Tilings of $p$-ary cyclic groups

A tiling of a finite abelian group $G$ is a pair $(T , A)$ of subsets of $G$ such that every element $g \in G$ can be uniquely represented as $t+a$ with $t \in T$ , $a \in A$. In this paper we consider tilings of groups $\mathbb{Z}_{p^n}$ ($p$ is prime) and give a description of a recurrent scheme embracing all tilings of such groups. Furthermore we count their number.