On the Existence of Galois Self-Dual GRS and TGRS Codes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912245878882304 |
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| author | Zhu, Shixin Wan, Ruhao |
| author_facet | Zhu, Shixin Wan, Ruhao |
| contents | Let $q=p^m$ be a prime power and $e$ be an integer with $0\leq e\leq m-1$. $e$-Galois self-dual codes are generalizations of Euclidean $(e=0)$ and Hermitian ($e=\frac{m}{2}$ with even $m$) self-dual codes. In this paper, for a linear code $\C$ and a nonzero vector $\bm{u}\in \F_q^n$, we give a sufficient and necessary condition for the dual extended code $\underline{\C}[\bm{u}]$ of $\C$ to be $e$-Galois self-orthogonal. From this, a new systematic approach is proposed to prove the existence of $e$-Galois self-dual codes. By this method, we prove that $e$-Galois self-dual (extended) generalized Reed-Solomon (GRS) codes of length $n>\min\{p^e+1,p^{m-e}+1\}$ do not exist, where $1\leq e\leq m-1$. Moreover, based on the non-GRS properties of twisted GRS (TGRS) codes, we show that in many cases $e$-Galois self-dual (extended) TGRS codes do not exist. Furthermore, we present a sufficient and necessary condition for $(\ast)$-TGRS codes to be Hermitian self-dual, and then construct several new classes of Hermitian self-dual $(+)$-TGRS and $(\ast)$-TGRS codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_10562 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Existence of Galois Self-Dual GRS and TGRS Codes Zhu, Shixin Wan, Ruhao Information Theory 94B05, 81p70 Let $q=p^m$ be a prime power and $e$ be an integer with $0\leq e\leq m-1$. $e$-Galois self-dual codes are generalizations of Euclidean $(e=0)$ and Hermitian ($e=\frac{m}{2}$ with even $m$) self-dual codes. In this paper, for a linear code $\C$ and a nonzero vector $\bm{u}\in \F_q^n$, we give a sufficient and necessary condition for the dual extended code $\underline{\C}[\bm{u}]$ of $\C$ to be $e$-Galois self-orthogonal. From this, a new systematic approach is proposed to prove the existence of $e$-Galois self-dual codes. By this method, we prove that $e$-Galois self-dual (extended) generalized Reed-Solomon (GRS) codes of length $n>\min\{p^e+1,p^{m-e}+1\}$ do not exist, where $1\leq e\leq m-1$. Moreover, based on the non-GRS properties of twisted GRS (TGRS) codes, we show that in many cases $e$-Galois self-dual (extended) TGRS codes do not exist. Furthermore, we present a sufficient and necessary condition for $(\ast)$-TGRS codes to be Hermitian self-dual, and then construct several new classes of Hermitian self-dual $(+)$-TGRS and $(\ast)$-TGRS codes. |
| title | On the Existence of Galois Self-Dual GRS and TGRS Codes |
| topic | Information Theory 94B05, 81p70 |
| url | https://arxiv.org/abs/2210.10562 |