Definition 1. The $n$-hypercube graph has vertices which are the elements of the set $\lbrace 0,1\rbrace^n$ of $n$-bit binary strings, and an edge is drawn between each pair of vertices representing a pair of strings differing on precisely $1$ bit.
Definition 2. The $(n,k)$-hypercube graph has vertices which are the elements of $\\lbrace 0,1\rbrace^n$, and an edge is drawn between each pair of strings differing on between $1$ and $k$ bits, inclusive.
The former is a classic of course, and the latter is a product of my imagination for the sake of some research I am doing in coding theory. For $k=n$, it would be the complete graph of order $2^n$.
Question. Has the latter structure been studied, for any $k\ge 2$? If so, I would be delighted to receive direction to known results, such as papers, theorems, keywords.