Hermitian Self-Dual Cyclic Codes of Length $p^a$ over $GR(p^2,s)$

In this paper, we study cyclic codes over the Galois ring ${\rm GR}({p^2},s)$. The main result is the characterization and enumeration of Hermitian self-dual cyclic codes of length $p^a$ over ${\rm GR}({p^2},s)$. Combining with some known results and the standard Discrete Fourier Transform decomposition, we arrive at the characterization and enumeration of Euclidean self-dual cyclic codes of any length over ${\rm GR}({p^2},s)$. Some corrections to results on Euclidean self-dual cyclic codes of even length over $\mathbb{Z}_4$ in Discrete Appl. Math. 128, (2003), 27 and Des. Codes Cryptogr. 39, (2006), 127 are provided.