# Consta-dihedral Codes over Finite Fields

## TL;DR

This paper studies the algebraic and asymptotic properties of consta-dihedral codes over finite fields, showing conditions under which self-dual or LCD codes are asymptotically good, and corrects previous errors in the literature.

## Contribution

It extends prior work by analyzing consta-dihedral codes, establishing new conditions for their asymptotic goodness, and correcting earlier inaccuracies.

## Key findings

- Self-dual consta-dihedral codes are asymptotically good if q is even or divisible by 4 minus 1.
- LCD consta-dihedral codes are asymptotically good if q is odd and not divisible by 4 minus 1.
- Provides a new technique to correct errors in previous literature.

## Abstract

It is proved in a reference (Fan, Lin, IEEE TIT, vol.67, pp.5016-5025) that the self-dual (LCD respectively) dihedral codes over a finite field~$F$ with ${|F|=q}$ are asymptotically good if $q$ is even (odd respectively). In this paper, we investigate the algebraic property and the asymptotic property of conta-dihedral codes over $F$, and show that: if $q$ is even or $4\,|\,(q-1)$, then the self-dual consta-dihedral codes are asymptotically good; otherwise, the LCD consta-dihedral codes are asymptotically good. And, with the help of a technique developed in this paper, some errors in the reference mentioned above are corrected.