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. 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.
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.
A generalization of the FGLM technique is given to compute Gröbner bases for two-sided ideals of free finitely generated algebras. Specializations of this algorithm are presented for the cases in which the ideal is determined by either functionals or monoid (group) presentations. Generalizations are discussed in order to compute Gröbner bases on (twisted) semigroup rings.
We show herein that a pattern based on FGLM techniques can be used for computing Gröbner bases, or related structures, associated to linear codes. This Gröbner bases setting turns out to be strongly related to the combinatorics of the codes.
In this paper we use the Gröbner representation of a binary linear code C to give efficient algorithms for computing the whole set of coset leaders, denoted by CL(C) and the set of leader codewords, denoted by L(C). The first algorithm could be adapted to provide not only the Newton and the covering radius of C but also to determine the coset leader weight distribution. Moreover, providing the set of leader codewords we have a test-set for decoding by a gradient-like decoding algorithm. Another contribution of this article is the relation stablished between zero neighbours and leader codewords.
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