On Euclidean, Hermitian and symplectic quasi-cyclic complementary dual codes

TL;DR

This paper introduces new criteria for identifying Euclidean, Hermitian, and symplectic quasi-cyclic LCD codes at the codeword level, and constructs codes with improved parameters.

Contribution

It provides a novel characterization for LCD codes and determines conditions for one-generator quasi-cyclic codes to be LCD under various inner products.

Findings

Constructed Euclidean, Hermitian, and symplectic LCD codes with excellent parameters.

Improved known codes, including a symplectic LCD code with higher distance.

Outperformed existing optimal quaternary Hermitian LCD codes.

Abstract

Linear complementary dual codes (LCD) intersect trivially with their dual. In this paper, we develop a new characterization for LCD codes, which allows us to judge the complementary duality of linear codes from the codeword level. Further, we determine the sufficient and necessary conditions for one-generator quasi-cyclic codes to be LCD codes involving Euclidean, Hermitian, and symplectic inner products. Finally, we constructed many Euclidean, Hermitian and symmetric LCD codes with excellent parameters, some improving the results in the literature. Remarkably, we construct a symplectic LCD $[28,6]_2$ code with symplectic distance $10$, which corresponds to an trace Hermitian additive complementary dual $(14,3,10)_4$ code that outperforms the optimal quaternary Hermitian LCD $[14,3,9]_4$ code.