Fix integer $n\ge 1$, and let $E=\{e\_1,...,e\_n\}$ denote the standard basis of the vector space ${\mathbb F}\_2^n$. Thus, for a set $A\subset{\mathbb F}\_2^n$, the sumset $A+E:=\{a+e\colon a\in A,\ e\in E\}$ consists of all those elements of ${\mathbb F}\_2^n$ which are at Hamming distance $1$ from an element of $A$. Now I wonder,



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> How small can $|A+E|$ be in terms of $|A|$? How to choose a set $A$ of prescribed size to minimize the size of $A+E$?
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I would expect the answer to be well-known -- any reference?


Somewhat closer to what I actually need is the situation where $A$ consists of even vectors only; that is, of vectors orthogonal to $e\_1+\dotsb+e\_n$. How to choose $A$ (of prescribed size) under this additional constrain to minimize $|A+E|$?



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]. 


