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 =( v  1 , v  2 ,..., vn ) ∈ GF(2) n . Finally, for  p   n , let \mmbΦ 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. 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 )]\mmbΦ C  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 )]\mmbΦ C  to the Marchenko-Pastur distribution as a function of  y  and  d  ⊥  is presented.
['Behtash Babadi', 'Vahid Tarokh']
Spectral Distribution of Random Matrices From Binary Linear Block Codes