AI 中文总结
研究4阶交换非酉环\(I_2\)上的循环码，通过相关剩余码和挠码，引入扭曲与非扭曲循环码概念，利用格雷映射建立与二元准循环码联系，研究对偶性质，对长度\(n\leq7\)的码分类并确定其结构性质。
AI 中文摘要
设\(I_2\)是在Fine分类中出现的4阶交换非酉环。本文通过\(I_2\)上循环码的相关剩余码和\(\mathbb{F}_2\)上的挠码来研究\(I_2\)上的循环码。引入了扭曲和非扭曲循环码的概念，并根据涉及扭曲映射和循环移位的相容性条件来刻画循环性。通过格雷映射建立了\(I_2\)上循环码与二元准循环码之间的联系。特别地，证明了\(I_2\)上循环码的格雷像是指数为2的二元准循环码。还研究了\(I_2\)上循环码的对偶性质，证明了循环码的对偶仍是循环的。最后，对长度\(n\leq7\)的\(I_2\)上置换不等价循环码进行了分类，并确定了这些码的各种结构性质。
英文摘要
Let $I_2$ be the commutative non-unitary ring of order $4$ arising in the classification of Fine. In this paper, we investigate cyclic codes over $I_2$ through their associated residue and torsion codes over $\mathbb{F}_2$. We introduce the notions of twisted and untwisted cyclic codes and characterize cyclicity in terms of a compatibility condition involving the twist map and the cyclic shift. Connections between cyclic codes over $I_2$ and binary quasi-cyclic codes are established via Gray maps. In particular, we show that the Gray image of a cyclic code over $I_2$ is a binary quasi-cyclic code of index $2$. We also study duality properties of cyclic codes over $I_2$ and prove that the dual of a cyclic code is again cyclic. Finally, we classify permutation inequivalent cyclic codes over $I_2$ for lengths $n \le 7$ and determine various structural properties of these codes.
Comments29 pages
