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