A036297 -id:A036297 - cp's OEIS frontend

A003432 Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.

1, 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, 25515, 131072, 327680, 1114112, 3411968, 19531250, 56640625, 195312500

Offset: 0

N. J. A. Sloane

The entries are restricted to 0 and 1; the determinant is computed in the field of real numbers.
Suppose M = (m(i,j)) is an n X n matrix of real numbers. Let
a(n) = max det M subject to m(i,j) = 0 or 1 [this sequence],
g(n) = max det M subject to m(i,j) = -1 or 1 [A003433],
h(n) = max det M subject to m(i,j) = -1, 0 or 1 [A003433],
F(n) = max det M subject to 0 <= m(i,j) <= 1 [this sequence],
G(n) = max det M subject to -1 <= m(i,j) <= 1 [A003433].
Then a(n) = F(n), g(n) = h(n) = G(n), g(n) = 2^(n-1)*a(n-1). Thus all five problems are equivalent.
Hadamard proved that a(n) <= 2^(-n)*(n+1)^((n+1)/2), with equality if and only if a Hadamard matrix of order n+1 exists. Equivalently, g(n) <= n^(n/2), with equality if and only if a Hadamard matrix of order n exists. It is believed that a Hadamard matrix of order n exists if and only if n = 1, 2 or a multiple of 4 (see A036297).
We have a(21) = 195312500?, a(22) = 662671875?, and a(36) = 1200757082375992968. Furthermore, starting with a(23), many constructions are known that attain the upper bounds of Hadamard, Barba, and Ehlich-Wojtas, and are therefore maximal. See the Orrick-Solomon web site for further information. [Edited by William P. Orrick, Dec 20 2011]
The entry a(21) = 195312500 is now known to be correct. [Edited by Richard P. Brent, Aug 17 2021]

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 32*x^7 + 56*x^8 + ...
One of 2 ways to get determinant 9 with a 6 X 6 matrix, found by Williamson:
  1 0 0 1 1 0
  0 0 1 1 1 1
  1 1 1 0 0 1
  0 1 0 1 0 1
  0 1 0 0 1 1
  0 1 1 1 1 0

J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240-246.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 54.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jan Brandts and A Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
J. Brenner, The Hadamard maximum determinant problem, Amer. Math. Monthly, 79 (1972), 626-630.
R. P. Brent, W. P. Orrick, J. Osborn, and P. Zimmermann, Maximal determinants and saturated D-optimal designs of orders 19 and 37, arXiv:1112.4160 [math.CO], 2011. [From William P. Orrick, Dec 20 2011]
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv preprint arXiv:1208.3819 [math.CO], 2012.
Swee Hong Chan and Igor Pak, Computational complexity of counting coincidences, arXiv:2308.10214 [math.CO], 2023. See p. 18.
V. Chasiotis, S. Kounias, and N. Farmakis, The D-optimal saturated designs of order 22, Discrete Mathematics 341 (2018), 380-387. Corrigendum ibid 342 (2019), 2161.
H. Ehlich, Determinantenabschätzungen für binäre Matrizen, Math. Z. 83 (1964), 123-132.
H. Ehlich and K. Zeller, Binäre Matrizen, Zeit. Angew. Math. Mech., 42 (1962), 20-21.
J. Freixas and S. Kurz, On alpha-roughly weighted games, arXiv preprint arXiv:1112.2861 [math.CO], 2011.
Matthew Hudelson, Victor Klee and David Larman, Largest j-simplices in d-cubes: some relatives of the Hadamard maximum determinant problem, Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994). Linear Algebra Appl. 241/243 (1996), 519-598.
J. Huttenhain and C. Ikenmeyer, Binary Determinantal Complexity, arXiv:1410.8202 [cs.CC], 2014-2015.
Yongbin Li, Junwei Zi, Yan Liu and Xiaojun Zhang, A note of values of minors for Hadamard matrices, arXiv:1905.04662 [math.CO], 2019.
William P. Orrick, The maximal {-1, 1}-determinant of order 15, arXiv:math/0401179 [math.CO], 2004.
William P. Orrick and B. Solomon, Large determinant sign matrices of order 4k+1, arXiv:math/0311292 [math.CO], 2003.
William P. Orrick and B. Solomon, The Hadamard Maximal Determinant Problem (website)
William P. Orrick, B. Solomon, R. Dowdeswell and W. D. Smith, New lower bounds for the maximal determinant problem, arXiv:math/0304410 [math.CO], 2003.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Hadamard's maximum determinant problem.
Eric Weisstein's World of Mathematics, (0, 1)-Matrix
Eric Weisstein's World of Mathematics, (-1, 0, 1)-Matrix
J. Williamson, Determinants whose elements are 0 and 1, Amer. Math. Monthly 53 (1946), 427-434. Math. Rev. 8,128g.
Luke Zeng, Shawn Xin, Avadesian Xu, Thomas Pang, Tim Yang and Maolin Zheng, Seele's New Anti-ASIC Consensus Algorithm with Emphasis on Matrix Computation, arXiv:1905.04565 [cs.CR], 2019.
Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181-211.
Index entries for sequences related to binary matrices
Index entries for sequences related to Hadamard matrices
Index entries for sequences related to maximal determinants

