Affine cartesian codes
Hiram H. Lopez, Carlos Renteria, Rafael H. Villarreal
TL;DR
The paper addresses the problem of determining the basic parameters (dimension, length, minimum distance) of affine cartesian evaluation codes on a cartesian product $X^*=A_1\times\cdots\times A_n$ and their projective closure $Y$. It leverages the vanishing ideal $I(Y)$, proves it is a complete intersection with generators of degrees $d_i=|A_i|$, and uses graded invariants (Hilbert function, regularity, degree) to derive explicit formulas for dimension and minimum distance. A central result is a closed-form minimum-distance formula $\delta_{X^*}(d)$ in terms of a partition of $d$ with parameters $(k,\ell)$, plus the threshold $r=\sum_i (d_i-1)$ beyond which the distance collapses to $1$, with $C_{X^*}(d)=C_Y(d)$. The paper also constructs cartesian codes over degenerate tori to realize prescribed parameters, linking to classical results for projective tori and affine spaces, and providing a unifying algebraic framework for evaluating codes on these sets.
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. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.