On the number of minimal and next-to-minimal weight codewords of toric codes over hypersimplices

Abstract

Toric codes are a type of evaluation code introduced by J.P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of $({\mathbb{F}}_q^{\ast }{)}^s$, 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 homogeneous monomially square-free 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, and has been determined in the cases where $3\le d\le \frac{s-2}{2}$ or $\frac{s+2}{2}\le d<s$, by Carvalho and Patanker in 2024. In this work we characterize and determine the number of minimal (respectively, next-to-minimal) weight codewords when $3\le d<s$ (respectively, $3\le d\le \frac{s-2}{2}$ or $\frac{s+2}{2}\le d<s$).