Given a subset $S$ of an abelian group $G$ and an integer $k\geq 1$, the $k$-deck of $S$ is the function that assigns to every $T\subseteq G$ with at most $k$ elements the number of elements $g\in G$ with $g+T\subseteq S$. The reconstruction problem for an abelian group $G$ asks for the minimal value of $k$ such that every subset $S$ of $G$ is determined, up to translation, by its $k$-deck. This minimal value is the set-reconstruction number$r_{\rm set}(G)$ of $G$; the corresponding value for multisets is the reconstruction number$r(G)$.

Previous work had given bounds for the set-reconstruction number of cyclic groups: Alon, Caro, Krasikov and Roditty [1] showed that $r_{\rm set}({\mathbb{Z}}_n)<\log_2n$ and Radcliffe and Scott [15] that $r_{\rm set}({\mathbb{Z}}_n)<9\frac{\ln n}{\ln\ln n}$. We give a precise evaluation of $r(G)$ for all abelian groups $G$ and deduce that $r_{\rm set}({\mathbb{Z}}_n)\leq 6$.