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Given $k,n\in\mathbb{N}$.

Let M:={0,..$k,m\in\mathbb{N}$ be given. Let $M:=\{0,... , m-1}m-1\}$. How to find a subset $T\subset M$, |T|=k with T={ $n_1$,...,$n_k$ | |T|=k$such that$n_i\in M$} and T+T|={ (a+b)%m | |T+T|$ is maximal, where $T+T=\{ (a+b)\mathbin\%m \mid a\in N,b\in N$ } T,b\in T \}$("%" “%” means modulo) such that |T+T| is max. modulo)? I tried to construct a sequence of numbers which maximize |T+T|.$|T+T|$. But I couln't couldn’t figure out: - is • Is it possible to cover the whole set M$M$for$k\leq \sqrt{n}$- sqrt{m}$?
• What is the best way to construct such a sequence in Theory.theory?

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

My Idea 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} $T=\{a_i\mid i=0,...,k-1\}$ to get as less small number of collisions as possible collision in among the sums of |T+T|. in $T+T$. But random subsets of M $M$ show me , that there are better subsets.

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

Sorry for my bad englishEnglish.

1

# Optimize / simple Set Covering Problem

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:
- is it possible to cover the whole set M for $k\leq \sqrt{n}$
- 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.