Size of biggest mutually 0-1 string with odd mutual 1 Let $q$ be an odd number...
consider $0-1$ strings of length $2q$ with $q$ ones. [with total number of $C(2q,q)$] 
I want to find an upper bound for a set of these strings such that the number of mutual ones in any two of them be an odd number...

i already wrote a code in MATLAB and it seems this number is not very big... (about $2q$)
can someone help me to find any efficient (OR NOT!) upper bound?
[if it helps you can think of even intersection strings as well]
 A: The answer is $2^{q+o(q)}$ (well, there are rooms for improvement). 
Example: consider all strings with $1$ at the last position and $(q-1)/2$ pairs of one's chosen from possible pairs of positions $(1,2)$, $(3,4),\dots$, $(2q-3,2q-2)$. It is a family consisting of $\binom{q-1}{(q-1)/2}$ strings.
For the estimate, choose index $j$ so that at least half of your strings has 1 at position $j$. Consider only them and remove this position from them. We have $2q-1$ positions, and intersection of any two strings is even. There may be at most $2^{q-1}$ such strings, this is already well known. Indeed, consider the space generated by these strings over $\mathbb{F}_2$. This space is contained in its own orthogonal complement, hence its dimension does not exceed $q-1$.
A: If q is odd and you want even intersection, then this is the Frankl-Wilson Theorem (and indeed the upper bound is 2q).  For odd intersection it seems it can be exponential.  See https://www.dpmms.cam.ac.uk/~dce27/chapter1version2.pdf Theorem 4 and remark following.
