Next-to-minimal weight of toric codes defined over hypersimplices

TL;DR

This paper determines the next-to-minimal weight of toric codes associated with hypersimplices for specific degrees using Gr"obner basis techniques, extending previous results beyond the case d=1.

Contribution

It introduces a novel application of Gr"obner basis theory to compute the next-to-minimal weight for a broader range of degrees in hypersimplex toric codes.

Findings

Next-to-minimal weight computed for 3 ≤ d ≤ (s-2)/2 and (s+2)/2 ≤ d < s.

Extension of previous results from d=1 to higher degrees.

Provides explicit weight values for these codes.

Abstract

Toric codes are a type of evaluation codes introduced by J.P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of $(\mathbb{F}_q^*)^s$, the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of square-free homogeneous polynomials of degree $d$. The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in 2021. The next-to-minimal weight in the case $d = 1$ has been determined by Jaramillo-Velez et al. in 2023. In this work we use tools from Gr\"obner basis theory to determine the next-to-minimal weight of these codes for $d$ such that $3 \leq d \leq \frac{s - 2}{2}$ or $\frac{s + 2}{2} \leq d < s$.