On Compressed Sensing Matrices Breaking the Square-Root Bottleneck

Compressed sensing is a celebrated framework in signal processing and has many practical applications. One of the challenging problems in compressed sensing is to construct deterministic matrices having the restricted isometry property (RIP). So far, there are only a few publications providing deterministic RIP matrices beating the square-root bottleneck on the sparsity level. In this paper, we investigate RIP of certain matrices defined by higher power residues modulo primes. Moreover, we prove that the widely-believed generalized Paley graph conjecture implies that these matrices have RIP breaking the square-root bottleneck. Also the compression ratio realized by these RIP matrices is significantly larger than 2.