The Hadamard maximum determinant problem: (1997)
Matrixes from { +1, -1 } with maximal determinant and reaching
the Hadamard bound are known as Hadamard-Matrices. It it well
known that a neccessary condition is: there order is 1, 2, or 4*n.
The Hadamard conjecture says that all cases are possible.
Questions:
- What is the order of the smallest unknown Hadamard-Matrices?
- Is there any hope to settle the Hadamard conjecture within
say 10 years?
- What is with case were no Hadamard-Matrix can exist. Are
there conjectures about the maximal reachable determinant?
- Are there better bounds?
Achim Flammenkamp tried all orders upto 16. He found the
following maximal determinants. Do you found bigger cases?
(mailto:Achim@uni-bielefeld.de)
order maximal value found by Achim Hadamard bound
N=1 max= 1 = 1 1
N=2 max= 2 = 2 2
N=3 max= 2^2 = 4 5.2
N=4 max= 2^4 = 16 16
n=5 max= 2^4 * 3 = 48 55.9
n=6 max= 2^5 * 5 = 160 216
n=7 max= 2^6 * 3^2 = 576 907.5
N=8 max= 2^12 = 4096 4096
n=9 max= 2^11 * 7 = 14336 19683
n=10 max= 2^13 * 3^2 = 73728 100000
n=11 max= 2^16 * 5 = 327680 534145.7
N=12 max= 2^12 * 3^6 = 2985984 2985984
n=13 max= 2^12 * 3^6 * 5 = 14929920 17403307.3
n=14 max= 2^13 * 3^6 * 13 = 77635584 105413504
n=15 max= 2^14 * 3^6 * 5*7 = 418037760 661735513.9
N=16 max= 2^32 = 4294967296 4294967296
The maximum is reached for symmetric matrices for n=1..8.
The maximum found by Achim for n=14 has been conjectured
to be maximal by several researchers but it is still not proven.
references (Hadamard matrices, The Hadamard maximum determinant problem):
S. S. Agaian;
Hadamard Matrices and Their Application,
Springer Verlag, Berlin (1985)
Beth, Jungnickel, Lenz;
Design Theory
BI-Verlag (1985)
Hedayat, Wallis;
Hadamard matrices and their applications
Ann. Stat. 6 (1978) 1184-1238
van Lint and Wilson;
A Course in Combinatorics
(They gave 4*107 as smallest unknown case. [Hadamard matrices])
H. L"uneburg;
Kombinatorik, Birkh"auser, Basel-Stuttgart (1971)
((I.9) Hadamard's inequality on determinants (1893))
Peter Gritzmann, Victor Klee;
On the complexity of some basic problems in computational
convexity: I. Containment problems
Discrete Mathematics 136 (1994) 129-174
Section 9.6: Hadamard maximum determinant problem
J. Brenner, L Cummings;
The Hadamard maximum determinant problem,
American Math. Monthly 79 (1972) 626-630
J. H. E. Cohn;
Hadamard matrices and some generalizations,
American Math. Monthly 72 (1965) 515-518
J. H. E. Cohn;
On determinants with elements +1, -1,
London Math. Soc. 42 (1967) 436-442
W. D. Smith;
Polytope triangulations in d-space, improving Hadamard's inequality,
and maximal volumes of regular polytopes in hyperbolic d-space,
unpublished, Dept. of Math., Princton Univ., 1987
M. Hudelson, V. Klee, D. Larman;
Largest j-Simplices in d-Cubes:
Some Relatives of the Hadamard Maximum Determinant Problem,
manuscript, (1996)
(The Hadamard maximum determinant problem: bound for n=14 not proven.)
J. Hadamard;
Resolution d'une question relativ aux determinants,
Bull. Sci. Math. 28 (1893) 240-246
H. S. Wilf;
Some examples of combinatorial averaging,
American Math. Monthly 92 (1985) 250-261
Sect 2: In search of the biggest determinant