  We propose a superfast method for constructing orthogonal matrices
$M\in\mathcal{O}(n,q)$ in finite fields $GF(q)$. It can be used to construct
$n\times n$ orthogonal matrices in $Z_p$ with very high values of $n$ and $p$,
and also orthogonal matrices with a certain circulant structure. Equally well
one can construct paraunitary filter banks or wavelet matrices over finite
fields. The construction mechanism is highly efficient, allowing for the
complete screening and selection of an orthogonal matrix that meets specific
constraints. For instance, one can generate a complete list of orthogonal
matrices with given $n$ and $q=p^m$ provided that the order of
$\mathcal{O}(n,q)$ is not too large. Although the method is based on
randomness, isolated cases of failure can be identified well in advance of the
basic procedure's start.
  The proposed procedures are based on the Janashia-Lagvilava method which was
developed for an entirely different task, therefore, it may seem somewhat
unexpected.
