Affine cartesian codes

Abstract

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 of a certain type. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.

Minimum distance functions of complete intersections

“…. , A s of a field K, we denote the image of [21]. The basic parameters of the projective Reed-Muller-type code C X (d) are equal to those of C X * (d) [22].…”

Generalized minimum distance functions and algebraic invariants of Geramita ideals

“…There are some families of Reed-Muller type codes where the minimum distance and its index of regularity are known [22,31]. In these cases one can determine whether or not the corresponding sets of points are Cayley-Bacharach.…”

Relative generalized Hamming weights of affine Cartesian codes

“…These codes were introduced in 2013 by Geil and Thomsen [12] in a more general setting of weighted Reed-Muller codes. The name "affine Cartesian codes" was coined by López, Rentería-Márquez and Villarreal [17] in 2014. Since then several articles have appeared where the parameters of these codes were studied extensively.…”

“…Like in the case of Reed-Muller codes, the problem of computing parameters such as minimum distance, generalized Hamming weights etc., of affine Cartesian codes translates to the problem of determination of the maximum number of common zeroes of systems of polynomials satisfying certain properties in a subset of an affine space over a finite field. The fundamental properties of affine Cartesian codes, such as their dimensions and the minimum distances, were obtained in [17]. Later in 2018, the generalized Hamming weights [1] of the affine Cartesian codes were completely determined.…”