有限链环上的二维常循环码
Two-dimensional constacyclic codes over finite chain rings
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中文总结 AI 辅助
研究有限链环上二维\((\lambda,\mu)\)-常循环码的代数结构，通过分析本原幂等元确定码的生成元以获其结构，还找出二维常循环码在有限链环上关于秩是最大汉明距离的条件。
中文摘要 AI 辅助
本文主要关注具有剩余域\(\mathbb{F}_q\)的有限链环上长度为\(\ell\mathrm{m}\)的二维\((\lambda,\mu)\)-常循环码的代数结构，其中\(q \equiv 1 \pmod{r\mathrm{m}}\)且\(r\)表示\(\bar{\mu}\)的乘法阶。通过分析有限链环内的本原幂等元来确定码的生成元，得到了二维\((\lambda,\mu)\)-常循环码的结构，还找到了二维常循环码在有限链环上关于秩是最大汉明距离的条件。
英文摘要
The main focus of this paper is on the algebraic structure of two-dimensional $(λ,μ)$-constacyclic codes of length $\ell\mathrm{m}$ over finite chain rings with residue field $\mathbb{F}_q$, where $q \equiv 1 \pmod{r\mathrm{m}}$ and $r$ denotes the multiplicative order of $\barμ$. In this paper, the structure of two-dimensional $(λ,μ)$-constacyclic codes is obtained. Our approach relies on analysing primitive idempotents within the finite chain ring to determine the generators of these codes. We also find the condition under which two-dimensional constacyclic codes are maximum Hamming distance with respect to rank (MHDR) over finite chain rings.
