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

2026-07-12

AI Summary

This study addresses the incomplete classification and existing inaccuracies concerning self-orthogonal, quasi-self-dual, and self-dual codes over the commutative non-unital ring \( I_5 \) of order 25 for the prime \( p = 5 \). The authors systematically develop a theoretical framework for linear codes over \( I_5 \), introducing structural analyses via residue and torsion codes, and combine mass formulas with constructive methods to achieve a complete monomial equivalence classification of these three code families for lengths up to 4 and prescribed types \( \{k_1, k_2\} \). In doing so, the work corrects erroneous claims by Alahmadi et al. regarding the automorphism group orders of quasi-self-dual codes in the cases \( n=2 \), type \( \{1,0\} \) and \( n=3 \), type \( \{1,1\} \), ensuring consistency with mass formulas and thereby advancing the classification theory of codes over non-unital rings.

Abstract

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.

Problem

• self-orthogonal codes
• non-unitary ring
• mass formula
• quasi self-dual codes
• monomial equivalence

Innovation

• self-orthogonal codes
• non-unitary ring
• mass formula
• monomial equivalence
• residue and torsion codes