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.

翻译：我们提出了一种超快速方法，用于在有限域$GF(q)$中构造正交矩阵$M\in\mathcal{O}(n,q)$。该方法可用于在$Z_p$中构造$n\times n$的正交矩阵，且支持非常大的$n$和$p$值，同时也能构造具有特定循环结构的正交矩阵。同样地，还可以在有限域上构造么正滤波器组或小波矩阵。该构造机制效率极高，能够对满足特定约束条件的正交矩阵进行完全筛选和选择。例如，当$\mathcal{O}(n,q)$的阶数不太大时，可以生成给定$n$和$q=p^m$的全部正交矩阵列表。尽管该方法基于随机性，但失败孤立案例可在基本过程开始前被提前识别。所提出的过程基于Janashia-Lagvilava方法，该方法原本是为完全不同的问题而开发的，因此其在此处的应用可能显得出人意料。