Some MDS and ACD codes over commutative non-unital rings of orders 4 and 9 (Revision)

RESUMO

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.