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M. Borges-Quintana

Publications and source records attributed to M. Borges-Quintana.

7 recordsLinked to original sources
arXiv2023

Complete Gröbner basis for lattice codes

In this work, two algorithms are developed related to lattice codes. In the first one, an extended complete Gröbner basis is computed for the label code of a lattice. This basis supports all term orderings associated with a total degree order offering information about de label code of the lattice. The second one is a decoding algorithm that uses an extended complete Gröbner basis of the label code of the lattice for monomial reduction, this provides all the lattice vectors that constitute candidates for the solution of the Close Vector Problem for a given vector.

cs.IT↗
arXiv2017

On the weak order ideal associated to linear codes

In this work we study a weak order ideal associated with the coset leaders of a non-binary linear code. This set allows the incrementally computation of the coset leaders and the definitions of the set of leader codewords. This set of codewords has some nice properties related to the monotonicity of the weight compatible order on the generalized support of a vector in $\mathbb F_q^n$ which allow us to describe a test set, a trial set and the set of zero neighbours of a linear code in terms of the leader codewords.

cs.IT↗
arXiv2014

Computing coset leaders and leader codewords of binary codes

In this paper we use the Gröbner representation of a binary linear code $\mathcal C$ to give efficient algorithms for computing the whole set of coset leaders, denoted by $\mathrm{CL}(\mathcal C)$ and the set of leader codewords, denoted by $\mathrm L(\mathcal C)$. The first algorithm could be adapted to provide not only the Newton and the covering radius of $\mathcal C$ 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.

cs.IT↗
arXiv2010

Computing coset leaders of binary codes

We present an algorithm for computing the set of all coset leaders of a binary code $\mathcal C \subset \mathbb{F}_2^n$. The method is adapted from some of the techniques related to the computation of Gröbner representations associated with codes. The algorithm provides a Gröbner representation of the binary code and the set of coset leaders $\mathrm{CL}(\mathcal C)$. Its efficiency stands of the fact that its complexity is linear on the number of elements of $\mathrm{CL}(\mathcal C)$, which is smaller than exhaustive search in $\mathbb{F}_2^n$.

cs.IT↗
arXiv2005

Groebner 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. 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.

math.CO↗
arXiv2005

On a Grobner bases structure associated to linear codes

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.

math.AC↗