2025/02/25 by Cícero Carvalho, Carvalho, Cícero, Nupur Patanker +1
Computer Science · Engineering · #11T71 #14G50 #94B05 #Algebraic Geometry (math.AG) #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2502.18084
openalex publication_date 2025/02/25 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
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 (\mathbbFq^*)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 ≤ d ≤ (s - 2)/(2) or (s + 2)/(2) ≤ 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 ≤ d < s (respectively, 3 ≤ d ≤ (s - 2)/(2) or (s + 2)/(2) ≤ d < s).