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

Abstract

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.

21 pages