A low-complexity algorithm to search for Legendre pairs

• Published in: Linear algebra and its applications Vol. 721; pp. 149 - 171
• Main Authors: Perera, Sirani M., Kotsireas, Ilias S.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.09.2025
• Subjects: Complexity and performance of algorithms; Fast Fourier transforms; Legendre pairs; Matrix analysis; Matrix equations; Sparse and orthogonal factors; Structured matrices; Matrix analysis; 65T50; 15B05; 65Y20; 65Y04; Legendre pairs; 42C10; 65F40; Sparse and orthogonal factors; Fast Fourier transforms; Complexity and performance of algorithms; Structured matrices; 15A23; 15B10; 65F15; Matrix equations; 65F45; 65F35
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2025.01.010

Legendre pairs constitute an important combinatorial object that can be used to construct Hadamard matrices. In this paper, we search for Legendre pairs exploring matrix structures and obtain a low arithmetic and time complexity algorithm instead of conventional combinatorial algorithms. First, we explore the structure of the Legendre pair matrix equation and study its properties. Next, we study the boundaries of the spectra of the matrices appearing in the Legendre pair matrix equation, using Gershgorin circles. After stating an invariant that characterizes Legendre pairs and their relation to the discrete Fourier transform (DFT) matrix, we propose a low-complexity algorithm, i.e., a fast Fourier transform (FFT)-like algorithm, to compute the product of the DFT matrix with each sequence of the Legendre pair having any odd length. By utilizing the FFT-like algorithm, we present a low-time complexity algorithm associated with searching for Legendre pairs. Finally, we show numerical results based on the C implementation of the FFT-like algorithm yielding a low time complexity in searching for Legendre pairs as opposed to conventional combinatorial algorithms. This leads us to demonstrate that the proposed FFT-like algorithm significantly accelerates the search for Legendre pairs of orders 45 and 63, achieving at least 99% improvement in speed compared to the conventional algorithms.