Some MDS and ACD codes over commutative non-unital rings of orders 4 and 9 (Revision)
Abstract
There are eleven finite rings of order p2 denoted by Ap to Kp in alphabetical order. In particular, we consider I2 and I3 which are commutative non-unital rings of orders 4 and 9 defined by generators and relations as Ip=⟨ a,b| pa=pb=0, a2=b, ab=0⟩ 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 I2 and I3. We show relations between ACD codes over I2 and binary linear complementary dual (LCD) codes using a reduction map from I2 to 𝔽2, and between ACD codes over I3 and ternary LCD codes using a reduction map from I3 to 𝔽3. Using the first relation, we classify ACD codes over I2 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 I3 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 Ip and p-ary LCD codes using a reduction map from Ip to 𝔽p. 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.
Versions
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BibTeX
@misc{kim2026some,
title = {{Some MDS and ACD codes over commutative non-unital rings of orders 4 and 9 (Revision)}},
author = {Jon-Lark Kim and Marvin Olavides and Young Gun Roe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20190},
eprint = {2609.20190},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20190},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Some MDS and ACD codes over commutative non-unital rings of orders 4 and 9 (Revision)
AU - Jon-Lark Kim
AU - Marvin Olavides
AU - Young Gun Roe
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20190
UR - https://arxiv.org/abs/2609.20190
AB - 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=⟨ a,b| pa=pb=0, a^2=b, ab=0⟩ 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 𝔽_2, and between ACD codes over I_3 and ternary LCD codes using a reduction map from I_3 to 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 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.
ER -