Hadamard Matrices
can be constructed using Galois Field GF() when
and is Odd. Pick a representation Relatively Prime to . Then by coloring white
(where
is the Floor Function) distinct equally spaced Residues mod (, ,
, ...; , , , ...; etc.) in addition to 0, a Hadamard Matrix is obtained if the
Powers of (mod ) run through
. For example,
can be trivially constructed from . cannot be built up from smaller Matrices, so use . Only the first form can be used, with and . We therefore use GF(19), and color 9 Residues plus 0 white. can be constructed from .
Now consider a more complicated case. For , the only form having is the first, so use the GF() field. Take as the modulus the Irreducible Polynomial , written 1021. A four-digit number can always be written using only three digits, since and . Now look at the moduli starting with 10, where each digit is considered separately. Then
Taking the alternate terms gives white squares as 000, 001, 020, 021, 022, 100, 102, 110, 111, 120, 121, 202, 211, and 221.
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover,
pp. 107-109 and 274, 1987.
Beth, T.; Jungnickel, D.; and Lenz, H. Design Theory, 2nd ed. rev. Cambridge, England: Cambridge University Press, 1998.
Geramita, A. V. Orthogonal Designs: Quadratic Forms and Hadamard Matrices.
New York: Marcel Dekker, 1979.
Kitis, L. ``Paley's Construction of Hadamard Matrices.''
http://www.mathsource.com/cgi-bin/MathSource/Applications/Mathematics/0205-760.
© 1996-9 Eric W. Weisstein