I'm interested in counting the number of partitions of the space of binary strings of length $k$, subject to a condition. The condition is: strings with Hamming distance equal to $k$ can't be in the same partition element. For example, if $k = 2$, 01 and 10 can't be in the same partition element.
I've been looking at the literature and I haven't been able to find a resource for this. I've also been unable to generalize this from the simple $k=2$ and $k=3$ cases that I've been able to work out by enumeration.
Any pointers, suggestions or references are much appreciated. Thanks.