On the spectral norms of pseudo-wigner and related matrices

We investigate the spectral norms of symmetric N × 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
^p
, where p ϵ (0,1] and ε > 0, the norm of the matrices is almost surely within o(log
^{1 + ε}
N/N
^min
[ρ, 2/3]) distance from 1. Numerical simulations verifying the obtained results are provided.