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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 thatEuclidean self-dual cyclic codes of length $n$ over $\mathbb{F}_q$ exist if andonly 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 withgenerator polynomial $x^{\frac{n}{2}}+1$ called the \textit{trivial self-dualcyclic code}. In this paper we prove the existence of nontrivial self-dualcyclic 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 $Z, X_0,X_1$ areunions of $2^r$-cyclotomic cosets mod $\bar{n}.$ We also express the formulafor 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 whichare analogous to those of Euclidean self-dual cyclic codes. That is, theexistence 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 atwhich nontrivial Hermitian self-dual cyclic codes exist and the formula for thenumber of Hermitian self-dual cyclic codes for each $n$.

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