I've come across a seemingly natural question in additive combinatorics to which I can't find the answer in the literature. I would like to know if, given an $\alpha > 0$ and large $n$, there are interesting examples of sets $X \subseteq \mathbb{Z}_2^n$ such that $|X| \ge \alpha 2^n$ and $X + X \ne \mathbb{Z_2}^n$.

There are certainly uninteresting examples: say, the set of points in some suitably large proper subspace of $\mathbb{Z_2}^n$, or some union of cosets of such a subspace, etc etc. I would like to know if there are examples that do not have an obvious linear-algebraic structure of this sort. specifically, are there examples that are 'pseudorandom', say such that every codimension-1 subspace contains roughly $|X|/2$ elements of $X$?

In the language of additive theory, we know that $X$ has 'doubling constant' $\frac{|X+X|}{|X|} < \alpha^{-1}$, and there are many results in additive theory describing sets with small doubling constant. However, the same inequality also holds for an arbitrary set strictly larger than $X$, so no result on doubling constant alone will give sensible structure. Is there a well-known theorem or observation somewhere that I'm missing?