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
ization of the so-called projective Reed-Muller codes. We calculate the length and the dimension of such codes, a lower bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed-Muller codes). At the end we show some relations between the parameters of these codes and those of the affine cartesian codes.
We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to nu
merical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup.
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.
For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set theoretic complete inters
ections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set theoretic complete intersection is a complete intersection.
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 pres
cribed parameters. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.
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
t X*. If Y is the projective closure of X*, it is shown that C_X^*(d) has the same basic parameters that C_Y(d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of C_X*(d). We show how to compute the vanishing ideals of X* and Y.
