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Random Generator of Orthogonal Matrices in Finite Fields
Information Theory · Submitted 14 May 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We propose a superfast method for constructing orthogonal matrices $M\in\mathcal{O}(n,q)$ in finite fields $GF(q)$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose a superfast method for constructing orthogonal matrices $M\in\mathcal{O}(n,q)$ in finite fields $GF(q)$. | contribution 1.00 | 0.90 | 0.68 | asserted 0.99 |
| s1 | 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. | contribution 0.75 | 0.53 | 0.12 | hedged possibility 0.65 |
| s2 | Equally well one can construct paraunitary filter banks or wavelet matrices over finite fields. | contribution 0.55 | 0.37 | 0.07 | asserted 0.60 |
| s3 | The construction mechanism is highly efficient, allowing for the complete screening and selection of an orthogonal matrix that meets specific constraints. | method 0.49 | 0.73 | 0.17 | asserted 1.00 |
| s4 | 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. | limitation 0.35 | 0.53 | 0.09 | hedged possibility 0.77 |
| s5 | Although the method is based on randomness, isolated cases of failure can be identified well in advance of the basic procedure's start. | limitation 0.63 | 0.72 | 0.05 | asserted 0.84 |
| s6 | The proposed procedures are based on the Janashia-Lagvilava method which was developed for an entirely different task, therefore, it may seem somewhat unexpected. | background 0.52 | 0.19 | 0.04 | hedged possibility 0.77 |
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The paper
Original abstract
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.
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