By C. C. Lin

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P. Street, and J. S. Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Mathematics, 292, Springer, New York (1972). 3. Y. X. Yang, Theory and Applications of Higher-Dimensional Hadamard Matrices, Kluwer Academic and Science Press, Beijing/New York (2001). 4. R. Damaschini, “Binary encoding image based on original Hadamard matrices,” Opt. Commun. 90, 218–220 (1992). 5. D. C. Tilotta, R. M. Hammaker, and W. G. Fateley, “A visible near-infrared Hadamard transform in spectrometer based on a liquid crystal spatial light modulator array: a new approach in spectrometry,” Appl.

Generally, the templates into which suitable matrices are plugged are orthogonal designs. 31 • Paley’s methods: Paley’s “direct” construction presented in 193336 gives Hadamard matrices of the order (pi + 1)(q j + 1), where pi ≡ 3 (mod 4), q j ≡ 1 i, j (mod 4) are prime powers. Paley’s theorem states that Hadamard matrices can be constructed for all positive orders divisible by 4 except those in the following sequence: multiples of 4 not equal to a power of 2 multiplied by q + 1, for some power q of an odd prime.

K. Rao, K. Reveluni, M. Narasimhan, and N. Ahmed, “Complex Haar transform,” IEEE Trans. Acoust. Speech Signal Process. 2 (1), 102–104 (1976). 45. H. G. Sarukhanyan, “Hadamard matrices: construction methods and applications,” Proc. 1st Int. Workshop on Transforms and Filter Banks, TICSP Ser. 1, Tampere University, Finland, pp. 95–130 (1998). 46. N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal Processing, Springer-Verlag, New York (1975). 47. S. S. Agaian and H. G. Sarukhanyan, “Recurrent formulae of the construction Williamson type matrices,” Math.