arXiv subjects

Marvin Olavides

Publications and source records attributed to Marvin Olavides.

3 recordsLinked to original sources

arXiv2026

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

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.

cs.IT↗

arXiv2026

Cyclic codes over a commutative non-unitary ring of order 4

Let $I_2$ be the commutative non-unitary ring of order $4$ arising in the classification of Fine. In this paper, we investigate cyclic codes over $I_2$ through their associated residue and torsion codes over $\mathbb{F}_2$. We introduce the notions of twisted and untwisted cyclic codes and characterize cyclicity in terms of a compatibility condition involving the twist map and the cyclic shift. Connections between cyclic codes over $I_2$ and binary quasi-cyclic codes are established via Gray maps. In particular, we show that the Gray image of a cyclic code over $I_2$ is a binary quasi-cyclic code of index $2$. We also study duality properties of cyclic codes over $I_2$ and prove that the dual of a cyclic code is again cyclic. Finally, we classify permutation inequivalent cyclic codes over $I_2$ for lengths $n \le 7$ and determine various structural properties of these codes.

cs.IT↗

arXiv2026

Construction of self-orthogonal codes over a commutative non-unitary ring of order 25

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.

cs.IT↗