Let's $$ C\subseteq F^n_2, $$ $$ 2C=C+C=\{\bar\alpha+\bar\beta\ | \bar\alpha,\bar\beta\in C\}. $$ I need to find $C$ such that $2C=F_2^n$ and $|C|$ is minimal. I have found the following bounds:$$|C|\ge\frac{1+\sqrt{2^{n+3}-7}}{2}=B(n)$$and $$|C|\le2^{\frac{n+2}{2}}-2=A(n),(n=2k,n\ge4),$$$$|C|\le2^{\frac{n+1}{2}}+2^{\frac{n-1}{2}}-2=A(n),(n=2k+1,n\ge5).$$It is easy to see that $$\lim_{n\rightarrow\infty}{\frac{A(2n)}{B(2n)}=\sqrt{2}},\lim_{n\rightarrow\infty}{\frac{A(2n+1)}{B(2n+1)}}=\frac{3}{2}.$$The last equations show that there might be some improvements for lower or/and upper bounds.
Actually this problem is related with my thesis and I understand that I should do it myself but I have been thinking about it for about two months and I can't find any clever method to get better bounds. Any hints and suggestions would be very appreciated.
Thanks!
NOTE: I have also asked this question at MathOverflow and there was an answer. You can find it here https://mathoverflow.net/questions/188767/minimal-sumset-basis-in-the-discrete-linear-space-mathbb-f-2n . Also it gives some improvements for upper bounds, it doesn't give asymtotics.