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Paley Construction

Hadamard matrices can be constructed using finite 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,

(1)

is of this form with and . Since , we are dealing with GF(11), so pick and compute its residues (mod 11), which are

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

Picking the first residues and adding 0 gives: 0, 1, 2, 4, 5, 8, which should then be colored in the matrix obtained by writing out the residues increasing to the left and up along the border (0 through , followed by ), then adding horizontal and vertical coordinates to get the residue to place in each square.

[infty infty infty infty infty infty infty infty infty infty infty infty; 10 0 1 2 3 4 5 6 7 8 9 infty; 9 10 0 1 2 3 4 5 6 7 8 infty; 8 9 10 0 1 2 3 4 5 6 7 infty; 7 8 9 10 0 1 2 3 4 5 6 infty; 6 7 8 9 10 0 1 2 3 4 5 infty; 5 6 7 8 9 10 0 1 2 3 4 infty; 4 5 6 7 8 9 10 0 1 2 3 infty; 3 4 5 6 7 8 9 10 0 1 2 infty; 2 3 4 5 6 7 8 9 10 0 1 infty; 1 2 3 4 5 6 7 8 9 10 0 infty; 0 1 2 3 4 5 6 7 8 9 10 infty]

(13)

To construct , consider the representations . Only the first form can be used, with and . We therefore use GF(19), and color 9 residues plus 0 white.

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

x^0=1 x^1=10 x^2=100 ; x^3=1000=12 x^4=120 x^5=1200=212 ; x^6=2120=111 x^7=1100=122 x^8=1220=202 ; x^9=2020=11 x^(10)=110 x^(11)=1100=112 ; x^(12)=1120=102 x^(13)=1020=2 x^(14)=20 ; x^(15)=200 x^(16)=2000=21 x^(17)=210 ; x^(18)=2100=121 x^(19)=1210=222 x^(20)=2220=211 ; x^(21)=2110=101 x^(22)=101=22 x^(23)=220 ; x^(24)=2200=221 x^(25)=2210=201 x^(26)=2010=1

(14)

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: Dekker, 1979.

Kitis, L. "Paley's Construction of Hadamard Matrices." http://library.wolfram.com/infocenter/MathSource/499/.

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Weisstein, Eric W. "Paley Construction." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PaleyConstruction.html