Suppose $A$ is a finite alphabet, $|A| = n$. Suppose $m \in \mathbb{N}$. Let’s define the Hamming metric on $A^m$ as $d_m(a_1…a_m, b_1…b_m) = |\{i \leq m|a_i \neq b_i\}|$.
Now, let’s define $s(n, m, k)$ as the maximal possible cardinality of a subset $L \subset A^m$ such that $\forall u, v \in L$ either $u = v$ or $d_m(u, v) = k$.
Is there some sort of general formula for $s(n, m, k)$?
It is not hard to see, that $s(n, m , k) = 1$ for $k > m$. Also, $s(n, m, 1) = n^m$. However I do not know how to deal with $1 < k \leq m$.
EDIT:
There is also a probably relevant combinatorial theorem called Singleton inequality which states:
Suppose $L \subset A^m$ such that $\forall u, v \in L$ either $u = v$ or $d_m(u, v) \geq k$. Then $|L| \leq |A|^{m - k + 1}$
However, I currently do not know, will it be useful to solve this problem or not...