Linear complementary dual quasi-cyclic codes of index 2

TL;DR

The paper develops a polynomial, CRT-based framework for linear complementary dual (LCD) quasi-cyclic codes of index $2$ over finite fields, deriving necessary and sufficient conditions for Euclidean, Hermitian, and symplectic LCD codes and specializing to one-generator QC codes. By decomposing QC codes into constituents via $x^m-1$ factorization and analyzing generator polynomials, it provides explicit criteria in terms of gcds and reciprocity for each inner product, supported by concrete constructions over binary, ternary, and quaternary fields. The results extend prior work on index-$2$ one-generator QC LCD codes to the full two-generator index-$2$ setting, and offer practical code constructions and tables, with potential applications to quantum and additive codes and cryptographic security. The approach is amenable to generalization to broader quasi-twisted families and may facilitate systematic design of LCD QC codes with optimal or best-known parameters.

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. Furthermore, using these characterizations, we construct some new quasi-cyclic LCD codes over small fields.

Theorems & Definitions (42)

• Lemma 2.1: lally2001ling2001
• Remark 1
• Theorem 2.2
• Theorem 2.3
• Remark 2
• Lemma 2.4
• proof
• Remark 3
• Theorem 3.1
• Lemma 3.2