This question derives from a PPCG coding challenge I posed previously.
For a given positive integer $n$, consider all binary strings of length $2n-1$. For a given string $S$, let $L$ be an array of length $n$ which contains the count of the number of $1$s in each substring of length $n$ of $S$. For example, if $n=3$ and $S = 01010$ then $L=[1,2,1]$. We call $L$ the counting array of $S$.
We say that two strings $S1$ and $S2$ of the same length match if their respective counting arrays $L1$ and $L2$ have the property that $L1[i] \leq 2*L2[i]$ and $L2[i] \leq 2*L1[i]$ for all $i$.
Problem
For increasing $n$ starting at $n=1$, we want to compute the size of the largest set of strings, each of length $2n-1$ so that no two strings match.
Known answers
For $n=1,2,3,4,5$ the optimal answers are $2,4,10,16,31$.
For $n=6,7,8,9,10$, the best known values are $47, 76, 112, 168, 235$ from a combination of the answers of joriki and Peter Taylor.
The question
Plotting these values they appear to be subexponential but it is hard to say much more than that. This leads to the question:
How do optimal solutions scale with $n$?
Question now posed at https://mathoverflow.net/questions/215343/counting-the-size-of-the-largest-sets-of-independent-strings as well.