Dihedral group codes over finite fields

TL;DR

This paper extends the understanding of dihedral group codes over finite fields, demonstrating their asymptotic goodness and constructing specific types of codes depending on the field's characteristic.

Contribution

It proves that dihedral group codes over any finite field are asymptotically good and constructs self-dual, self-orthogonal, and LCD dihedral group codes based on the field's characteristic.

Findings

Dihedral group codes are asymptotically good over any finite field.

Constructs self-dual dihedral group codes for even characteristic fields.

Constructs self-orthogonal and LCD dihedral group codes for odd characteristic fields.

Abstract

Bazzi and Mitter [3] showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with good mathematical properties are asymptotically good. If the characteristic of the field is even, we construct asymptotically good self-dual dihedral group codes. If the characteristic of the filed is odd, we construct both the asymptotically good self-orthogonal dihedral group codes, and the asymptotically good LCD dihedral group codes.