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
Table of maximal determinants, orders 0 - 39
Det should be multiplied by 2N-1. Refer to key
for more information.
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
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
For a list of recent changes to these pages, see the update
- 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
- 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
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
Other useful sites:
- 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
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
- Jennifer Seberry's home
page contains libraries of
matrices of orders 32 and 36, as well as a nice collection of
with most of the circulant block designs listed in third column of the
above table and many of higher order too. She also lists
Hadamard matrices and good
- 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.
Koukouvinos lists large numbers of Hadamard matrices and D-optimal
designs in many orders.
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.
Credits and acknowledgments
Page created 26 January 2002.
Last modified 17 October 2012.