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