A003433(n) = 2^(n-1)*a(n-1). Cf. A013588, A036297, A051752.

a(18)-a(20) added by William P. Orrick, Dec 20 2011

a(21) added by Richard P. Brent, Aug 16 2021

A007299 Number of Hadamard matrices of order 4n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

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.

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

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

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

A048615 Numerator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.

Original entry on oeis.org

1, 1, 1, 27, 127, 34603, 278617, 264156210586399

Offset: 1

N. J. A. Sloane, E. M. Rains

This list starts with the Hadamard matrix of order 4. The masses of the Hadamard matrices of orders 1 and 2 are 1/2 and 1/8 respectively.

			1/192, 1/21504, 1/190080, 27/573440, 127/218880, 34603/40480, 278617/2106, 264156210586399/53329920, ...

V. Alvarez, 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.
Index entries for sequences related to Hadamard matrices

Cf. A048616, A007299, A036297.

a(8) from Brendan McKay, Feb 11 2012

A048616 Denominator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.

Original entry on oeis.org

192, 21504, 190080, 573440, 218880, 40480, 2106, 53329920

Offset: 1

N. J. A. Sloane, E. M. Rains

			1/192, 1/21504, 1/190080, 27/573440, 127/218880, 34603/40480, 278617/2106, 264156210586399/53329920, ...

V. Alvarez, 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.
Index entries for sequences related to Hadamard matrices

Cf. A048615, A007299, A036297.

a(8) from Brendan McKay, Feb 11 2012

A019442 Numbers m such that a Hadamard matrix of order m exists.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240

Offset: 1

N. J. A. Sloane, Oct 16 2008

It is conjectured that this sequence consists of 1, 2 and all multiples of 4.
Already in 1992 Hadamard matrices were known of all orders 4t up through 424.
The old entry with this sequence number was a duplicate of A007740.
Integers m such that a simplex of dimension m - 1 can be inscribed in a hypercube of dimension m - 1. - Violeta Hernández Palacios, Oct 23 2020
Integers m such that an orthoplex of dimension m can be inscribed in a hypercube of dimension m. - Violeta Hernández Palacios, Dec 05 2020
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 (see comment in A007299). - Bernard Schott, Apr 25 2022; Mar 03 2023

J. Hadamard, Résolution d'une question relative aux déterminants. Bull. des Sciences Math. (2), 17, 1893, pp. 240-246.
M. Hall, Jr., Hadamard matrices of order 16. Research Summary No. 36-10, Jet Propulsion Lab., Pasadena, CA, Vol. 1, 1961, pp. 21-26.
M. Hall, Jr., Hadamard matrices of order 20. Technical Report 32-761, Jet Propulsion Lab., Pasadena, CA, 1965.
M. Hall, Jr., Combinatorial Theory. 2nd edn. New York: Wiley, 1986.
S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, Chapter 7.
Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences and block designs, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431-560, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
W. D. Wallis, Anne Penfold Street, and Jennifer Seberry Wallis; Combinatorics: Room squares, sum-free sets, Hadamard matrices. Lecture Notes in Mathematics, Vol. 292. Springer-Verlag, Berlin-New York, 1972. iv+508 pp.

