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

Problem Statement
This paper addresses the classification of linear codes over the commutative non-unitary ring $I_5$ of order 25. While coding theory over rings has expanded significantly since the discovery of links between codes over $\mathbb{Z}_4$ and nonlinear binary codes, the study of codes over non-unitary rings remains a developing field. Specifically, the authors focus on the ring $I_p$ (where $p$ is a prime) from Fine's classification. Previous work by Alahmadi et al. had established mass formulas and classifications for $I_3$, $I_5$, and $I_7$ for short lengths. However, the authors identify specific errors in the existing classification of quasi self-dual codes over $I_5$, particularly regarding the automorphism group orders and consistency with mass formulas for lengths $n=2$ and $n=3$. The primary goal is to provide a corrected and complete classification of self-orthogonal, quasi self-dual, and self-dual codes over $I_5$ for lengths up to 4.

Methodology
The study employs a combination of algebraic characterization, recursive construction, and computational enumeration:

1. Algebraic Framework: The authors define linear codes over $I_5$ as $I_5$-submodules of $I_5^n$. They utilize the residue code ($\text{res}(C)$) and torsion code ($\text{tor}(C)$) associated with the reduction map modulo the unique maximal ideal $J = \{0, b, d, i, n\}$. The size of a code of type $\{k_1, k_2\}$ is determined by the dimensions of these associated $\mathbb{F}_5$ codes.
2. Propagation Rules (Building-Up Constructions): The paper derives specific propagation rules to construct codes of length $n+h$ from codes of length $n$.
   • Self-Orthogonal Codes: Rules of order 2, 4, and 5 are established. These involve extending generator matrices with new rows constructed from vectors in $\mathbb{F}_5^n$ and coefficients in $I_5$ satisfying specific orthogonality conditions (e.g., $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = 0$).
   • Quasi Self-Dual Codes: Propagation rules of order 2, 4, and 5 are derived, ensuring the resulting code maintains the size $5^n$ while preserving self-orthogonality.
   • Self-Dual Codes: A propagation rule of order 6 is presented, showing that a self-dual code over $I_5$ exists only for even lengths and is constructed from a self-dual residue code over $\mathbb{F}_5$.
3. Mass Formulas: The authors utilize mass formulas (adapted from Alahmadi et al.) as stopping criteria. These formulas calculate the total number of distinct codes of a given length and type, allowing the authors to verify the completeness of their classification.
4. Computational Classification: Using the MAGMA computer algebra system, the authors generate codes using the derived propagation rules. They classify these codes up to monomial equivalence (permutation and scaling by units $\{1, -1\}$) for lengths $n \le 4$.

Key Contributions

• Correction of Previous Results: The paper explicitly corrects errors found in the classification of quasi self-dual codes over $I_5$ by Alahmadi et al. [5]. Specifically, it rectifies incorrect automorphism group orders for codes of length $n=2$ (type $\{1, 0\}$) and resolves inconsistencies in the mass formula for length $n=3$ (type $\{1, 1\}$).
• Complete Classification: The authors provide a complete classification of self-orthogonal, quasi self-dual, and self-dual codes over $I_5$ for all lengths $n \le 4$ and all types $\{k_1, k_2\}$.
• Structural Characterization: The paper establishes that a linear code over $I_5$ is self-orthogonal if and only if its residue code over $\mathbb{F}_5$ is self-orthogonal. It also characterizes self-dual codes over $I_5$ as those where the residue code is self-dual over $\mathbb{F}_5$ and the torsion code is the full space $\mathbb{F}_5^n$.
• Explicit Constructions: The paper provides explicit generator matrices and weight distributions for the classified codes, offering concrete representatives for each equivalence class.

Results
The computational results are summarized in Table 3 of the paper, listing the number of distinct inequivalent codes, their automorphism group orders, minimum distances, and weight distributions for lengths 1 through 4.

• For length 1, the paper identifies one quasi self-dual code of type $\{0, 1\}$.
• For length 2, it classifies codes of types $\{0, 1\}$, $\{0, 2\}$, $\{1, 0\}$, and $\{1, 1\}$, noting corrections to the $\{1, 0\}$ and $\{1, 1\}$ types compared to previous literature.
• For length 3 and length 4, the paper lists numerous inequivalent codes across various types, including free codes (type $\{k, 0\}$) and non-free codes.
• The study confirms that self-dual codes over $I_5$ only exist for even lengths, consistent with the derived theoretical constraints.

Significance
The paper claims that the case $p=5$ represents a critical threshold where the algebraic and combinatorial structure of $I_p$-codes becomes sufficiently rich to produce a broad range of monomially inequivalent codes, more so than the $p=3$ case. The work provides a necessary correction to the existing literature, ensuring the accuracy of mass formulas and automorphism group calculations. By establishing rigorous propagation rules and a complete classification for short lengths, the paper lays the groundwork for extending these techniques to longer lengths and other non-unitary rings. The authors suggest that the methods developed here can be applied to study other categories of codes, such as cyclic codes, over rings of the form $I_p$.