4阶和9阶交换非幺环上的一些MDS码和ACD码（修订版）
Some MDS and ACD codes over commutative non-unital rings of orders 4 and 9 (Revision)
- Sogang University（西江大学）
- Kangwon National University（江原国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究4阶和9阶交换非幺环上的加性互补对偶码，建立其与二元、三元及p元LCD码的联系，分类高最小距离的ACD码并证明其为MDS码，同时更正了先前论文中的定理。
AI中文摘要：
阶为$p^{2}$的有限环共有十一个，按字母顺序记为$A_p$到$K_p$。我们特别考虑$I_{2}$和$I_{3}$，它们分别是阶为4和9的交换非幺环，由生成元和关系定义为\\[I_{p}=\left\langle a,b\mid pa=pb=0,\\:a^{2}=b,\\:ab=0\right\rangle\\]其中$p=2, 3$。Alahmadi等人研究了这些环上的码。本文研究环$I_{2}$和$I_{3}$上的加性互补对偶（ACD）码。我们利用从$I_{2}$到$\mathbb{F}_{2}$的约化映射，建立了$I_{2}$上ACD码与二元线性互补对偶（LCD）码之间的关系；利用从$I_{3}$到$\mathbb{F}_{3}$的约化映射，建立了$I_{3}$上ACD码与三元LCD码之间的关系。利用第一个关系，我们对$n=1, 2, 3$时$I_{2}$上具有最高最小距离的ACD码进行了分类，并对$n=4, 5$时进行了部分分类。结果表明它们是最大距离可分（MDS）码。利用第二个关系，我们对$n=1, 2$时$I_{3}$上具有最高最小Lee距离的ACD码进行了分类，并对$n=3$时进行了部分分类。我们将这两个关系推广为利用从$I_{p}$到$\mathbb{F}_{p}$的约化映射，建立$I_{p}$上ACD码与$p$元LCD码之间的关系。这是对发表在《Advances in Mathematics of Communications》第24卷，第61-76页，2026年上的论文的更正。我们特别更正了定理3.5、4.7、4.8的陈述及其证明。
英文摘要：
There are eleven finite rings of order $p^{2}$ denoted by $A_p$ to $K_p$ in alphabetical order. In particular, we consider $I_{2}$ and $I_{3}$ which are commutative non-unital rings of orders 4 and 9 defined by generators and relations as \[I_{p}=\left\langle a,b\mid pa=pb=0,\:a^{2}=b,\:ab=0\right\rangle\] for $p=2, 3,$ respectively. Alahmadi et al. studied codes over these rings. In this paper, we study additive complementary dual (ACD) codes over the rings $I_{2}$ and $I_{3}$. We show relations between ACD codes over $I_{2}$ and binary linear complementary dual (LCD) codes using a reduction map from $I_{2}$ to $\mathbb{F}_{2}$, and between ACD codes over $I_{3}$ and ternary LCD codes using a reduction map from $I_{3}$ to $\mathbb{F}_{3}$. Using the first relation, we classify ACD codes over $I_{2}$ with the highest minimum distances for $n=1, 2, 3$ and partially for $n=4, 5$. It turns out that they are maximum distance separable (MDS) codes. Using the second relation, we classify ACD codes over $I_{3}$ with the highest minimum Lee distances for $n=1, 2$ and partially for $n=3$. We generalize the two relations into a relation between ACD codes over $I_{p}$ and $p$-ary LCD codes using a reduction map from $I_{p}$ to $\mathbb{F}_{p}$. This is a correction of the paper published in Advances in Mathematics of Communications, Volume 24, pages 61-76, 2026. In particular, we corrected the statements of Theorems 3.5, 4.7, 4.8, and their proofs.
