Next-to-minimal weight of toric codes defined over hypersimplices
Cícero Carvalho, Nupur Patanker
TL;DR
This work determines the next-to-minimal weight of toric codes defined over hypersimplices for a broad range of degrees $d$, building on known dimension and minimum-distance results. By applying Gröbner-bases techniques, the authors analyze remainders of $S$-polynomials modulo a footprint-based generating set and identify four possible leading-monomial types for these remainders. They derive explicit lower bounds on the next-to-minimal weight and show tight attainability in the regime $2d+2 \le s$ via a carefully constructed square-free homogeneous polynomial, establishing concrete weight formulas and, in many cases, a code isomorphism $\mathcal{C}(d) \cong \mathcal{C}(s-d)$ when $2d\le s$. The results provide exact weight information for $3 \le d \le \frac{s-2}{2}$ and $\frac{s+2}{2} \le d < s$, advancing the understanding of weight distributions in hypersimplex toric codes and highlighting the efficacy of Gröbner-bases methods in coding theory.
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 Gröbner 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 $\frac{s + 2}{2} \leq d < s$.