On the ideal associated to a linear code

This article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal $I_+(\mathcal C)$ to an arbitrary linear code. The binomials involved in the reduced Gr\"obner basis of such an ideal relative to a degree-compatible ordering induce a uniquely defined test-set for the code, and this allows the description of a Hamming metric decoding procedure. Moreover, the binomials involved in the Graver basis of $I_+(\mathcal C)$ provide a universal test-set which turns out to be a set containing the set of codewords of minimal support of the code

Computing syzygies in finite dimension using fast linear algebra

We consider the computation of syzygies of multivariate polynomials in afinite-dimensional setting: for a $\mathbb{K}[X_1,\dots,X_r]$-module$\mathcal{M}$ of finite dimension $D$ as a $\mathbb{K}$-vector space, andgiven elements $f_1,\dots,f_m$ in $\mathcal{M}$, the problem is to computesyzygies between the $f_i$'s, that is, polynomials $(p_1,\dots,p_m)$ in$\mathbb{K}[X_1,\dots,X_r]^m$ such that $p_1 f_1 + \dots + p_m f_m = 0$ in$\mathcal{M}$. Assuming that the multiplication matrices of the $r$variables with respect to some basis of $\mathcal{M}$ are known, we give analgorithm which computes the reduced Gr\"obner basis of the module of thesesyzygies, for any monomial order, using $O(m D^{\omega-1} + r D^\omega\log(D))$ operations in the base field $\mathbb{K}$, where $\omega$ is theexponent of matrix multiplication. Furthermore, assuming that $\mathcal{M}$is itself given as $\mathcal{M} = \mathbb{K}[X_1,\dots,X_r]^n/\mathcal{N}$,under some assumptions on $\mathcal{N}$ we show that these multiplicationmatrices can be computed from a Gr\"obner basis of $\mathcal{N}$ within thesame complexity bound. In particular, taking $n=1$, $m=1$ and $f_1=1$ in$\mathcal{M}$, this yields a change of monomial order algorithm along thelines of the FGLM algorithm with a complexity bound which is sub-cubic in$D$

Algorithms for Linearly Recurrent Sequences of Truncated Polynomials

Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding recurrence relations is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements are vectors over the ring A = K[x]/ of truncated polynomials. Finding the ideal of their recurrence relations has applications such as the computation of minimal polynomials and determinants of sparse matrices over A. We present three methods for finding this ideal: a Berlekamp-Massey-like approach due to Kurakin, one which computes the kernel of some block-Hankel matrix over A via a minimal approximant basis, and one based on bivariate Pade approximation. We propose complexity improvements for the first two methods, respectively by avoiding the computation of redundant relations and by exploiting the Hankel structure to compress the approximation problem. Then we confirm these improvements empirically through a C++ implementation, and we discuss the above-mentioned applications

A Combinatorial Commutative Algebra Approach to Complete Decoding

Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, Geometría y Topologí

Solving a Multivariable Congruence By Change of Term Order

this paper we propose algorithms for determining Grobner bases of the solution module M of (??), relative to arbitrary term orders in the case that I is a zero dimensional ideal. We take as our starting point the algorithm of Faug`ere et al. (1993) (often denoted by the initials FGLM) whose purpose is to convert a Grobner basis for a zero dimensional ideal of A with respect to one term order into a Grobner basis with respect to another. In Section 2 we give the (straightforward) generalization of this algorithm to submodules of finite codimension in