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

TL;DR

This paper investigates the number of minimal and next-to-minimal weight codewords in toric codes over hypersimplices, extending previous results to new parameter ranges and providing explicit characterizations.

Contribution

It characterizes and counts minimal and next-to-minimal weight codewords for a broader range of parameters in toric codes over hypersimplices.

Findings

Explicit formulas for the number of minimal weight codewords.

Explicit formulas for the number of next-to-minimal weight codewords.

Extended understanding of codeword structure in hypersimplex toric codes.

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 $(\mathbb{F}_q^*)^s$, 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 $3 \leq d \leq \frac{s - 2}{2}$ or $\frac{s + 2}{2} \leq d < s$, by Carvalho and Patanker in 2024. In this work we characterize and determine the number of minimal (respectively, next-to-minimal) weight codewords when $3 \leq d < s$ (respectively, $3 \leq d \leq \frac{s - 2}{2}$ or $\frac{s + 2}{2} \leq d < s$).