Complete classification for simple root cyclic codes over local rings ${\mathbb{Z}}_{p^s}[v]\mathrm{/}⟨v^2-pv⟩$

Let $p$ be a prime integer, $n,s\ge 2$ be integers satisfying ${\rm gcd}(p,n)=1$,anddenote$R=$\mathbb{Z}$_{p^s}[v]/\langle v^2-pv\rangle$\mathrm{.}Then$R$isalocalnon-principalidealringof$p^{2s}$ elements. First, the structure of any cyclic code over $R$ of length $n$ and a complete classification of all these codes are presented. Then the cardinality of each code and dual codes of these codes are given. Moreover, self-dual cyclic codes over $R$ of length $n$ are investigated. Finally, we list some optimal $2$-quasi-cyclic self-dual linear codes over ${\mathbb{Z}}_4$ of length $30$ and extremal $4$-quasi-cyclic self-dual binary linear $[60,30,12]$ codes derived from cyclic codes over ${\mathbb{Z}}_4[v]\mathrm{/}⟨v^2+2v⟩$ of length $15$.