On Perfect Bases in Finite Abelian Groups

Abstract

Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of $A$; similarly, we say that $A$ is a {\em perfect restricted $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ distinct elements of $A$. We prove that perfect $s$-bases exist only in the trivial cases of $s=1$ or $|A|=1$. The situation is different with restricted addition where perfection is more frequent; here we treat the case of $s=2$ and prove that $G$ has a perfect restricted $2$-basis if, and only if, it is isomorphic to $\mathbb{Z}_2$, $\mathbb{Z}_4$, $\mathbb{Z}_7$, $\mathbb{Z}_2^2$, $\mathbb{Z}_2^4$, or $\mathbb{Z}_2^2 \times \mathbb{Z}_4$.