Subsets in Linear Spaces over the Finite Field F 3 Uniquely Determined by Their Pairwise Sums Collection

Let F3n be an n-dimensional linear space over the finite field F3. Let A = {a1, a2,..., aN} be a set in F3n and A + A be the collection of sums of pairs of distinct elements in A. It is said, that A is uniquely determined by A + A if for any B ⊆ F3n such that |A| = |B| and A + A = B + B it follows that A = B. In this paper, we find those values of N for which any set A ⊂ F3n containing N elements is determined uniquely by A + A.