An isoperimetric problem on the hypercube

When A consists of even vectors only, the problem was probably solved by Korner and Wei [Odd and even Hamming spheres also have minimum boundary, Discrete Math. 51 (1984), 147–165]. See also Lemma 1.10 in [D. Galvin, On homomorphisms from the Hamming cube to Z, Israel J. of Math 138 (2003), 189-213].

If $E$ consists not only of unit vectors, but also of the zero vector, then according to Alon & Spencer ``Probabilistic method'' chapter 7, the sharp isoperimetric inequality was proved by Harper. It asserts that the Hamming ball minimizes $A+E$. In that chapter they show how to get a very good asymptotic bound for the same problem.