在阶为25的交换非酉环上自正交码的构造
Construction of self-orthogonal codes over a commutative non-unitary ring of order 25
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中文总结 AI 辅助
研究阶为25的交换非酉环\(I_5\)上的线性码，定义三类线性\(I_5\) - 码并研究其结构，对长度至多为4的三类码进行单项式等价完全分类，同时纠正了\(I_5\)上拟自对偶码分类的错误。
中文摘要 AI 辅助
非酉环上的码最近受到研究。特别地，考虑在Fine分类中阶为\(p^2\)的交换非酉环\(I_p\)（\(p\)为素数）上的码。对于\(p = 2\)（分别地，\(p = 3\)），已研究了\(I_p\)上的三类码：自正交码、拟自对偶码和自对偶码。利用相关质量公式和构造方法，对某些小长度的这些码进行了置换等价（分别地，单项式等价）分类。本文取素数\(p = 5\)并考虑环\(I_5\)。引入\(I_5\)上线性码的概念，定义相同的三类线性\(I_5\) - 码，研究其结构并与相关剩余码和挠码联系起来。对于给定类型\(\{ k_1, k_2 \}\)，在长度至多为\(4\)时，对这三类码进行了单项式等价的完全分类。此外，在Alahmadi等人关于\(I_p\)上自正交码质量公式的论文中，\(I_5\)上拟自对偶码的分类存在错误，如某些码的自同构群阶不正确，以及对于长度\(n = 2\)和类型\(\{ 1, 0 \}\)以及长度\(n = 3\)和类型\(\{ 1, 1 \}\)与\(I_p\)上自正交码质量公式不一致。本文纠正并改进了这些结果。
英文摘要
Codes over non-unitary rings have been studied recently. In particular, codes over the commutative non-unitary ring $I_p$ (in the classification of Fine) of order $p^2$ where $p$ is a prime are being considered. For $p=2$ (resp. $p=3$), three categories of codes over $I_p$ have been studied: self-orthogonal codes, quasi self-dual codes, and self-dual codes over $I_p$. Using some related mass formulas and building-up constructions, classifications of these codes have been done up to the permutation equivalence (resp. the monomial equivalence) for certain small lengths. In this paper, we take the prime $p=5$ and consider the ring $I_5$. We introduce the notion of linear codes over $I_5$. We also define the same three categories of linear $I_5$-codes, study the structures of these $I_5$-codes and relate them to their associated residue and torsion codes. We classify the three categories of codes completely in lengths at most $4$ up to the monomial equivalence for a given type $\{ k_1 , k_2 \}$. Moreover, in the paper of Alahmadi et al. regarding the mass formula for self-orthogonal codes over $I_p$, mistakes in the classification of quasi self-dual codes over $I_5$ had been made such as incorrect automorphism group order of some codes or inconsistency with the mass formula for self-orthogonal codes over $I_p$ for length $n=2$ and type $\{ 1 , 0 \}$ and for length $n=3$ and type $\{ 1, 1 \}$. We correct and improve such results.
