On metacyclic p-group codes

TL;DR

This paper investigates metacyclic p-group codes derived from finite semisimple group algebras, extending previous results to broader classes of groups and constructing new codes with optimal parameters.

Contribution

It extends prior work on group codes to include all metacyclic p-groups and constructs novel non-central codes with superior parameters.

Findings

Extended results to metacyclic groups with arbitrary prime divisors.

Constructed non-central codes using Bass and bicyclic units.

Provided explicit left codes for the studied group algebras.

Abstract

In this article, we study the metacyclic p-group codes arising from finite semisimple group algebras. In [CM25], we studied group codes arising from metacyclic groups with order divisible by two distinct odd primes. In the current work, we focus on metacyclic p-group codes, as a result of which we are also able to extend the results of [CM25] for metacyclic groups with order divisible by any two primes, not necessarily odd or distinct. Consequently, existing results on group algebras of some important classes of groups, including dihedral and quaternion groups, have been extended. Additionally, we provide left codes for the undertaken group algebras. Finally, we construct non-central codes using units motivated by Bass and bicyclic units, which are inequivalent to any abelian group codes and yield best known parameters.