A007299 Number of Hadamard matrices of order 4n.

1, 1, 1, 1, 5, 3, 60, 487, 13710027

Offset: 0

Author

N. J. A. Sloane

Keywords

• hard
• nonn
• nice

Comments

More precisely, number of inequivalent Hadamard matrices of order n if two matrices are considered equivalent if one can be obtained from the other by permuting rows, permuting columns and multiplying rows or columns by -1.

The Hadamard conjecture is that a(n) > 0 for all n >= 0. - Charles R Greathouse IV, Oct 08 2012

From Bernard Schott, Apr 24 2022: (Start)

A brief historical overview based on the article "La conjecture de Hadamard" (see link):

1893 - J. Hadamard proposes his conjecture: a Hadamard matrix of order 4k exists for every positive integer k (see link).

As of 2000, there were five multiples of 4 less than or equal to 1000 for which no Hadamard matrix of that order was known: 428, 668, 716, 764 and 892.

2005 - Hadi Kharaghani and Behruz Tayfeh-Rezaie publish their construction of a Hadamard matrix of order 428 (see link).

2007 - D. Z. Djoković publishes "Hadamard matrices of order 764 exist" and constructs 2 such matrices (see link).

As of today, there remain 12 multiples of 4 less than or equal to 2000 for which no Hadamard matrix of that order is known: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964. (End)

By private email, Felix A. Pahl informs that a Hadamard matrix of order 1004 was constructed in 2013 (see link Djoković, Golubitsky, Kotsireas); so 1004 is deleted from the last comment. - Bernard Schott, Jan 29 2023

References

• J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240-246.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• S. Wolfram, A New Kind of Science. Champaign, IL: Wolfram Media, p. 1073, 2002.

Links

• V. Álvarez, J. A. Armario, M. D. Frau, and F. Gudiel, The maximal determinant of cocyclic (-1, 1)-matrices over D_{2t}, Linear Algebra and its Applications, 2011.
• F. J. Aragon Artacho, J. M. Borwein, and M. K. Tam, Douglas-Rachford Feasibility Methods for Matrix Completion Problems, March 2014.
• Dragomir Z. Djoković, Hadamard matrices of order 764 exist, arXiv:math/0703312 [math.CO], 2007.
• Dragomir Z. Djoković, Oleg Golubitsky, and Ilias S. Kotsireas, Some new orders of Hadamard and skew-Hadamard matrices, arXiv:1301.3671 [math.CO], 2013.
• Shalom Eliahou, La conjecture de Hadamard (I), Images des Mathématiques, CNRS, 2020.
• Shalom Eliahou, La conjecture de Hadamard (II), Images des Mathématiques, CNRS, 2020.
• Jacques Salomon Hadamard, Sur le module maximum que puisse atteindre un déterminant, C. R. Acad. Sci. Paris 116 (1893) 1500-1501 (link from Gallica).
• Hadi Kharaghani, Home Page
• Hadi Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32, J. Combin. Designs 21 (2013) no. 5, 212-221. [DOI]
• Hadi Kharaghani and B. Tayfeh-Rezaie, A Hadamard matrix of order 428, Journal of Combinatorial Designs, Volume 13, Issue 6, November 2005, pp. 435-440 (First published: 13 December 2004).
• H. Kharaghani and B. Tayfeh-Rezaie, On the classification of Hadamard matrices of order 32, J. Combin. Des., 18 (2010), 328-336.
• H. Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32, 2012.
• H. Kimura, Hadamard matrices of order 28 with automorphism groups of order two, J. Combin. Theory (1986), A 43, 98-102.
• H. Kimura, New Hadamard matrix of order 24, Graphs Combin. (1989), 5, 235-242.
• H. Kimura, Classification of Hadamard matrices of order 28 with Hall sets, Discrete Math. (1994), 128, 257-268.
• H. Kimura, Classification of Hadamard matrices of order 28, Discrete Math. (1994), 133, 171-180.
• W. P. Orrick, Switching operations for Hadamard matrices, arXiv:math/0507515 [math.CO], 2005-2007. (Gives lower bounds for a(8) and a(9))
• N. J. A. Sloane, Tables of Hadamard matrices
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• Warren D. Smith, C program for generating Hadamard matrices of various orders
• Edward Spence, Classification of Hadamard matrices of order 24 and 28, Discrete Math. 140 (1995), no. 1-3, 185-243.
• Eric Weisstein's World of Mathematics, Hadamard Matrix
• Index entries for sequences related to Hadamard matrices

Crossrefs

Cf. A019442, A096201, A036297, A048615, A048616, A003432, A048885.

Extensions

a(8) from the H. Kharaghani and B. Tayfeh-Rezaie paper. - N. J. A. Sloane, Feb 11 2012