Let $n\in \mathbb N$ be an integer, and consider the field $\mathbb K=GF(2^n)$.
For $c\in \mathbb N$, let $S_c$ be a set of $n$ elements from $\mathbb K$ such that:
- Every element $e$ from $S$ is balanced: its weight $|e|=n/2$ (there are as many $1$s as $0$s).
- Every pair of distinct elements $e,e'\in S, e\neq e'$ are at distance a multiple of $c$. That is: $$\forall (e,e')\in S^2, e\neq e', \exists k \in \mathbb N, |e\oplus e'|=k\cdot c$$
- If the set $S_c$ could contain 0, 1, or 2 elements, its construction is trivial.
- For some values of $c$, there are no solutions.
- Does this set structure has a name?
- Are there algorithms to construct $S_c$?
- For fixed $(n,c)$, how many sets $S_c$ exist?
- This question seems related to binary coding theory where the minimal distance is replaced by codewords evenly located in the space. Is there a way to express the problem into a code problem?