Subsets in Linear Spaces over the Finite Field F3 Uniquely Determined by Their Pairwise Sums Collection

Abstract

Let F₃ⁿ be an n-dimensional linear space over the finite field F₃. Let A = {a₁, a₂,..., a_N} 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.