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We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus, with prescribed parameters. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation 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 Reed--Muller-typ
Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Groebner bases, to compute the length and the dimension of C_X*(d), the parameterized affine code of degree d on the se
We show that the degree of a graded lattice ideal of dimension 1 is the order of the torsion subgroup of the quotient group of the lattice. This gives an efficient method to compute the degree of this type of lattice ideals.
In this paper we introduce a new type of code, called projective nested cartesian code. It is obtained by the evaluation of homogeneous polynomials of a fixed degree on a certain subset of $mathbb{P}^n(mathbb{F}_q)$, and they may be seen as a general
Let $mathbb{F}_{q}$ denote the finite field of order $q,$ let $m_1,m_2,cdots,m_{ell}$ be positive integers satisfying $gcd(m_i,q)=1$ for $1 leq i leq ell,$ and let $n=m_1+m_2+cdots+m_{ell}.$ Let $Lambda=(lambda_1,lambda_2,cdots,lambda_{ell})$ be fixe