

A003432


Hadamard maximal determinant problem: largest determinant of a (real) {0,1}matrix of order n.
(Formerly M0720)


22



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

0,4


COMMENTS

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^(n1)*a(n1). 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 EhlichWojtas, and are therefore maximal. See the OrrickSolomon 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]


REFERENCES

J. Hadamard, Résolution d'une question relative aux déterminants, Bull. des Sciences Math. 2 (1893), 240246.
F. J. MacWilliams and N. J. A. Sloane, The Theory of ErrorCorrecting Codes, ElsevierNorth Holland, 1978, p. 54.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

H. Ehlich and K. Zeller, Binäre Matrizen, Zeit. Angew. Math. Mech., 42 (1962), 2021.


EXAMPLE

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


CROSSREFS



KEYWORD

nonn,hard,more,nice


AUTHOR



EXTENSIONS



STATUS

approved



