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Let A and B be two affinely generating sets of (Z_2)^n. As usual, we denotetheir Minkowski sum by A+B. How small can A+B be, given the cardinalities of Aand B? We give a tight answer to this question. Our bound is attained when bothA and B are unions of cosets of a certain subgroup of (Z_2)^n. These cosets arearranged as Hamming balls, the smaller of which has radius 1.  By similar methods, we re-prove the Freiman-Ruzsa theorem in (Z_2)^n, with anoptimal upper bound. Denote by F(K) the maximal spanning constant |<A>|/|A|,over all subsets A of (Z_2)^n with doubling constant |A+A|/|A| < K. Weexplicitly calculate F(K), and in particular show that 4^K / 4K < F(K) (1+o(1))< 4^K / 2K. This improves the estimate F(K) = poly(K) 4^K, found recently byGreen and Tao and by Konyagin.

On Sums of Generating Sets in (Z_2)^n