Construction of Regular Hadamard Matrices From Circulant Hadamard Matrices and Ryser’s Conjecture
DOI:
https://doi.org/10.26713/cma.v16i3.3215Keywords:
Hadamard matrix, Partial Hadamard matrix, Circulant matrixAbstract
This article presents a simplified proof of Ryser’s conjecture by building upon and extending the modular arithmetic framework introduced by Luis Henri Gallardo (On Ryser’s conjecture: modulo 2 approach, Applied Mathematics E-Notes 21 (2021), 220 – 224,), specifically leveraging the modulo 2 approach. The methodology highlights an accessible and robust pathway to understanding the conjecture’s constraints, providing new insights into the structural properties of circulant and regular Hadamard matrices.
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References
R. A. Brualdi, A note on multipliers of difference sets, Journal of Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics 69B(1-2) (1965), 87 – 89.
P. J. Davis, Circulant Matrices, 2nd edition, AMS Chelsea Publishing: An Imprint of the American Mathematical Society, New York, 240 pages (1994).
P. Erdös, On the coefficients of the cyclotomic polynomial, Bulletin of the American Mathematical Society 52 (1946), 179 – 184, DOI: 10.1090/S0002-9904-1946-08538-9.
L. H. Gallardo, On Ryser’s conjecture: modulo 2 approach, Applied Mathematics E-Notes 21 (2021), 220 – 224, https://www.math.nthu.edu.tw/~amen/2021/AMEN-200412.pdf.
A. Hedayat and W. D. Wallis, Hadamard matrices and their applications, The Annals of Statistics 6(6) (1978), 1184 – 1238, DOI: 10.1214/aos/1176344370.
K. H. Leung, S. L. Ma and B. Schmidt, New Hadamard matrices of order 4p2 obtained from Jacobi sums of order 16, Journal of Combinatorial Theory, Series A 113(5) (2006), 822 – 835, DOI: 10.1016/j.jcta.2005.07.011.
P. K. Manjhi, On permutation groups and Fourier matrices, International Journal for Research in Engineering Application & Management 04(04) (2018), 197 – 200, URL: http://ijream.org/papers/IJREAMV04I0440039.pdf.
D. B. Mcisncr, On a construction of regular Hadamard matrices, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Serie 9 3(4) (1992), 233 – 240, URL: http://www.bdim.eu/item?fmt=pdf&id=RLIN_1992_9_3_4_233_0.
H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, Vol. 14, The Mathematical Association of America, xiv + 154 pages (1963).
B. Schmidt, Cyclotomic integers and finite geometry, Journal of the American Mathematical Society 12 (1999), 929 – 952, DOI: 10.1090/S0894-0347-99-00298-2.
R. J. Turyn, Character sums and difference sets, Pacific Journal of Mathematics 15(1) (1965), 319 – 346, DOI: 10.2140/pjm.1965.15.319.
R. Turyn, Sequences with small correlation, in: Error Correcting Codes: Symposium Proceedings, H. B. Mann (editor), John Wiley & Sons Inc., New York, pp. 195 – 228 (1968).
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