# Extended Hypercube Graph

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.

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A coclique in your $(n,k)$-hypercube is a set of binary vectors with the property that any two distinct vectors are distance at least $k+1$ (in the Hamming metric), and of course these are exactly the binary codes with minimum distance $k+1$. In his Ph.D. thesis Delsarte showed how to use linear programming, in combination with the theory of association schemes, to derive useful bounds on the size of codes.