On the Spectral Norms of Pseudo-Wigner and Related Matrices

We investigate the spectral norms of symmetric N x N matrices from two pseudo-random ensembles. The first is the pseudo-Wigner ensemble introduced in "Pseudo-Wigner Matrices" by Soloveychik, Xiang and Tarokh and the second is its sample covariance-type analog defined in this work. Both ensembles are defined through the concept of r-independence by controlling the amount of randomness in the underlying matrices, and can be constructed from dual BCH codes. We show that when the measure of randomness r grows as N-rho, where rho is an element of (0, 1] and epsilon > 0, the norm of the matrices is almost surely within o(log(1+epsilon)N/N-min[rho,N-2/3] ) distance from 1. Numerical simulations verifying the obtained results are provided.