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

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