Construction of Cyclic Codes over a Class of Matrix Rings

2026-02-20

AI Summary

This work investigates the construction of cyclic codes over the matrix ring $\mathcal{R} = M_4(\mathbb{F}_2[u]/\langle u^k \rangle)$ and their linear representations over finite fields. By establishing an isomorphism between $\mathcal{R}$ and an extended polynomial ring, the ideal structure and explicit form of cyclic codes are characterized. Notably, this study pioneers the integration of extension rings with zero divisors and matrix rings to construct cyclic codes decomposable as direct sums. Employing the Bachoc map and Gray map, these codes are effectively mapped to $\mathbb{F}_{16}$, enabling a systematic analysis of their Euclidean and Hermitian dual structures. A cardinality formula for cyclic codes is derived, leading to the construction of several $\mathbb{F}_{16}$-linear codes with favorable parameters, some of which outperform existing codes in the literature.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $ \mathbb F_2[u]/ \langle u^k \rangle= \mathbb F_2+u\mathbb F_2+u^2\mathbb F_2+\cdots+u^{k-1}\mathbb F_2 ,$ where $u^k=0$ for a positive integer $k$, and $\mathcal{R}=M_4 (\mathbb F_2( u)/ \langle u^k \rangle)$ be the finite noncommutative non-chain matrix ring of order $4\times4$. This paper presents the construction of cyclic codes over the finite field $\mathbb F_{16}$ via the considered matrix ring $\mathcal{R}$. In this connection, first, we discuss the structure of the ring $\mathcal{R}$ and show that $\mathcal{R}$ is isomorphic to the ring $( \mathbb F_{16}+ v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16}) + u(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16}) + u^2(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16}+ v^3\mathbb F_{16}) + \cdots + u^{k-1}(\mathbb F_{16} + v\mathbb F_{16} + v^2\mathbb F_{16} + v^3\mathbb F_{16})$ where $v^4=0, u^k=0, u^iv^j=v^ju^i$ for $i \in \{1,\dots, k-1\}$ and $j \in \{1, 2, 3\}$. Then, we establish the form of ideals of the ring $\mathcal{R}$ and related cyclic codes over $\mathcal{R}$. Further, we show that these cyclic codes can be written as the direct sums of $\mathcal{R}$-submodules of $\frac{\mathcal{R}[x]}{}$, and derive the formula for the cardinality of cyclic codes over $\mathcal{R}$. Then, we consider the Euclidean and Hermitian duals of the derived cyclic codes over $\mathcal{R}$. Under the module isometry for $\mathcal{R}$, we use the Bachoc map and the Gray map, which takes a derived cyclic code over $\mathcal{R}$ to $\mathbb F_{16}$. Finally, we provide some non-trivial examples of linear codes over $\mathbb F_{16}$ with good parameters that support our derived results and compare a few codes with existing codes in the literature.

Problem

• cyclic codes
• matrix rings
• finite fields
• noncommutative rings
• code construction

Innovation

• cyclic codes
• matrix rings
• Gray map
• Bachoc map
• noncommutative rings