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.

翻译：本文提出了一种研究有限域上线性互补对偶(LCD)准循环码的多项式方法。针对欧几里得内积、埃尔米特内积和辛内积，我们建立了指数为2的LCD准循环码的充要条件。作为这些刻画结果的推论，我们推导出了单生成元LCD准循环码的充要条件。此外，基于这些刻画结果，我们在小域上构造了若干新型准循环LCD码。