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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.00243 |
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| _version_ | 1866915107757359104 |
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| author | Bell, Jason P. Monahan, Sean Satriano, Matthew Situ, Karen Xie, Zheng |
| author_facet | Bell, Jason P. Monahan, Sean Satriano, Matthew Situ, Karen Xie, Zheng |
| contents | Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemerédi-type result: for all $c\in(0,1]$ and all positive integers $N$, subsets of density at least $c$ in $\{0,1,\dots,N-1\}^n$ contain hypercubes of arbitrarily large dimension as $n$ grows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00243 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | There are no good infinite families of toric codes Bell, Jason P. Monahan, Sean Satriano, Matthew Situ, Karen Xie, Zheng Combinatorics Information Theory Algebraic Geometry 14G50, 14M25, 11B30, 94B05 Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemerédi-type result: for all $c\in(0,1]$ and all positive integers $N$, subsets of density at least $c$ in $\{0,1,\dots,N-1\}^n$ contain hypercubes of arbitrarily large dimension as $n$ grows. |
| title | There are no good infinite families of toric codes |
| topic | Combinatorics Information Theory Algebraic Geometry 14G50, 14M25, 11B30, 94B05 |
| url | https://arxiv.org/abs/2406.00243 |