Explicit matrices with low coherence based on algebraic geometric codes

Abstract

A key element in the performance of a compressed sensing (CS) setup is the so called sensing matrix. It is known that the success of CS-based methods strongly rely on the properties of the employed sensing matrix. Random matrices are the widely-adopted choice due to their order-optimal performance and flexibility in size. In real world applications, however, random structures are rarely feasible. For this reason, the deterministic design of sensing matrices has been an ongoing research topic. In this paper, we introduce new classes of deterministic complex-valued sensing matrices based on algebraic curves. In particular, we design a number of algebraic-geometric codes with large minimum distances specifically for the construction of sensing matrices. Our approach is to find maximal curves in the Galois field Fpm‾, and transform them into Fp codes by a trace map. Invoking the Riemann-Roch theorem, we demonstrate that the resulting code has a large minimum distance compared to its length, which leads to a sensing matrix with small coherence value. For general m×n matrices, the Welch bound (≈1m) sets a universal lower-bound on the coherence value, and the bound is achievable only when n≤m2. In our designs, we are able to construct m×n matrices with n ranging from around 8m to values larger than O(m2) by tuning the parameters. Meanwhile, the coherence of the designed matrices differ from the Welch bound by only an O(logm) factor. Simulation results indicate that the performance of our matrices in recovering sparse vectors from compressed measurements is superior or equivalent to Gaussian random matrices.