Spectral Distribution of Random Matrices From Binary Linear Block Codes

Abstract

Let C be a binary linear block code of length n, dimension k and minimum Hamming distance d over GF(2)n. Let d⊥ denote the minimum Hamming distance of the dual code of C over GF(2)n. Let e:GF(2)n→{-1,1}n be the component-wise mapping e(vi):=(-1)vi, for v=(v1,v2,...,vn) ∈ GF(2)n. Finally, for p <; n, let \mmbΦC 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. It is shown that for d⊥ large enough and y:=p/n ∈ (0,1) fixed, as n→∞ the empirical spectral distribution of the Gram matrix of [1/(√n)]\mmbΦC resembles that of a random i.i.d. Rademacher matrix (i.e., the Marchenko-Pastur distribution). Moreover, an explicit asymptotic uniform bound on the distance of the empirical spectral distribution of the Gram matrix of [1/(√n)]\mmbΦC to the Marchenko-Pastur distribution as a function of y and d⊥ is presented.