The pleasure to find new matrix: Hadamard 92, 1961

© Nickolay A. Balonin, 16.10.2014

From left to right, Solomon Golomb (Assistant Chief of the Communications Systems Research Section), Leonard Baumert (a postdoc student at Caltech), and Marshall Hall, Jr. (Caltech mathematics professor) hold a framed representation of the matrix.

The first things first

John Williamson, Note on Hadamard's determinant theorem, Bull. Amer. Math. Soc. 53, 1947, pp. 608-613.

Belevitch V. Theory of 2n-terminal networks with application to conference telephony // Electr. Commun. 1950. Vol. 26. P. 231–244.

Marshall Hall, A survey of difference sets, Proc. Amer. Math. Soc. 7, 1956, pp. 975-986.

R. G. Stanton and D.A. Sprott, A family of difference sets, Canad. J. Math. 10, 1958, pp. 73-77.

Damaraju Raghavarao, Some optimum weighing design. University of Bombey. Ann. Math. Statist., v. 30, 1959, pp. 295-303.

Leonard Baumert, S. W. Golomb and Marshall Hall, Discovery of an Hadamard Matrix of order 92. JR. Communicated by F. Bohnenblust, California Institute of Technology. Bull. Amer. Math. Soc. 68, 1962, pp. 237-238.

C. H. Yang, A Construction for Maximal (+1, -1)-Matrix of Order 54, Math. Comp., v. 20, 1965, P. 293.

C. H. Yang, Some Designs for Maximal (+1, -1)-Determinant of Order n-2 (mod 4), Math. Comp., v. 20, 1966, pp. 147-148.

Philippe Delsarte, Jean-Marie Goethals, Tri-weight Codes and Generalized Hadamard Matrices. Information and Control 15(2), 1969, pp. 196-206.

J. M. Goethals and J. J. Seidel, Orthogonal matrices with zero diagonal. To H. S. M. Coxeter on the occasion of his sixtieth birthday. 1967, pp. 1001-1010.

J. M. Goethals and J. J. Seidel, A skew-Hadamard matrix of order 36, 1969, pp. 343-344.

F. C. Bussemaker and J. J. Seidel, Symmetric Hadamard matrices of order 36. 1970, technical report, 68 p.

Jennifer Seberry Wallis, A Class of Hadamard Matrices. Communicated by Marshall Hall. Journal of combinatorial theory 6, 1969, pp. 40-44.

Jennifer Seberry Wallis, A skew-Hadamard matrix of order 92, Bulletin of the Australian Mathematical Society, 5, 1971, pp. 203-204.

P. Delsarte, J. M. Goethals and J. J. Seidel. Orthogonal matrices with zero diagonal. II. Can. J. Math., Vol. XXIII, No. 5, 1971, pp. 816-832.

Jennifer Seberry Wallis, On supplementary difference sets, Aequationes mathematicae, Vol. 8, 1972, pp. 242-257.

R. J. Turyn, Hadamard Matrices, Baumert-Hall Units, Four-Symbol Sequences, Pulse Compression, and Surface Wave Encodings. Journal of combinatorial theory, Series A 16, 1974, pp. 313-333.

H. Kharaghani, Note. A Construction of D-Optimal Designs for N=2 mod 4. Journal of combinatorial theory, Series A 46, 1987, pp. 156-158.

Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences, and block designs, Contemporary Design Theory: A Collection of Surveys, J. H. Dinitz and D. R. Stinson, eds., John Wiley and Sons, Inc., 1992. pp. 431–560

Warwick de Launey, A Product for Twelve Hadamard Matrices. Australasian Journal of Combinatorics 7, 1993, pp. 123-127.

Dieter Jungnickel, Bernhard Schmidt, Difference Sets: a Second Update. Augsburg Inst. für Mathematik, Preprint, 1998, pp. 1-26.

P. Borwein and M.J. Mossinghoff, Barker sequences and flat polynomials. Research of P. Borwein supported in part by NSERC of Canada and MITACS. Date: January 21, 2007.

