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 80 - 119

Det should be multiplied by 2N-1. Refer to key for more information.
N Det R N Det R N Det R N Det R
80 40 × 2039 1 81 196 × 2039 ?? .77 82 1620 × 2039 1 83 5.070994773E54 ?? .86
84 42 × 2141 1 85 4509 × 2140 ?? .79 86 1785 × 2141 1 87 1.685070861E58 ?? .85
88 44 × 2243 1 89 220 × 2243 ?? .75 90 1958 × 2243 1 91 6.144764944E61 ?? .85
92 46 × 2345 1 93 239 × 2345 ?? .76 94 3.931163325E64 ?? .97 95 2.438687439E65 ?? .84
96 48 × 2447 1 97 256 × 2447 ?? .77 98 2328 × 2447 1 99 114487974849 × 2442 ?? .85
100 50 × 2549 1 101 7110 × 2548 ?? .80 102 2525 × 2549 1 103 4.982877917E72 ?? .84
104 52 × 2651 1 105 3.847437473E74 ?? .70 106 2704 × 2651 ?? .99 107 2.529473175E76 ?? .84
108 54 × 2753 1 109 1.877483842E78 ?? .65 110 2943 × 2753 1 111 1.380383133E80 ?? .84
112 56 × 2855 1 113 420 × 2855 1 114 3164 × 2855 1 115 8.112413396E83 ?? .83
116 58 × 2957 1 117 7.231499425E85 ?? .72 118 3393 × 2957 1 119 5.122227512E87 ?? .83
<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:

Related links:

Other useful sites:

Credits and acknowledgments


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