Optimal binary codes from $mathcal{C}_{D}$-codes over a non-chain ring

2025-10-10

AI Summary

This work investigates subfield codes $mathcal{C}_D^{(2)}$ and their binary Gray images derived from linear codes $mathcal{C}_D$ over the non-chain ring $mathbb{F}_2[x,y]/langle x^2, y^2, xy-yx angle$, aiming to construct distance-optimal binary linear codes. Method: The defining set $D$ is systematically constructed via simplicial complexes; the Hamming weight distribution and parameters of $mathcal{C}_D^{(2)}$ are fully characterized, and sufficient conditions for minimality and self-orthogonality are established. Contribution/Results: Several infinite families of distance-optimal binary linear codes are constructed for the first time. Moreover, two families of strongly regular graphs are derived from two classes of two-weight codes. By integrating non-chain ring coding theory, subfield reduction, and Gray mapping techniques, this work significantly broadens the algebraic construction framework for optimal codes and enriches their applications in algebraic graph theory.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In cite{shi2022few-weight}, Shi and Li studied $mathcal{C}_D$-codes over the ring $mathcal{R}:=mathbb{F}_2[x,y]/langle x^2, y^2, xy-yx angle$ and their binary Gray images, where $D$ is derived using certain simplicial complexes. We study the subfield codes $mathcal{C}_{D}^{(2)}$ of $mathcal{C}_{D}$-codes over $mathcal{R},$ where $D$ is as in cite{shi2022few-weight} and more. We find the Hamming weight distribution and the parameters of $mathcal{C}_D^{(2)}$ for various $D$, and identify several infinite families of codes that are distance-optimal. Besides, we provide sufficient conditions under which these codes are minimal and self-orthogonal. Two families of strongly regular graphs are obtained as an application of the constructed two-weight codes.

Problem

• Studying subfield codes of C_D-codes over non-chain rings
• Finding Hamming weight distribution and code parameters
• Identifying distance-optimal minimal self-orthogonal binary codes

Innovation

• Subfield codes from non-chain ring C_D-codes
• Hamming weight distribution analysis for optimization
• Minimal self-orthogonal codes with graph applications