Given $k,n\in\mathbb{N}$. Let M:={0,... , m-1}. How to find a subset $T\subset M$, |T|=k with T={ $n_1$,...,$n_k$ | $n_i\in M$ } and T+T|={ (a+b)%m | $a\in N,b\in N$ } ("%" means modulo) such that |T+T| is max. 

I tried to construct a sequence of numbers which maximize |T+T|. But I couln't figure out:<br>
- is it possible to cover the whole set M for $k\leq \sqrt{n}$<br>
- What is the best way to construct such a sequence in Theory.

I am looking for papers which deals with this topic or any word to find those papers. I don't think this problem is running under the ordinary topicname "set covering problems".

My Idea to construct such a sequence is $a_0=0;a_{i+1}=a_i+(k-i)$ for T={$a_i$, i=0,...,k-1} to get as less as possible collision in the sums of |T+T|. But random subsets of M show me, that there are better subsets.

In my opinion it is hard to find such a optimal subset T.

Sorry for my bad english.