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A subset S of the Boolean hypercube F₂ⁿ is a sumset if S = A+A = \a + b \ | \ a, b A\ for some A F₂ⁿ. We prove that the number of sumsets in F₂ⁿ is asymptotically (2ⁿ-1) 2^2^{n-1}. Furthermore, we show that the family of sumsets in F₂ⁿ is almost identical to the family of all subsets of F₂ⁿ that contain a complete linear subspace of co-dimension 1.
Alon et al. (Mon,) studied this question.