Edgar Martínez-Moro

Gröbner bases and combinatorics for binary codes

In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gröbner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for the code. In the second one, by associating the code with the set of cycles in a graph, we can solve the problem of finding all codewords of minimal length (minimal cycles in a graph), and show how to find a minimal cycle basis. Finally we discuss some results on the computation of the Gröbner basis.

On a Gröbner bases structure associated to linear codes

Abstract We present a structure associated to the class of linear codes. The properties of that structure are similar to some structures in the linear algebra techniques into the framework of the Gröbner bases tools. It allows to get some insight in the problem of determining whether two codes are permutation equivalent or not. Also an application to the decoding problem is presented, with particular emphasis on the binary case.

Algebraic structure of the minimal support codewords set of some linear codes

It has been widely known that complete decoding for binary linear codes can be regarded as a linear integer programming problem with binary arithmetic conditions. Conti and Traverso [9] have proposed an algorithm which uses Grobner bases to solve integer programming with ordinary integer arithmetic conditions. Ikegami and Kaji [12] extended the Conti-Traverso algorithm to solve integer programming with modulo arithmetic conditions. It is natural to consider for those problems the Graver basis associated to them which turns out to be the minimal cycles of the matroid associated to the code, i.e. minimal support codewords in the binary case and its geometry. This provides us a universal test set for the programs considered.

A general framework for applying FGLM techniques to linear codes

We show herein that a pattern based on FGLM techniques can be used for computing Grobner bases, or related structures, associated to linear codes. This Grobner bases setting turns out to be strongly related to the combinatorics of the codes.

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple

On repeated-root multivariable codes over a finite chain ring

{(\mathcal X,\mathcal P, E)}

An algebraic view to gradient descent decoding

from the algebraic geometry code