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 $$\theta .$$