  We investigate the spectral norms of symmetric $N \times N$ 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 $\rho \in
(0,1]$ and $\varepsilon > 0$, the norm of the matrices is almost surely within
$o\left(\frac{\log^{1+\varepsilon} N}{N^{\min[\rho,2/3]}}\right)$ distance from
$1$. Numerical simulations verifying the obtained results are provided.
