On the number of minimal and next-to-minimal weight codewords of toric codes over hypersimplices
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arXiv
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| Natura: | Preprint |
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2025
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| author | Carvalho, Cícero Patanker, Nupur |
| author_facet | Carvalho, Cícero Patanker, Nupur |
| contents | 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 $\frac{s + 2}{2} \leq d < s$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18084 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of minimal and next-to-minimal weight codewords of toric codes over hypersimplices Carvalho, Cícero Patanker, Nupur Information Theory Commutative Algebra Algebraic Geometry 94B05, 11T71, 14G50 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 $\frac{s + 2}{2} \leq d < s$). |
| title | On the number of minimal and next-to-minimal weight codewords of toric codes over hypersimplices |
| topic | Information Theory Commutative Algebra Algebraic Geometry 94B05, 11T71, 14G50 |
| url | https://arxiv.org/abs/2502.18084 |