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...




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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]



Size of biggest mutually 0-1 string with odd mutual 1
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 $q-1$.


