From ds-Bounds for Cyclic Codes to True Minimum Distance for Abelian Codes (original) (raw)

From ds-bounds for cyclic codes to true distance for abelian codes

ArXiv, 2017

In this paper we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes through the notion of mathbbB\mathbb{B}mathbbB-apparent distance. We use this technique to improve the searching for new bounds for the minimum distance of abelian codes. We also study conditions for an abelian code to verify that its mathbbB\mathbb{B}mathbbB-apparent distance reaches its (true) minimum distance. Then we construct some tables of such codes as an application

Ds-bounds for cyclic codes: new bounds for abelian codes

2016

In this paper we develop a technique to extend any bound for cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes. We use this technique to improve the searching of new bounds for abelian codes.

Generalizing Bounds on the Minimum Distance of Cyclic Codes Using Cyclic Product Codes

2013

Two generalizations of the Hartmann-Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved.

A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes

2012

A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann-Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes.

Cyclic and BCH codes whose minimum distance equals their maximum BCH bound

Advances in Mathematics of Communications, 2016

In this paper we study the family of cyclic codes such that its minimum distance reaches the maximum of its BCH bounds. We also show a way to construct cyclic codes with that property by means of computations of some divisors of a polynomial of the form x n − 1. We apply our results to the study of those BCH codes C, with designed distance δ, that have minimum distance d(C) = δ. Finally, we present some examples of new binary BCH codes satisfying that condition. To do this, we make use of two related tools: the discrete Fourier transform and the notion of apparent distance of a code, originally defined for multivariate abelian codes.

On the Minimal Distance of Binary Self-Dual Cyclic Codes

IEEE Transactions on Information Theory, 2000

In this paper, an explicit construction of binary selfdual cyclic codes of length n going to infinity with a minimal distance at least half the square root of n is presented. The same idea is also used to construct more general binary cyclic codes with a large minimal distance. Finally, in the special case of self-dual cyclic codes, a simplified version of a proof by Conway and Sloane is given, showing an upper bound for the distance of binary selfdual codes.

Decoding Cyclic Codes up to a New Bound on the Minimum Distance

IEEE Transactions on Information Theory, 2012

A new lower bound on the minimum distance of qary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula. Index Terms-Bose-Chaudhuri-Hocquenghem (BCH) bound, cyclic codes, Forney's formula, Hartmann-Tzeng (HT) bound, Roos bound.

Describing A Cyclic Code by Another Cyclic Code

2012

A new approach to bound the minimum distance of q-ary cyclic codes is presented. The connection to the BCH and the Hartmann-Tzeng bound is formulated and it is shown that for several cases an improvement is achieved.