In this paper, we study quasi-twisted codes and their relationship with additive constacyclic codes through a polynomial-based approach. We first present a polynomial characterization of quasi-twisted codes over finite fields analogous to quasi-cyclic codes and determine Euclidean, Hermitian, and symplectic duals of quasi-twisted codes with index $2$. Additionally, we provide necessary and sufficient conditions for the self-orthogonality of appropriate quasi-twisted codes. Next, we explore a one-to-one correspondence between quasi-twisted codes of length $lm$ with index $l$ over $\mathbb{F}_q$ and additive constacyclic codes of length $m$ over $\mathbb{F}_{q^l}$. We establish relationships between trace inner products in the additive setting and Euclidean, symplectic inner products in the quasi-twisted setting. Using these relations and the correspondence, we determine the dual of additive constacyclic codes with respect to the trace inner products. As a consequence, we conclude that determining the trace Euclidean dual and trace Hermitian dual of an additive constacyclic code is equivalent to determining the Euclidean and symplectic dual of the corresponding quasi-twisted code.

翻译：本文通过多项式方法研究准扭码及其与加性常循环码的关系。我们首先给出了有限域上准扭码的多项式刻画，类似于准循环码的情形，并确定了指标为$2$的准扭码的欧几里得对偶、厄米特对偶及辛对偶。此外，我们给出了适当准扭码自正交的充要条件。接着，我们探讨了$\mathbb{F}_q$上长度为$lm$、指标为$l$的准扭码与$\mathbb{F}_{q^l}$上长度为$m$的加性常循环码之间的一一对应关系。我们建立了加性设定下的迹内积与准扭码设定下的欧几里得内积、辛内积之间的联系。利用这些关系及对应性，我们确定了加性常循环码关于迹内积的对偶码。由此我们得出结论：确定加性常循环码的迹欧几里得对偶与迹厄米特对偶，等价于确定对应准扭码的欧几里得对偶与辛对偶。