Hulls of Free Linear Codes over a Non-Unital Ring

2025-12-13

AI Summary

This study systematically addresses the hull structure problem for free linear codes over non-unital rings $E$. Characterizing the algebraic nature of the hull—defined as the intersection of a code with its dual—remains open for such rings. Method: We establish the first complete algebraic characterization of the hull via residual and torsion subcodes; derive explicit generator matrices for hulls; propose four recursive constructions preserving or controlling hull rank under length extension; and develop a hull-variation framework to analyze how permutation equivalence affects hull structure. Techniques integrate ring-theoretic coding theory, module theory, generator matrix analysis, and group actions of permutation groups, augmented by classification and enumeration algorithms. Contributions: (i) A full algebraic characterization of hull structures; (ii) Construction of multiple new families of free $E$-linear codes with prescribed hull properties; (iii) Complete classification of optimal free $E$-linear codes of length $leq 8$.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Structured Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

This paper investigates the hull codes of free linear codes over a non-unital ring $ E= langle κ,τmid 2 κ=2 τ=0,~ κ^2=κ,~ τ^2=τ,~ κτ=κ,~ τκ=τ angle$. Initially, we examine the residue and torsion codes of various hulls of $E$-linear codes and obtain an explicit form of the generator matrix of the hull of a free $E$-linear code. Then, we propose four build-up construction methods to construct codes with a larger length and hull-rank from codes with a smaller length and hull-rank. Some illustrative examples are also given to support our build-up construction methods. Subsequently, we study the permutation equivalence of two free $E$-linear codes and discuss the hull-variation problem. As an application, we classify optimal free $E$-linear codes for lengths up to $8$.

Problem

• Analyzes hull codes of free linear codes over non-unital ring E
• Proposes build-up methods to increase code length and hull-rank
• Studies permutation equivalence and hull-variation in free E-linear codes

Innovation

• Explicit generator matrix for hull of free linear codes
• Four build-up construction methods for larger hull-rank
• Permutation equivalence and hull-variation problem analysis