On Euclidean and Hermitian Self-Dual Cyclic Codes over $\mathbb{F}_{2^r}$

Cyclic and self-dual codes are important classes of codes in coding theory. Jia, Ling and Xing \cite{Jia} as well as Kai and Zhu \cite{Kai} proved that Euclidean self-dual cyclic codes of length $n$ over $\mathbb{F}_q$ exist if and only if $n$ is even and $q=2^r$, where $r$ is any positive integer. For $n$ and $q$ even, there always exists an $[n, \frac{n}{2}]$ self-dual cyclic code with generator polynomial $x^{\frac{n}{2}}+1$ called the \textit{trivial self-dual cyclic code}. In this paper we prove the existence of nontrivial self-dual cyclic codes of length $n=2^\nu \cdot \bar{n}$, where $\bar{n}$ is odd, over $\mathbb{F}_{2^r}$ in terms of the existence of a nontrivial splitting $(Z, X_0, X_1)$ of $\mathbb{Z}_{\bar{n}}$ by $\mu_{-1}$, where $Z, X_0,X_1$ are unions of $2^r$-cyclotomic cosets mod $\bar{n}.$ We also express the formula for the number of cyclic self-dual codes over $\mathbb{F}_{2^r}$ for each $n$ and $r$ in terms of the number of $2^r$-cyclotomic cosets in $X_0$ (or in $X_1$). We also look at Hermitian self-dual cyclic codes and show properties which are analogous to those of Euclidean self-dual cyclic codes. That is, the existence of nontrivial Hermitian self-dual codes over $\mathbb{F}_{2^{2 \ell}}$ based on the existence of a nontrivial splitting $(Z, X_0, X_1)$ of $\mathbb{Z}_{\bar{n}}$ by $\mu_{-2^\ell}$, where $Z, X_0,X_1$ are unions of $2^{2 \ell}$-cyclotomic cosets mod $\bar{n}.$ We also determine the lengths at which nontrivial Hermitian self-dual cyclic codes exist and the formula for the number of Hermitian self-dual cyclic codes for each $n$.