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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.11461 |
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| _version_ | 1866912199162724352 |
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| author | Hegedüs, Gábor Suda, Sho Xiang, Ziqing |
| author_facet | Hegedüs, Gábor Suda, Sho Xiang, Ziqing |
| contents | For a code $C$ in a space with maximal distance $n$, we say that $C$ has symmetric distances if its distance set $S(C)$ is symmetric with respect to $n / 2$. In this paper, we prove that if $C$ is a binary code with length $2n$, constant weight $n$ and symmetric distances, then \[
|C| \leq \binom{2 n - 1}{|S(C)|}. \] This result can be interpreted using the language of Johnson association schemes. More generally, we give a framework to study codes with symmetric distances in Q-bipartite Q-polynomial association schemes, and provide upper bounds for such codes. Moreover, we use number theoretic techniques to determine when the equality holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Codes with symmetric distances Hegedüs, Gábor Suda, Sho Xiang, Ziqing Combinatorics For a code $C$ in a space with maximal distance $n$, we say that $C$ has symmetric distances if its distance set $S(C)$ is symmetric with respect to $n / 2$. In this paper, we prove that if $C$ is a binary code with length $2n$, constant weight $n$ and symmetric distances, then \[ |C| \leq \binom{2 n - 1}{|S(C)|}. \] This result can be interpreted using the language of Johnson association schemes. More generally, we give a framework to study codes with symmetric distances in Q-bipartite Q-polynomial association schemes, and provide upper bounds for such codes. Moreover, we use number theoretic techniques to determine when the equality holds. |
| title | Codes with symmetric distances |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.11461 |