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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.08402 |
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| _version_ | 1866910717337141248 |
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| author | Fan, Yun Leng, Yue |
| author_facet | Fan, Yun Leng, Yue |
| contents | Let $F$ be a field with cardinality $p^\ell$ and $0\neq λ\in F$, and $0\le h<\ell$. Extending Euclidean and Hermitian inner products, Fan and Zhang introduced Galois $p^h$-inner product (DCC, vol.84, pp.473-492). In this paper, we characterize the structure of $2$-quasi $λ$-constacyclic codes over $F$; and exhibit necessary and sufficient conditions for $2$-quasi $λ$-constacyclic codes being Galois self-dual. With the help of a technique developed in this paper, we prove that, when $\ell$ is even, the Hermitian self-dual $2$-quasi $λ$-constacyclic codes are asymptotically good if and only if $λ^{1+p^{\ell/2}}=1$. And, when $p^\ell\,{\not\equiv}\,3~({\rm mod}~4)$, the Euclidean self-dual $2$-quasi $λ$-constacyclic codes are asymptotically good if and only if $λ^{2}=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08402 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Galois Self-dual 2-quasi Constacyclic Codes over Finite Fields Fan, Yun Leng, Yue Information Theory Let $F$ be a field with cardinality $p^\ell$ and $0\neq λ\in F$, and $0\le h<\ell$. Extending Euclidean and Hermitian inner products, Fan and Zhang introduced Galois $p^h$-inner product (DCC, vol.84, pp.473-492). In this paper, we characterize the structure of $2$-quasi $λ$-constacyclic codes over $F$; and exhibit necessary and sufficient conditions for $2$-quasi $λ$-constacyclic codes being Galois self-dual. With the help of a technique developed in this paper, we prove that, when $\ell$ is even, the Hermitian self-dual $2$-quasi $λ$-constacyclic codes are asymptotically good if and only if $λ^{1+p^{\ell/2}}=1$. And, when $p^\ell\,{\not\equiv}\,3~({\rm mod}~4)$, the Euclidean self-dual $2$-quasi $λ$-constacyclic codes are asymptotically good if and only if $λ^{2}=1$. |
| title | Galois Self-dual 2-quasi Constacyclic Codes over Finite Fields |
| topic | Information Theory |
| url | https://arxiv.org/abs/2404.08402 |