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

Irwansyah, Intan Muchtadi-Alamsyah, Ahmad Muchlis, Aleams Barra, Djoko Suprijanto

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