A note on the construction and enumeration of Euclidean self-dual skew-cyclic codes

Abstract

Let $\theta$ be an automorphism on a finite field $\mathbb {F}_q.$ In this paper, we give a way to construct and enumerate Euclidean self-dual $\theta$-cyclic codes of length n over $\mathbb {F}_q$ when n is even and $\gcd (n,|\theta |)=1.$ The restriction $\gcd (n,|\theta |)=1$ implies that the $\theta$-cyclic codes are in fact cyclic codes and $q=2^m,$ for some integer $m\ge 1.$ The construction and enumeration are done by analyzing the orbits of cyclotomic cosets under a multiplier map induced by $\theta .$