Related papers: Generalized minimum distance functions
We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives…
Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function $\delta_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that…
In this paper we introduce the relative generalized minimum distance function (RGMDF for short) and it allows us to give an algebraic approach to the relative generalized Hamming weights of the projective Reed--Muller--type codes. Also we…
We study the footprint function, with respect to a monomial order, of complete intersection graded ideals in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection,…
For projective Reed--Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed--Muller-type codes…
Let $G$ be a connected graph and let $\mathbb{X}$ be the set of projective points defined by the column vectors of the incidence matrix of $G$ over a field $K$ of any characteristic. We determine the generalized Hamming weights of the…
Let $S$ be a polynomial ring over a field $K$, with a monomial order $\prec$, and let $I$ be an unmixed graded ideal of $S$. In this paper we study two functions associated to $I$: the minimum distance function $\delta_I$ and the footprint…
We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\rm gp},M}$ for a multivariate polynomial $f(x) \in \mathbb{R}[x]$ of degree $ \le 2d$ in $n$ variables $x = (x_1,...,x_n)$…
We comprehensively study weighted projective Reed-Muller (WPRM) codes on weighted projective planes $\mathbb{P}(1,a,b)$. We provide the universal Gr\"obner basis for the vanishing ideal of the set $Y$ of $\mathbb{F}_q$--rational points of…
In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to $d$, where $d$ is a positive integer. In addition, we prove the following result which…
The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound. We prove that this bound can be sharp. We compute the…
The generalized Hamming weights (GHWs) of a linear code C extend the concept of minimum distance, which is the minimum cardinality of the support of all one-dimensional subspaces of C, to the minimum cardinality of the support of all…
In this paper, we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial function, assuming that the minimizer ideal is zero-dimensional. Our method is a…
The aim of this work is to give degree formulas for the generalized Hamming weights of evaluation codes and to show lower bounds for these weights. In particular, we give degree formulas for the generalized Hamming weights of…
There is a nice combinatorial formula of P. Beelen and M. Datta for the $r$-th generalized Hamming weight of an affine cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the $r$-th generalized…
In this note, we provide a description of the elements of minimum rank of a generalized Gabidulin code in terms of Grassmann coordinates. As a consequence, a characterization of linearized polynomials of rank at most $n-k$ is obtained, as…
We develop an algebraic theory of supports for $R$-linear codes of fixed length, where $R$ is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a…
Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of…
In this article, we give the answer to the following question: Given a field $\mathbb{F}$, finite subsets $A_1,\dots,A_m$ of $\mathbb{F}$, and $r$ linearly independent polynomials $f_1,\dots,f_r \in \mathbb{F}[x_1,\dots,x_m]$ of total…
Projective metrics on vector spaces over finite fields, introduced by Gabidulin and Simonis in 1997, generalize classical metrics in coding theory like the Hamming metric, rank metric, and combinatorial metrics. While these specific metrics…