue of a lattice is a linear code which is a vector subspace of Hamming space $\mathbb{F}^n$. The set $\mathbb{F}$ is a finite field and $n \in \mathbb{Z}_{>0}$. Our attempt is to construct a class of…

Let $\mathcal{C}$ be a quasi-cyclic code of index $l(l\geq2)$. Let $G$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by $\rho^l$ and the scalar multiplications of $\mathcal{C}$, where $\rho$ denotes the standard…

We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime-power order. For that we associate to each such group $G$ a list $\inv(G)$ of numerical group invariants which determines the isomorphism type of $G$.…

Families of codes such as group codes, constacyclic and skew cyclic codes, some of which independently suggested in the literature, turn out to be special instances of the general family of crossed product codes. Hamming-metric is a main…