Coding for a multiple access OR channel: A survey

The signature coding for M active users out of T total users over a multiple access OR channel is considered. The mathematical problem is equivalent to the M  -cover-free problem of extremal set theory. We survey the upper and lower bounds on the minimal code word length n(T,M)n(T,M), and present some code constructions. According to the current state of the theory, for 1⪡M⪡T1⪡M⪡T12M2logMlogT⩽n(T,M)⩽1ln2M2logT,so there is a huge gap between the upper and lower bounds. Moreover, there is no known construction approaching the upper bound.