# Asymptotic performance of metacyclic codes

## TL;DR

This paper proves that metacyclic codes, a generalization of dihedral codes, form an asymptotically good family of error-correcting codes, under certain number-theoretic assumptions.

## Contribution

It establishes the asymptotic goodness of metacyclic codes using a number-theoretic approach assuming GRH.

## Key findings

- Metacyclic codes are asymptotically good.
- Proof relies on Artin's conjecture under GRH.
- Generalizes properties of dihedral codes.

## Abstract

A finite group with a cyclic normal subgroup N such that G/N is cyclic is said to be metacyclic. A code over a finite field F is a metacyclic code if it is a left ideal in the group algebra FG for 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 a version of Artin's conjecture for primitive roots in arithmetic progression being true under the Generalized Riemann Hypothesis (GRH).