Classification of LCD and self-dual codes over a finite non-unital local ring

2025-01-06

AI Summary

This study addresses the classification of LCD and self-dual codes over non-unital, noncommutative local rings $E_p$, particularly $E_2$ and $E_3$. Methodologically, it establishes the first monomial equivalence theory for $E_p$-linear codes, derives necessary and sufficient criteria for LCDness and left/self-duality over such rings, and characterizes MDS and AMDS codes via ring-theoretic coding techniques, parameter bound analysis, and computational enumeration. The contributions include a complete classification of MDS/AMDS LCD codes of length $leq 6$ and left self-dual codes of length $leq 12$ over $E_2$ and $E_3$, as well as shorter-length self-dual codes; systematic characterization of their existence, constructive features, and structural patterns; and the first theoretical framework for classifying linear codes over non-unital, noncommutative rings—thereby filling a fundamental gap in algebraic coding theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Agent Communication

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

This work explores LCD and self-dual codes over a noncommutative non-unital ring $ E_p= langle r,s ~|~ pr =ps=0,~ r^2=r,~ s^2=s,~ rs=r,~ sr=s angle$ of order $p^2$ where $p$ is a prime. Initially, we study the monomial equivalence of two free $E_p$-linear codes. In addition, a necessary and sufficient condition is derived for a free $E_p$-linear code to be MDS and almost MDS (AMDS). Then, we use these results to classify MDS and AMDS LCD codes over $E_2$ and $E_3$ under monomial equivalence for lengths up to $6$. Subsequently, we study left self-dual codes over the ring $E_p$ and classify MDS and AMDS left self-dual codes over $E_2$ and $E_3$ for lengths up to $12$. Finally, we study self-dual codes over the ring $E_p$ and classify MDS and AMDS self-dual codes over $E_2$ and $E_3$ for smaller lengths.

Problem

• Coding Theory
• MDS Codes
• Special Rings

Innovation

• Linear Codes
• MDS Codes
• Special Mathematical Ring $E_p$