Long module skew codes are good

Abstract

Module skew codes are one sided modules for (a quotient of) a skew polynomial ring where multiplication is twisted by an automorphism of the Galois group of the alphabet field. We prove that long module skew codes over a fixed finite field are asymptotically good by using a non-constructive counting argument. We show that for fixed alphabet size, and automorphism order and large length their asymptotic rate and relative distance satisfy a modified Varshamov–Gilbert bound.