The Hadamard maximal determinant problem

The Hadamard maximal determinant problem asks when a matrix of a given order with entries -1 and +1 has the largest possible determinant. Despite well over a century of work by mathematicians, beginning with Sylvester's investigations of 1867, the question remains unanswered in general.

The table lists current record determinants. Clicking on a determinant will display a maximal matrix or matrices and other relevant information including references to the literature.

Table of maximal determinants, orders 40 - 79

Det should be multiplied by 2^N-1. Refer to key for more information.

N	Det	R	N	Det	R	N	Det	R	N	Det	R
40	20 × 10¹⁹	1	41	90 × 10¹⁹	1	42	410 × 10¹⁹	1	43	1890 × 10¹⁹ ??	.97
44	22 × 11²¹	1	45	89 × 11²¹ ??	.86	46	495 × 11²¹	1	47	3037500 × 11¹⁸ ??	.92
48	24 × 12²³	1	49	96 × 12²³ ??	.81	50	588 × 12²³	1	51	1.724074864E28 ??	.85
52	26 × 13²⁵	1	53	105 × 13²⁵ ??	.79	54	689 × 13²⁵	1	55	2.257229820E31 ??	.86
56	28 × 14²⁷	1	57	133 × 14²⁷ ??	.89	58	6.577371626E33 ??	.93	59	4312 × 14²⁷ ??	.97
60	30 × 15²⁹	1	61	165 × 15²⁹	1	62	915 × 15²⁹	1	63	5115 × 15²⁹ ??	.97
64	32 × 16³¹	1	65	148 × 16³¹ ??	.81	66	1040 × 16³¹	1	67	1.140030090E41 ??	.86
68	34 × 17³³	1	69	155 × 17³³ ??	.78	70	1156 × 17³³ ??	.99	71	31186944 × 17³⁰ ??	.88
72	36 × 18³⁵	1	73	174 × 18³⁵ ??	.80	74	1314 × 18³⁵	1	75	6.151181225E47 ??	.86
76	38 × 19³⁷	1	77	183 × 19³⁷ ??	.78	78	2.904232591E50 ??	.96	79	1.679602142E51 ??	.86

<0-39> <40-79> <80-119> <Full table>

The aim of this page is to inspire people to try to improve the above numbers (where possible). If you are aware of better bounds or other constructions, please notify the authors of this page (email: maxdet at indiana dot edu). Likewise, if you feel your work, or somebody else's, is not properly credited, we want to hear from you!

A related (and even more difficult) problem is the determinant spectrum problem which asks, not just for the maximal determinant, but for the complete set of values taken by the determinant function.

For a list of recent changes to these pages, see the update log.

News:

6 October 2012 -- There is now a blog associated with this site.
2 October 2012 -- On 20 January 2011, Hiroki Tamura reported new records to this site in sizes 35, 39, 43, and 63. These records have been added to the table.
30 August 2012 -- Dragomir Djokovic and Ilias Kotsireas have found D-optimal designs of sizes 206, 242, 262, and 482 [DK].
20 December 2011 -- The record determinants in sizes 19 and 37 have been proved to be maximal by Richard Brent, Will Orrick, Judy-anne Osborn, and Paul Zimmermann [BOOZ]. In addition, a new record has been established in size 45.
11 April 2005 -- The programming contest organized by Lars Backstrom that relates to this problem ended on 3 April 2005. Many new records were set by a number of participants. A list of records found by Tomas Rokicki can be found at his web site.
21 June 2004 -- Hadi Kharaghani and Behruz Tayfeh-Rezaie have announced the discovery of a Hadamard matrix of order 428. This had been the lowest outstanding order since a matrix of order 268 was found by Sawade in 1985 [Sa].)

Related links:

Roland Dowdeswell, Michael Neubauer, Bruce Solomon and Kagan Tumer set up an earlier web site on a similar theme. The site is connected to a continuously running search program. The current best lower bounds for orders 22, 23, 29, 31, and 33 were discovered either by this program, or by Bruce Solomon using an improved version of the algorithm. (The site appears to be offline.)
Richard Brent's web site, The Hadamard maximal determinant problem, contains a wealth of information about the problem, including data collected in numerous exhaustive and non-exhaustive searches.
A Library of Hadamard Matrices by N. J. A. Sloane has all inequivalent Hadamard matrices up to order 28, and many of higher order as well. The sequence D(N) for N = 2, 3, 4, ... which starts 1, 1, 2, 3, 5, 9, 32, ... is Sequence A003432 in Sloane's Online Encyclopedia of Integer Sequences [Sl]. It has the further distinction of being one of his classic sequences.
Jennifer Seberry's home page contains libraries of Hadamard matrices of orders 32 and 36, as well as a nice collection of D-optimal designs with most of the circulant block designs listed in third column of the above table and many of higher order too. She also lists Williamson-type Hadamard matrices and good matrices.
Ted Spence's home page has libraries of all inequivalent Hadamard matrices up to order 28 and two collections of matrices of order 36. It also lists D-optimal designs of orders 25, 61, and 113.
Christos Koukouvinos lists large numbers of Hadamard matrices and D-optimal designs in many orders.
Hadamard matrices of sizes 32 and 36 with specified automorphisms, classified by Iliya Bouyukliev, Veerle Fack, and Joost Winne.
Karol Życzkowski and Wojciech Tadej have compiled a catalogue of complex Hadamard matrices. Their site is maintained by Wojciech Bruzda.

Other useful sites:

Design Resources on the Web maintained by Professor Peter Cameron of the School of Mathematical Sciences at Queen Mary, University of London.
Open Directory Project design theory page.
The La Jolla Difference Set Repository maintained by Dan Gordon.

Credits and acknowledgments

Page created 26 January 2002.
Last modified 17 October 2012.
Comments: maxdet@indiana.edu