A low-complexity algorithm to search for Legendre pairs
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 e...
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| Vydáno v: | Linear algebra and its applications Ročník 721; s. 149 - 171 |
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15.09.2025
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Perera, Sirani M. Kotsireas, Ilias S. |
| Author_xml | – sequence: 1 givenname: Sirani M. orcidid: 0000-0002-3975-3742 surname: Perera fullname: Perera, Sirani M. email: pereras2@erau.edu organization: Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA – sequence: 2 givenname: Ilias S. surname: Kotsireas fullname: Kotsireas, Ilias S. email: ikotsire@wlu.ca organization: Wilfrid Laurier University, Waterloo, Ontario, Canada |
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| Cites_doi | 10.1002/jcd.21806 10.1109/ACCESS.2021.3120051 10.1109/TAU.1970.1162132 10.1109/TSP.2021.3099623 10.1073/pnas.73.4.1005 10.1109/MCISE.2000.814652 10.1016/j.tcs.2008.06.002 10.1109/ACCESS.2020.2970342 10.1007/s00211-007-0114-x 10.1090/S0025-5718-1965-0178586-1 10.1109/PROC.1968.6477 10.1137/17M1114557 10.1109/JPROC.2004.840301 10.1016/j.laa.2017.05.050 10.1007/s10623-013-9862-z 10.1016/0024-3795(94)90189-9 10.1109/TSP.2022.3231182 10.1007/978-3-030-04088-8 10.1109/TSP.2006.882087 10.1137/1020096 |
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| Keywords | 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 |
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| Snippet | Legendre pairs constitute an important combinatorial object that can be used to construct Hadamard matrices. In this paper, we search for Legendre pairs... |
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| SubjectTerms | Complexity and performance of algorithms Fast Fourier transforms Legendre pairs Matrix analysis Matrix equations Sparse and orthogonal factors Structured matrices |
| Title | A low-complexity algorithm to search for Legendre pairs |
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