Efficient construction of a small hitting set for combinatorial rectangles in high dimension

We describe a deterministic algorithm which, on input integersd, m and real number ∈∃ (0,1), produces a subset S of [m]d={1,2,3,...,m}d that hits every combinatorial rectangle in [m]d of volume at least ∈, i.e., every subset of [m]d the formR1×R2×...×Rd of size at least ∈md. The cardinality of S is polynomial inm(logd)/∈, and the time to construct it is polynomial inmd/∈. The construction of such sets has applications in derandomization methods based on small sample spaces for general multivalued random variables.