Codes which have a finite field $\mathbb{F}_{q^m}$ as their alphabet but which are only linear over a subfield $\mathbb{F}_q$ are a topic of much recent interest due to their utility in constructing quantum error correcting codes. In this article, we find generators for trace dual spaces of different families of $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^2}$. In particular, given the field extension $\mathbb{F}_q\leq \mathbb{F}_{q^2}$ with $q$ an odd prime power, we determine the trace Euclidean and trace Hermitian dual codes for the general $\mathbb{F}_q$-linear cyclic $\mathbb{F}_{q^2}$-code. In addition, we also determine the trace Euclidean and trace Hermitian duals for general $\mathbb{F}_q$-linear skew cyclic $\mathbb{F}_{q^2}$-codes, which are defined to be left $\mathbb{F}_q[X]$-submodules of $\mathbb{F}_{q^2}[X;σ]/(X^n-1)$, where $σ$ denotes the Frobenius automorphism and $\mathbb{F}_{q^2}[X;σ]$ the induced skew polynomial ring.

翻译：以有限域$\mathbb{F}_{q^m}$为字母表但仅在其子域$\mathbb{F}_q$上保持线性的码，因在构建量子纠错码中的实用性而成为近期研究热点。本文研究了$\mathbb{F}_{q^2}$上不同族$\mathbb{F}_q$-线性码的迹对偶空间的生成元。具体而言，考虑$q$为奇素数幂时的域扩张$\mathbb{F}_q\leq \mathbb{F}_{q^2}$，我们确定了$\mathbb{F}_q$-线性循环$\mathbb{F}_{q^2}$-码的迹欧几里得对偶码与迹埃尔米特对偶码。此外，我们还确定了$\mathbb{F}_q$-线性斜循环$\mathbb{F}_{q^2}$-码的迹欧几里得对偶与迹埃尔米特对偶——这类码被定义为$\mathbb{F}_{q^2}[X;\sigma]/(X^n-1)$的左$\mathbb{F}_q[X]$-子模，其中$\sigma$表示Frobenius自同构，$\mathbb{F}_{q^2}[X;\sigma]$表示诱导的斜多项式环。