Toric surface codes and the periodicity of polytopes
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916970266361856 |
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| author | Gibbs, Amelia Hogan, Eliza Jabbusch, Kelly Plute, Jenna Toloczko, Nicholas |
| author_facet | Gibbs, Amelia Hogan, Eliza Jabbusch, Kelly Plute, Jenna Toloczko, Nicholas |
| contents | Toric codes are error-correcting codes that are derived from toric varieties, which hold a unique correspondence to integral convex polytopes. In this paper, we focus on integral convex polytopes $P \subseteq \mathbb{R}^2$ and the toric codes they define. We begin by studying period-1 polytopes -- polytopes satisfying the property $L(tP)$ = $tL(P)$ for all $t \in \mathbb{Z}^+$, where $tP$ is the $t$-dilate of $P$, and we prove an explicit formula for the minimum distance of toric codes associated to a particular class of period-1 polytopes. We also apply the methods of Little and Schwarz, using Vandermonde matrices, to compute the minimum distance of another class of period-1 polytopes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_21178 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toric surface codes and the periodicity of polytopes Gibbs, Amelia Hogan, Eliza Jabbusch, Kelly Plute, Jenna Toloczko, Nicholas Algebraic Geometry Combinatorics 94B27 (Primary), 52B20, 14M25 Toric codes are error-correcting codes that are derived from toric varieties, which hold a unique correspondence to integral convex polytopes. In this paper, we focus on integral convex polytopes $P \subseteq \mathbb{R}^2$ and the toric codes they define. We begin by studying period-1 polytopes -- polytopes satisfying the property $L(tP)$ = $tL(P)$ for all $t \in \mathbb{Z}^+$, where $tP$ is the $t$-dilate of $P$, and we prove an explicit formula for the minimum distance of toric codes associated to a particular class of period-1 polytopes. We also apply the methods of Little and Schwarz, using Vandermonde matrices, to compute the minimum distance of another class of period-1 polytopes. |
| title | Toric surface codes and the periodicity of polytopes |
| topic | Algebraic Geometry Combinatorics 94B27 (Primary), 52B20, 14M25 |
| url | https://arxiv.org/abs/2509.21178 |