Full Rank Tilings of Finite Abelian Groups

A tiling of a finite abelian group $G$ is a pair $(V,A)$ of subsets of $G$ such that $0$ is in both $V$ and $A$ and every $g\in G$ can be uniquely written as $g=v+a$ with $v\in V$ and $a\in A$. Tilings are a special case of normed factorizations, a type of factorization by subsets that was introduced by Hajos [Casopsis Puest Path. Rys., 74, (1949), pp. 157-162]. A tiling is said to be full rank if both $V$ and $A$ generate $G$. Cohen, Litsyn, Vardy, and Zemor [SIAM J. Discrete Math., 9 (1996), pp. 393-412] proved that any tiling of $\Z_2^n$ can be decomposed into full rank and trivial tilings. We generalize this decomposition from $\Z_2^n$ to all finite abelian groups. We also show how to generate larger full rank tilings from smaller ones, and give two sufficient conditions for a group to admit a full rank tiling, showing that many groups do admit them. In particular, we prove that if $p\geq 5$ is a prime and $n\geq 4$, then $\Z_p^n$ admits a full rank tiling. This bound on $n$ is tight for $5\leq p\leq 11$, and is conjectured to be tight for all primes $p$.