Self-dual Repeated Root Cyclic and Negacyclic Codes over Finite Fields

Abstract

In this paper we investigate repeated root cyclic and negacyclic codes of length $p^{r}m$ over ${\mathbb{F}}_{p^s}$ with $(m,p)=1$. In the case $p$ odd, we give necessary and sufficient conditions on the existence of negacyclic self-dual codes. When $m=2m^{\prime }$ with $m^{\prime }$ odd, we characterize the codes in terms of their generator polynomials. This provides simple conditions on the existence of self-dual negacyclic codes, and generalizes the results of Dinh \cite{dinh}. We also answer an open problem concerning the number of self-dual cyclic codes given by Jia et al. \cite{jia}.