Explicit deterministic constructions for membership in the bitprobe model

We look at time-space tradeoffs for the static membership problem in the bit-probe model. The problem is to represent a set of size up to n from a universe of size m using a small number of bits so that given an element of the universe, its membership in the set can be determined with as few bit probes to the representation as possible. We show several deterministic upper bounds for the case when the number of bit probes, is small, by explicit constructions, culminating in one that uses o(m) bits of space where membership can be determined with [lg lgn] + 2 adaptive bit probes. We also show two tight lower bounds on space for a restricted two probe adaptive scheme