J. Adams, P. Zvengrowski, and P. Laird, Vertex embeddings of regular polytopes, Expositiones Mathematicae, (4), 21, 2003, pp. 339-353.
D. Z. Djokovic, Construction of some new Hadamard matrices, Bull. Austral. Math. Soc., Volume 45, Issue 2, April 1992, pp. 327-332.
D. Z. Djokovic, Five new orders for Hadamard matrices of skew type. Australas. J. Combin., Vol. 10, 1994.
D. Z. Djokovic, Two Hadamard matrices of order 956 of Goethals-Seidel type, Combinatorica, 1994, 14, 375-377.
J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. (2), 17, 1893, 240-246. English translation.
Hiroshi Kimura, Hadamard matrices of order 28 with automorphism groups of order two, J. Combin. Theory, 1986, A 43, 98-102.
Hiroshi Kimura, New Hadamard matrix of order 24, Graphs Combin., 1989, 5, 235-242.
Hiroshi Kimura, Classification of Hadamard matrices of order 28 with Hall sets, Discrete Math., 1994, 128, 257-268.
Hiroshi 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.
R. E. A. C. Paley, On orthogonal matrices, J. Math. Phys., 12, 311-320.
R. L. Plackett and J. P. Burman, The design of optimum multifactorial experiments, Biometrika, 1946, 33, 305-325.
K. Sawade, A Hadamard matrix of order 268, Graphs Combin., 1985, 1, 185-187.
Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences and block designs, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431-560, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
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).
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.
J. Williamson, Hadamard's determinant theorem and the sum of four squares, Duke Math. J., 1994, 11, 65-81.
Index entries for sequences related to Hadamard matrices

Cf. A007299, A036297, A016742.

Conjectured g.f.: (2*x^3 + x^2 + 1)/(x - 1)^2. - Jean-François Alcover, Oct 03 2016

A206711 Total number of distinct Hadamard matrices of order 4n.

Original entry on oeis.org

768, 4954521600, 20251509535014912000, 88526812916367202104587059200000, 3776127947893930552689423154306445475840000000, 92624181047745713568610317051197596401168530978226831360000000, 886156947284057553944669848348035536068124589065755283423684984832000000000000

Offset: 1

Brendan McKay, Feb 11 2012 (entered by N. J. A. Sloane)

This is the total number of distinct Hadamard matrices of order 4n, ignoring all equivalences.

Brendan McKay, Table of n, a(n) for n = 1..8
J. Dong, V. Elser, G. Gyawali, K. Y. Jee, J. Kent-Dobias, A. Mandaiya, M. Renz, Y. Su, Glass phenomenology in the hard matrix model, arXiv:1912.07558 [cond-mat.stat-mech], 2019.
H. Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32.
Index entries for sequences related to Hadamard matrices

Cf. A007299, A036297, A206712.

a(n) = A048615(n)/A048616(n) * (2^n * n!)^2.

a(n) = A206712(4n).

A206712 Total number of distinct Hadamard matrices of order n.

Original entry on oeis.org

2, 8, 0, 768, 0, 0, 0, 4954521600, 0, 0, 0, 20251509535014912000, 0, 0, 0, 88526812916367202104587059200000, 0, 0, 0, 3776127947893930552689423154306445475840000000, 0, 0, 0, 92624181047745713568610317051197596401168530978226831360000000, 0, 0, 0, 886156947284057553944669848348035536068124589065755283423684984832000000000000, 0, 0, 0

Offset: 1

Brendan McKay, Feb 11 2012 (entered by N. J. A. Sloane)

a(n) is the total number of distinct Hadamard matrices of order n, ignoring all equivalences.

Brendan McKay, Table of n, a(n) for n = 1..35
F. J. Aragon Artacho, J. M. Borwein, M. K. Tam, Douglas-Rachford Feasibility Methods for Matrix Completion Problems, March 2014.
H. Kharaghani and B. Tayfeh-Rezaie, Hadamard matrices of order 32
Index entries for sequences related to Hadamard matrices

Cf. A007299, A036297, A206711.

a(4n) = A206711(n) = A048615(n)/A048616(n) * (2^n * n!)^2.

For n>1, a(4n+1) = a(4n+2) = a(4n+3) = 0.

cp's OEIS Frontend

A003432 Hadamard maximal determinant problem: largest determinant of a (real) {0,1}-matrix of order n.

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A007299 Number of Hadamard matrices of order 4n.

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A048615 Numerator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.

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A048616 Denominator of mass (Sum 1/|Aut(H)|) of Hadamard matrices of order 4n.

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A019442 Numbers m such that a Hadamard matrix of order m exists.

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A206711 Total number of distinct Hadamard matrices of order 4n.

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A206712 Total number of distinct Hadamard matrices of order n.

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A206704 Number of Hadamard matrices of order 4n that are equivalent to their transpose.

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A206705 Number of Hadamard matrices of order n that are equivalent to their transpose.

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