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 ɛ : GF(2ⁿ) → {−1, 1}n be the component-wise mapping ɛ(vi) := (−1)^vi , for v =(v1, v2, … , vn) ∈ GF(2ⁿ). Finally, for p ≪ n, let φC 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 [3] for d^⊥ large enough and y := p/n ∈ (0, 1) fixed, as n → ∞ the empirical eigen-distribution of the Gram matrix of 1 over √n times φC 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.