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

2025/02/11 by Cícero Carvalho, Carvalho, Cícero, Nupur Patanker +1 · 1 voice
Computer Science · Mathematics · #11T71 #14G50 #94B05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.IT #math.AC #math.AG

paper · pdf · doi:10.48550/arxiv.2502.07718

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 (\mathbbF_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 ≤ d ≤ (s - 2)/(2) or (s + 2)/(2) ≤ d < s.

Discussions

• C\'icero Carvalho, Nupur Patanker: Next-to-minimal weight of toric codes defined over hypersimplices https://arxiv.org/abs/2502.07718 https://arxiv.org/pdf/2502.07718 https://arxiv.org/html/2502.07718 [bsky, 0 points, 1 comments]

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