Paley Construction
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 thePowers of (mod ) run through 
. For example,
is of this form with
and 
. Since , we are dealing with GF(11), so pick 
and compute itsResidues (mod 11), which are |  |  | |
 | |  | |
 | |  | |
 | |  | |
 | |  | |
 | |  | |
 | |  | |
 | |  | |
 | |  | |
 | |  | |
 | |  |
Picking the first
Residues and adding 0 gives: 0, 1, 2, 4, 5, 8, which shouldthen be colored in the Matrix obtained by writing out the Residues increasing to theleft and up along the border (0 through 
, followed by 
), then adding horizontal and vertical coordinates to getthe residue to place in each square.
can be trivially constructed from 
. 
cannot be built up from smallerMatrices, so use 
. Only the first form can be used, with 
and . We therefore use GF(19), and color 9 Residues plus 0 white. 
can beconstructed from 
.
Now consider a more complicated case. For 
, the only form having is the first, so use theGF(
) field. Take as the modulus the Irreducible Polynomial 
, written 1021. A four-digit number canalways be written using only three digits, since 
and 
. Now look at the modulistarting 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, and221.
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.
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