Consider the set of binary strings of length $n$, that is $A_n = \{0,1\}^n$. I am trying to partition $A_n$ into $n$ subsets $B_n^1,\ldots,B_n^n$ such for all $x,x’\in B_n^i$ we have $d_H(x,x’) \neq \frac{n}{2}$, where $d_H$ denotes the Hamming distance.
This is not possible for all $n\in\mathbb{N}$, as it contradicts https://arxiv.org/abs/quant-ph/0607174, however, it is in fact possible for $n=2$, $n=4$ and $n=8$ (I looped through all partitions on a computer). I am particularly interested in the situation, where $n$ is a power of $2$, and so my question boils down to: Is this possible for $n=16$?
At this stage I have not been able to loop through all partitions, so I was wondering if anyone has any thoughts on identifying the least such $n = 2^k$, where it is impossible to partition $A_n$ subject to the Hamming distance constraint?
Any help is appreciated!