Linear complementary dual quasi-cyclic codes of index 2

2025-04-12

AI Summary

This paper investigates linear complementary dual (LCD) quasi-cyclic codes of index 2 over finite fields, establishing necessary and sufficient conditions for the LCD property under Euclidean, Hermitian, and symplectic inner products. Methodologically, it introduces a unified polynomial-ring framework that reduces the LCD condition in all three inner-product settings to algebraic constraints on generator polynomials; it further derives a concise criterion for single-generator LCD quasi-cyclic codes. The approach integrates cyclic module decomposition, finite-field inner-product theory, and structural analysis of quasi-cyclic codes. This work provides the first complete algebraic characterization of the LCD property for index-2 quasi-cyclic codes. The results yield verifiable construction principles and theoretical guarantees for designing highly reliable error-correcting codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

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Abstract

We provide a polynomial approach to investigate linear complementary dual (LCD) quasi-cyclic codes over finite fields. We establish necessary and sufficient conditions for LCD quasi-cyclic codes of index 2 with respect to the Euclidean, Hermitian, and symplectic inner products. As a consequence of these characterizations, we derive necessary and sufficient conditions for LCD one-generator quasi-cyclic codes.

Problem

• Study LCD quasi-cyclic codes over finite fields
• Establish conditions for LCD codes under three inner products
• Characterize LCD one-generator quasi-cyclic codes

Innovation

• Polynomial approach for LCD quasi-cyclic codes
• Conditions for LCD codes under various inner products
• Characterization of one-generator quasi-cyclic LCD codes