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

2025-02-11

AI Summary

This paper investigates the second minimum weight of toric codes defined on hypersimplices, focusing on square-free homogeneous polynomials of degree $d$ satisfying either $3 leq d leq frac{s-2}{2}$ or $frac{s+2}{2} leq d < s$. Employing a synthesis of Gröbner basis theory, algebraic combinatorics, and polynomial coding analysis over finite fields, the authors fully determine the second minimum weight for this parameter range—achieving the first complete characterization. This resolves a longstanding theoretical gap for $d geq 3$, substantially generalizing prior work by Jaramillo-Velez et al., which addressed only the case $d = 1$. The result provides a key structural characterization of the weight distribution for high-dimensional combinatorial codes, particularly toric codes on hypersimplices, and establishes foundational support for further study of their coding-theoretic properties.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

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

Problem

• Determine next-to-minimal weight
• Toric codes over hypersimplices
• Using Gröbner basis theory

Innovation

• Gröbner basis theory
• Toric codes analysis
• Next-to-minimal weight