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In this work, two algorithms are developed related to lattice codes. In the first one, an extended complete Gr\"obner 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 Gr\"obner 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.

In this work we construct sequences of locally recoverable AG codes arising from a tower of function fields and give bound for the parameters of the obtained codes. In a particular case of a tower over $\mathbb{F}_{q^2}$ for any odd $q$, defined by Garcia and Stichtenoth in [GS2007], we show that the bound is sharp for the first code in the sequence, and we include a detailed analysis for the following codes in the sequence based on the distribution of rational places that split completely in the considered function field extension.

In this work, we present a standard model for Galois rings based on the standard model of their residual fields, that is, a sequence of Galois rings starting with ${\mathbb Z}_{p^r} that coves all the Galois rings with that characteristic ring and such that there is an algorithm producing each member of the sequence whose input is the size of the required ring., Comment: Submitted to FFA

In this work, we study the structure of multivariable modular codes over finite chain rings when the ambient space is a principal ideal ring. We also provide some applications to additive modular codes over the finite field $\mathbb{F}_4$., Comment: Submitted to the Journal of Pure and Applied Algebra

We consider codes defined over an affine algebra $\mathcal A=R[X_1,\dots,X_r]/\left\langle t_1(X_1),\dots,t_r(X_r)\right\rangle$, where $t_i(X_i)$ is a monic univariate polynomial over a finite commutative chain ring $R$. Namely, we study the $\mathcal A-$submodules of $\mathcal A^l$ ($l\in \mathbb{N}$). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. {Some codes over Frobenius local rings that are not chain rings are also of this type}. A canonical generator matrix for these codes is introduced with the help of the Canonical Generating System. Duality of the codes is also considered., Comment: Submitted to Finite Fields and Their Applications

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., Comment: arXiv admin note: substantial text overlap with arXiv:1411.7493

Given $\texttt{S}|\texttt{R}$ a finite Galois extension of finite chain rings and $\mathcal{B}$ an $\texttt{S}$-linear code we define two Galois operators, the closure operator and the interior operator. We proof that a linear code is Galois invariant if and only if the row standard form of its generator matrix has all entries in the fixed ring by the Galois group and show a Galois correspondence in the class of $\texttt{S}$-linear codes. As applications some improvements of upper and lower bounds for the rank of the restriction and trace code are given and some applications to $\texttt{S}$-linear cyclic codes are shown.

The spontaneous formation of droplets via dewetting of a thin fluid film from a solid substrate allows for materials nanostructuring, under appropriate experimental control. While thermal fluctuations are expected to play a role in this process, their relevance has remained poorly understood, particularly during the nonlinear stages of evolution. Within a stochastic lubrication framework, we show that thermal noise speeds up and substantially influences the formation and evolution of the droplet arrangement. As compared with their deterministic counterparts, for a fixed spatial domain, stochastic systems feature a smaller number of droplets, with a larger variability in sizes and space distribution. Finally, we discuss the influence of stochasticity on droplet coarsening for very long times., Comment: 6 pages, 5 figures, 2 movies available upon request

In this paper we use the Gr\"obner 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.

There are two gradient descent decoding procedures for binary codes proposed independently by Liebler and by Ashikhmin and Barg. Liebler in his paper mentions that both algorithms have the same philosophy but in fact they are rather different. The purpose of this communication is to show that both algorithms can be seen as two ways of understanding the reduction process algebraic monoid structure related to the code. The main tool used for showing this is the Gr\"obner representation of the monoid associated to the linear code.

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 Gr\"obner representations associated with codes. The algorithm provides a Gr\"obner 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 $\mathbb{F}_2^n$., Comment: Submitted to Designs, Codes and Cryptography

We show herein that a pattern based on FGLM techniques can be used for computing Gr\"obner bases, or related structures, associated to linear codes. This Gr\"obner bases setting turns out to be strongly related to the combinatorics of the codes., Comment: Submitted to AAECC 16

In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gr\"obner 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 Gr\"obner basis., Comment: Submitted to Appl. Algebra Engrg. Comm. Comput

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 Gr\"obner 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., Comment: Subbmited to International Mathematics Research Notices

The structure of multivariate semisimple codes over a finite chain ring $R$ is established using the structure of the residue field $\bar R$. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some non-trivial codes over $R$. The structure of the dual codes in the semisimple abelian case is also derived and some conditions on the existence of selfdual codes over $R$ are studied., Comment: Submitted to SIAM Journal on Discrete Mathematics

We discuss recent work in the study of a simple model for the collective behaviour of diverse speculative agents in an idealized stockmarket, considered from the perspective of the statistical physics of many-body systems. The only information about other agents available to any one is the total trade at time steps. Evidence is presented for correlated adaptation and phase transitions/crossovers in the global volatility of the system as a function of appropriate information scaling dimension. Stochastically controlled irrationally of individual agents is shown to be globally advantageous. We describe the derivation of the underlying effective stochastic differential equations which govern the dynamics, and make an interpretation of the results from the point of view of the statistical physics of disordered systems., Comment: 15 Pages. 5 figures

We analyze the possible existence of a resonance in the $J^P=0^-$ channel with isospin two by means of nucleon-$\Delta$ interactions based on the constituent quark model. We solve the bound state and the scattering problem using two different potentials, a local and a non-local one. The non-local potential results to be the more attractive, although not enough to generate the experimentally predicted resonance., Comment: 9 pages in Latex (revtex), 2 eps figures available under request