On the permutation automorphisms of binary cubic codes
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911356580528128 |
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| author | Altunbulak, Murat Aksu, Fatma Altunbulak Hafezieh, Roghayeh Tuvay, İpek |
| author_facet | Altunbulak, Murat Aksu, Fatma Altunbulak Hafezieh, Roghayeh Tuvay, İpek |
| contents | A binary linear code whose permutation automorphism group has a fixed point free permutation of order $3$ is called a binary cubic code. The scope of this paper is to investigate the structural properties of binary cubic codes. Let $C$ be a binary cubic $[n,k]$ code. In this paper, we prove that if $n\geq 30$ and $C$ has permutation automorphism group of order three, then $k\geq 6$. Additionally, we show that if $n < 30$ and $k\leq 4$, then the permutation automorphism group of $C$ has order greater than three. Moreover, along the way, we provide some results on the structure of the higher dimensional cubic codes. In particular, we present some results concerning the structure of the putative extremal self-dual $[72,36,16]$ code under the assumption that it is cubic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the permutation automorphisms of binary cubic codes Altunbulak, Murat Aksu, Fatma Altunbulak Hafezieh, Roghayeh Tuvay, İpek Information Theory 94B05, 11T71, 20B05 A binary linear code whose permutation automorphism group has a fixed point free permutation of order $3$ is called a binary cubic code. The scope of this paper is to investigate the structural properties of binary cubic codes. Let $C$ be a binary cubic $[n,k]$ code. In this paper, we prove that if $n\geq 30$ and $C$ has permutation automorphism group of order three, then $k\geq 6$. Additionally, we show that if $n < 30$ and $k\leq 4$, then the permutation automorphism group of $C$ has order greater than three. Moreover, along the way, we provide some results on the structure of the higher dimensional cubic codes. In particular, we present some results concerning the structure of the putative extremal self-dual $[72,36,16]$ code under the assumption that it is cubic. |
| title | On the permutation automorphisms of binary cubic codes |
| topic | Information Theory 94B05, 11T71, 20B05 |
| url | https://arxiv.org/abs/2402.10667 |