Covering Point-Sets with Parallel Hyperplanes and Sparse Signal Recovery

We give a new deterministic construction of integer sensing matrices that can be used for the recovery of integer-valued signals in compressed sensing. This is a family of n x d integer matrices, d >= n, with bounded sup-norm and the property that no l column vectors are linearly dependent, l <= n. Further, if l <= o(log n) then d/n -> infinity as n -> infinity. Our construction comes from particular sets of difference vectors of point- sets in R-n that cannot be covered by few parallel hyperplanes. We construct examples of such sets on the 0, +/- 1 grid and use them for the matrix construction. We also show a connection of our constructions to a simple version of the Tarski's plank problem.