Determining the covering radius of all generalized Zetterberg codes in odd characteristic
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913582604615680 |
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| author | Shi, Minjia Li, Shitao Helleseth, Tor Ozbudak, Ferruh |
| author_facet | Shi, Minjia Li, Shitao Helleseth, Tor Ozbudak, Ferruh |
| contents | For an integer $s\ge 1$, let $\mathcal{C}_s(q_0)$ be the generalized Zetterberg code of length $q_0^s+1$ over the finite field $\F_{q_0}$ of odd characteristic. Recently, Shi, Helleseth, and Özbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of $\mathcal{C}_s(q_0)$ for $q_0^s \not \equiv 7 \pmod{8}$, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for $q_0^s \equiv 7 \pmod{8}$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length $\frac{q_0^s+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14087 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Determining the covering radius of all generalized Zetterberg codes in odd characteristic Shi, Minjia Li, Shitao Helleseth, Tor Ozbudak, Ferruh Information Theory For an integer $s\ge 1$, let $\mathcal{C}_s(q_0)$ be the generalized Zetterberg code of length $q_0^s+1$ over the finite field $\F_{q_0}$ of odd characteristic. Recently, Shi, Helleseth, and Özbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of $\mathcal{C}_s(q_0)$ for $q_0^s \not \equiv 7 \pmod{8}$, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for $q_0^s \equiv 7 \pmod{8}$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length $\frac{q_0^s+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes. |
| title | Determining the covering radius of all generalized Zetterberg codes in odd characteristic |
| topic | Information Theory |
| url | https://arxiv.org/abs/2411.14087 |