Given a finite extension $K/F$ of degree $r$ of a finite field $F$, we enumerate all selfdual skew cyclic codes in the Ore quotient ring $K[X;\text{Frob}]/(X^{rk}-1)$ for any positive integer $k$ coprime to the characteristic $p$ (separable case). We also provide an enumeration algorithm when $k$ is a power of $p$ (purely inseparable case), at the cost of some redundancies. Our approach is based on an explicit bijection between skew cyclic codes, on the one hand, and certain families of $F$-linear subspaces of some extensions of $K$. Finally, we report on an implementation in SageMath.

翻译：给定有限域$F$的有限扩张$K/F$，其扩张次数为$r$，本文枚举了Ore商环$K[X;\text{Frob}]/(X^{rk}-1)$中所有自对偶斜循环码，其中$k$为任意与特征$p$互质的正整数（可分离情形）。当$k$为$p$的幂次（纯不可分情形）时，我们亦提出一种枚举算法，但需容忍部分冗余。该方法基于斜循环码与$K$的某些扩张中特定$F$-线性子空间族之间的显式双射。最后，我们报告了在SageMath中的实现情况。