Two classes of linear codes and their generalized Hamming weights

The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. In this paper, we investigate the generalized Hamming weights of two classes of linear codes constructed from defining sets and determine them completely employing a number-theoretic approach

On the generalized Hamming weights of certain Reed-Muller-type codes

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 Hamming weight for a family of affine cartesian codes. If $\mathbb{X}$ is a set of projective points over a finite field we determine the basic parameters and the generalized Hamming weights of the Veronese type codes on $\mathbb{X}$ and their dual codes in terms of the basic parameters and the generalized Hamming weights of the corresponding projective Reed--Muller-type codes on $\mathbb{X}$ and their dual codes.Comment: An. \c{S}tiin\c{t}. Univ. "Ovidius'' Constan\c{t}a Ser. Mat., to appea

Generalized Hamming weights of projective Reed--Muller-type codes over graphs

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 Reed--Muller-type code over the set $\mathbb{X}$ in terms of graph theoretic invariants. As an application to coding theory we show that if $G$ is non-bipartite and $K$ is a finite field of ${\rm char}(K)\neq 2$, then the $r$-th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of $G$ is the $r$-th weak edge biparticity of $G$. If ${\rm char}(K)=2$ or $G$ is bipartite, we prove that the $r$-th generalized Hamming weight of that code is the $r$-th edge connectivity of $G$.Comment: Discrete Math., to appea

Relative Generalized Minimum Distance Function

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 introduce the relative generalized footprint function and it gives a tight lower bound for the RGMDF which is much easier to compute

Generalized minimum distance functions

Using commutative algebra methods we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set of projective points is expressed as the degree of a graded ideal. If $\mathbb{X}$ is a set of projective points over a finite field and $I$ is its vanishing ideal, we show that the gmd function and the Vasconcelos function of $I$ are equal to the $r$-th generalized Hamming weight of the corresponding Reed-Muller-type code $C_\mathbb{X}(d)$ of degree $d$. We show that the generalized footprint function of $I$ is a lower bound for the $r$-th generalized Hamming weight of $C_\mathbb{X}(d)$. Then we present some applications to projective nested cartesian codes. To give applications of our lower bound to algebraic coding theory, we show an interesting integer inequality. Then we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine cartesian code.Comment: J. Algebraic Combin., to appear. arXiv admin note: text overlap with arXiv:1512.0686