Warwick De Launey, Dane Laurence Flannery. Algebraic Design Theory. Mathematical Surveis ans Monorgaph, v. 175. AMS (Americam Mathematical Society) 2011. 300 p. (AMS-books)

N. A. Balonin and Jennifer Seberry, A Review and New Symmetric Conference Matrices // Informatsionno-upravliaiushchie sistemy, 2014, № 4 (71), pp. 2–7.

Балонин Н. А., Джокович Д. Ж. Симметрия двуциклических матриц Адамара и периодические пары Голея // Информационно-управляющие системы. 2015. № 3. С. 2–16.

Balonin N. A., Djocovich D. Z. Negaperiodic Golay pairs and Hadamard matrices // Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2015, no. 5, pp. 2–17. doi:10.15217/issn1684-8853.2015.5.2

Jennifer Seberry. Orthogonal Designs: Hadamard Matrices, Quadratic Forms and Algebras. 2017. 452 p. Springer (BOOK) (array Balonin-Seberry P. 152)


Moon Ho Lee, Ferenz Szo"llo"sy. Toeplitz Jacket matrices are circulant. Electronic Letters, 2012.

Moon Ho Lee, Md. Hashem Ali Khan, M. A. Latif Sarker, Ying Guo, Kyeong Jin Kim A MIMO LTE Precoding Based on Fast Diagonal Weighted Jacket Matrices pp. 1-10.


N. A. Balonin and Jennifer Seberry, Remarks on extremal and maximum determinant matrices with moduli of real entries ≤ 1 // Informatsionno-upravliaiushchie sistemy, 2014, № 5 (71), pp. 2–4.

Балонин Н. А., Мироновский Л. А. Матрицы Адамара нечетного порядка // Информационно-управляющие системы. 2006, № 3. C. 46–50.

Балонин Н. А. О существовании матриц Мерсенна 11-го и 19-го порядков // Информационно-управляющие системы. 2013. № 2. С. 90–91.

Балонин Н. А., Сергеев М. Б. К вопросу существования матриц Адамара и Мерсенна // Информационно-управляющие системы. 2013. № 5. С. 2–8.

Балонин Н. А., Сергеев М. Б. Матрица золотого сечения G10 // Информационно-управляющие системы. 2013. № 6. С. 2–5.

Балонин Н. А., Сергеев М. Б. Матрицы локального максимума детерминанта // Информационно-управляющие системы. 2014. № 1. С. 2–15.

K.T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026

K.T. Arasu, Manil T. Mohan. Entropy of orthogonal matrices and minimum distance orthostochastic matrices from the uniform van der Waerden matrices // Discrete Optimization Received 13 September 2017, Accepted 4 October 2018, Available online 29 October 2018. doi:


Balonin N. A., Seberry, Jennifer. Visualizing Hadamard Matrices: the Propus Construction. Electronic edition, 2014.

N. A. Balonin, Jennifer Seberry Conference Matrices with Two Borders and Four Circulants. Electronic edition, 2014.

Ж. Адамар Исследование психологии процесса изобретения в области математики. МЦНМО, 2001.


There is a skew-Hadamard matrix of order 92 (Jennifer Seberry, 1971). Previously the smallest order for which a skew-Hadamard matrix was not known was 92. The existence of any Hadamard matrix of order 92 was unknown until 1962 (see photo below).


Райзер Г. Дж. Комбинаторная математика. / Пер. с англ. Пер. Рыбникова К.А. –М.: Мир, 1965. 154 с. [PDF]

Холл М. Комбинаторика. / Пер. с англ. под ред. Гельфонда А.О., Тараканова В.Е. –М.: Мир, 1970. 448 с. [djvu]

Холл М. Комбинаторика. / Пер. с англ. под ред. Гельфонда А.О., Тараканова В.Е. –М.: Мир, 1970. 448 с. [pdf]


Wikipedia, Vitold Belevitch.


Mersenne matrices M35 and M63 constructed by SBIBDs from M. Hall's paper (1955)

Williamson, John Hadamard's determinant theorem and the sum of four squares.
Duke Math. J. 11, 1944. pp. 65–81.

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