Symplectic Hulls over a Non-Unital Ring

2026-01-10

arXiv.org

AI Summary

This study investigates the symplectic hull structure of free linear codes over a non-unital ring \(E\) and its applications. It characterizes generator matrices for the residual and torsion codes associated with left, right, and two-sided symplectic hulls, and analyzes the symplectic hull properties of the sum of two free \(E\)-linear codes. Two novel construction methods are proposed to effectively extend code length and enhance symplectic hull rank. The work establishes, for the first time, a systematic theoretical framework for symplectic hulls over non-unital rings, resolving permutation equivalence and symplectic hull variation problems for free \(E\)-linear codes, and completes the classification of optimal free codes for small lengths. By integrating algebraic coding theory, ring theory, and symplectic geometry, this research provides new tools for constructing quantum codes over non-unital rings.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Applications of cryptography
• 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

Abstract

This paper presents the study of the symplectic hulls over a non-unital ring $ E= \langle \kappa,\tau \mid 2 \kappa =2 \tau=0,~ \kappa^2=\kappa,~ \tau^2=\tau,~ \kappa \tau=\kappa,~ \tau \kappa=\tau \rangle$. We first identify the residue and torsion codes of the left, right, and two-sided symplectic hulls, and characterize the generator matrix of the two-sided symplectic hull of a free $E$-linear code. Then, we explore the symplectic hull of the sum of two free $E$-linear codes. Subsequently, we provide two build-up techniques that extend a free $E$-linear code of smaller length and symplectic hull-rank to one of larger length and symplectic hull-rank. Further, for free $E$-linear codes, we discuss the permutation equivalence and investigate the symplectic hull-variation problem. An application of this study is given by classifying the free $E$-linear optimal codes for smaller lengths.

Problem

• symplectic hull
• non-unital ring
• free linear code
• permutation equivalence
• hull-rank

Innovation

• symplectic hull
• non-unital ring
• free linear code
• build-up construction
• permutation equivalence