This paper generalizes the main results of {\it S. T. Dougherty} and {\it S. Ling} see [Des. Codes Cryptography 39, No. 2, 127‒153 (2006; Zbl 1172.94637)] to the ring $R=\mathbb F_p +u \mathbb F_p +\ldots+u^{k-1}\mathbb F_p$ (with $p$ prime) and continues the study of repeated-root cyclic codes of {\it Ping} and {\it Zhu} and of {\it Zhu, Ling} and {\it Wu} [Electron. Inform.Technol. 29, 1124‒1126 (2007) or 30, 1394‒1396 (2008)] respectively. All cyclic codes of length $p^sn$ (with $n$ prime to $p$) over $R$ are classified by determining the structure of ideals over an extension ring $S_{u^k}(m,ω)=$ GR$(u^k,m)/<ω^{p^s}-1>$ of the Galois ring GR$(u^k,m)$ and by constructing a ring isomorphism $γ$ between $R[X]/(X^N-1)$ and $\bigoplus\limits_{h\in I} S_{u^k}(m_h,ω)$ using discrete Fourier transform; (here $I$ is a complete set of $p$-cyclotomic coset representatives modulo $n$). Via $γ^{-1}$, the cyclic codes of over $R$ are characterized in terms of their generator polynomials.
Reviewer: Ralph-Hardo Schulz (Berlin)