On the second Hamming weight of some Reed-Muller type codes

TL;DR

This paper investigates the second Hamming weight of affine cartesian codes, a class of Reed-Muller type evaluation codes, using Gr"obner basis methods to determine their next-to-minimal weights in specific cases.

Contribution

It provides new results on the second Hamming weight of affine cartesian codes, extending understanding of their weight distribution using algebraic techniques.

Findings

Determined the second Hamming weight for certain affine cartesian codes.

Identified higher Hamming weights for these codes.

Applied Gr"obner basis methods to analyze code weights.

Abstract

We study affine cartesian codes, which are a Reed-Muller type of evaluation codes, where polynomials are evaluated at the cartesian product of n subsets of a finite field F_q. These codes appeared recently in a work by H. Lopez, C. Renteria-Marquez and R. Villareal and, in a generalized form, in a work by O. Geil and C. Thomsen. Using methods from Gr\"obner basis theory we determine the second Hamming weight (also called next-to-minimal weight) for particular cases of affine cartesian codes and also some higher Hamming weights of this type of code.