Asymptotic performance of metacyclic codes

A finite group is said to be metacyclic if it has a cyclic normal subgroup N such that G∕N is cyclic. A code over a finite field F is called metacyclic if it is a left ideal in the group algebra FG, with G a metacyclic group. Metacyclic codes are generalizations of dihedral codes, and can be constructed as quasi-cyclic codes with an extra automorphism. In this paper, we prove that metacyclic codes form an asymptotically good family of codes. Our proof relies on Artin’s primitive root conjecture for primes in certain arithmetic progressions being true, something that currently can only be guaranteed assuming the Generalized Riemann Hypothesis (GRH).