Random frames from binary linear block codes

Let C be an [n, k, d] binary linear block code of length n, dimension k and minimum Hamming distance d over GF(2ⁿ). Let d^¿ denote the minimum Hamming distance of the dual code of C over GF(2ⁿ). Let ¿ : GF(2ⁿ) ¿ {-1, 1}n be the component-wise mapping ¿(v_i) := (-1)^vi , for v =(v₁, v₂, ... , v_n) ¿ GF(2ⁿ). Finally, for p < n, let ¿_C be a p × n random matrix whose rows are obtained by mapping a uniformly drawn set of size p of the codewords of C under e. Recently, the authors have established that for d^¿ large enough and y := p/n ¿ (0, 1) fixed, as n ¿ ¿ the empirical eigen-distribution of the Gram matrix of 1/¿n times ¿_C resembles that of a random i.i.d. Rademacher matrix (i.e., the Marchenko-Pastur distribution). In this paper, we overview this result and discuss its implications on the design of frames for compressed sensing applications.