String Quartets in Binary

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) ≤ 2^0.78n 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) ≤ 2^cn for all sufficiently large n. Some related questions are discussed as well.