Let M(n, A) denote the maximum possible cardinality of a family of binary strings of length
n, such that for every four distinct members of the family there is a coordinate in which
exactly two of them have a 1. We prove that M(n, A) [les ] 20.78n
for all sufficiently large n. Let
M(n, C) denote the maximum possible cardinality of a family of binary strings of length
n, such that for every four distinct members of the family there is a coordinate in which
exactly one of them has a 1. We show that there is an absolute constant c < 1/2 such that
M(n, C) [les ] 2cn for all sufficiently large n.
Some related questions are discussed as well.