  Matrices with the restricted isometry property (RIP) are of particular
interest in compressed sensing. To date, the best known RIP matrices are
constructed using random processes, while explicit constructions are notorious
for performing at the "square-root bottleneck," i.e., they only accept sparsity
levels on the order of the square root of the number of measurements. The only
known explicit matrix which surpasses this bottleneck was constructed by
Bourgain, Dilworth, Ford, Konyagin and Kutzarova. This chapter provides three
contributions to further the groundbreaking work of Bourgain et al.: (i) we
develop an intuition for their matrix construction and underlying proof
techniques; (ii) we prove a generalized version of their main result; and (iii)
we apply this more general result to maximize the extent to which their matrix
construction surpasses the square-root bottleneck.
