Sumsets in the Hypercube

A subset S of the Boolean hypercube 𝔽_2^n is a sumset if S = A+A = {a + b | a, b∈ A} for some A ⊆𝔽_2^n. We prove that the number of sumsets in 𝔽_2^n is asymptotically (2^n-1)2^2^n-1. Furthermore, we show that the family of sumsets in 𝔽_2^n is almost identical to the family of all subsets of 𝔽_2^n that contain a complete linear subspace of co-dimension 1.