This list contains the Williamson matrices of order m, i.e. four circulant symmetric (1,-1) matrices A,B,C,D of order m which satisfy the Williamson matrix equation
A^2 + B^2 + C^2 + D^2 = 4m I_m
Then using A,B,C,D in the following Williamson array, Turyn constructed Hadamard matrices of order 4m.
A B C D
-B A D -C
-C -D A B
-D C -B A
In the following list - stands for -1.
We define Williamson matrices of type 2 or Turyn type if: 4n=(x+1)^2+(x-1)^2+y^2+y^2 and the Williamson equation is satisfied.
The following list contains the Williamson matrices of Turyn's type of order up to 63. Turyn in "An infinite class of Williamson matrices, J. Combin. Theory Ser. A, 12 (1972), 319-321", proved the following theorem:
THEOREM (Turyn (1972)): If q is a prime power = 1(mod4), q+1=2t, then there exist Williamson matrices of order 4t; we have C=D, and A and B differ only on the main diagonal.
In fact it was shown in "J.M.Goethals and J.J.Seidel, Orthogonal matrices with zero diagonal, Canad. J. Math., 19 (1967), 1001-1010" that if q is a prime power the quadratic character matrix of order q+1 (with zero main diagonal) can be realized as P=[\chi (det(x_i,x_j))], where \chi is the quadratic character and the x_i are a set of tw0-dimensional vectors over GF(q), no two linearly dependent. Also, if q=1(mod4) this matrix can be put in the form
R S
P=
S -R
with R,S symmetric circulants of order (q+1)/2. Since PP^T=P^2=qI_{q+1},
we obtain that R^2+S^2=qI_{(q+1)/2}. Multiplying on the left by e^T (the
1x(q+1)/2 vector of one's) and on the right by e both sides of the last
relation, we obtain that x^2+y^2=q=2t-1, where x and y are the sums of the
elements of the first row of matrices R and S respectively. Thus, the required
Williamson matrices are A=R+I, B=R-I, C=D=S and the corresponding decomposition
of 4t into four odd squares is: 4t=(x+1)^2 + (x-1)^2 + y^2 + y^2.
We now give the Williamson matrices of Turyn's type of order up to 63:
t=1, q=1, 4t=1^2+(-1)^2+1^2+1^2
1
-
1
1
t=3, q=5, 4t=3^2+1^2+1^2+1^2
111
-11
-11
-11
t=5, q=9, 4t=1^2+(-1)^2+3^2+3^2
11--1
-1--1
-1111
-1111
t=7, q=13, 4t=3^2+1^2+3^2+3^2
111--11
-11--11
1-1111-
1-1111-
t=9, q=17,
4t=5^2+3^2+1^2+1^2
1-111111-
--111111-
1-11--11-
1-11--11-
t=13, q=25,
4t=5^2+3^2+3^2+3^2
1-111-11-111-
--111-11-111-
-1--111111--1
-1--111111--1
t=15, q=29,
4t=3^2+1^2+5^2+5^2
1--11-1111-11--
---11-1111-11--
-1-1111--1111-1
-1-1111--1111-1
t=19,
q=37,
4t=7^2+5^2+1^2+1^2
1--1111-1111-1111--
---1111-1111-1111--
-1-111--1--1--111-1
-1-111--1--1--111-1
t=21, q=41,
4t=5^2+3^2+5^2+5^2
1-1-11--111111--11-1-
--1-11--111111--11-1-
1-1111-11----11-1111-
1-1111-11----11-1111-
t=25, q=49,
4t=1^2+(-1)^2+7^2+7^2
1---1-11-111--111-11-1---
----1-11-111--111-11-1---
--11111-1-11--11-1-11111-
--11111-1-11--11-1-11111-
t=27, q=53,
4t=3^2+1^2+7^2+7^2
1-1--1-11--111111--11-1--1-
--1--1-11--111111--11-1--1-
111-111---1-1111-1---111-11
111-111---1-1111-1---111-11
t=31, q=61,
4t=7^2+5^2+5^2+5^2
11--1-1111--11-11-11--1111-1--1
-1--1-1111--11-11-11--1111-1--1
----111-111-1-1111-1-111-111---
----111-111-1-1111-1-111-111---
t=37, q=73,
4t=9^2+7^2+3^2+3^2
1111-1111-1--1---1111---1--1-1111-111
-111-1111-1--1---1111---1--1-1111-111
--1-1-1111--11-11----11-11--1111-1-1-
--1-1-1111--11-11----11-11--1111-1-1-
t=41, q=81,
4t=1^2+(-1)^2+9^2+9^2
11----1-111-11-11---11---11-11-111-1----1
-1----1-111-11-11---11---11-11-111-1----1
1-1---111-11-1--1111111111--1-11-111---1-
1-1---111-11-1--1111111111--1-11-111---1-
t=45, q=89,
4t=9^2+7^2+5^2+5^2
111---1--11-11111-1-1-11-1-1-11111-11--1---11
-11---1--11-11111-1-1-11-1-1-11111-11--1---11
11--1-1--11----11111-1111-11111----11--1-1--1
11--1-1--11----11111-1111-11111----11--1-1--1
t=49, q=97,
4t=5^2+3^2+9^2+9^2
1----1-1111--1-111-1-111--111-1-111-1--1111-1----
-----1-1111--1-111-1-111--111-1-111-1--1111-1----
1111-11-1---11-111---11-11-11---111-11---1-11-111
1111-11-1---11-111---11-11-11---111-11---1-11-111
t=51, q=101,
4t=9^2+7^2+1^2+1^2
1---111-11-1-111--11111--11--11111--111-1-11-111---
----111-11-1-111--11111--11--11111--111-1-11-111---
-1--1----1-111-1-11111--1--1--11111-1-111-1----1--1
-1--1----1-111-1-11111--1--1--11111-1-111-1----1--1
t=55, q=109,
4t=11^2+9^2+3^2+3^2
1-1--1-1-11--1-11111-111--1111--111-11111-1--11-1-1--1-
--1--1-1-11--1-11111-111--1111--111-11111-1--11-1-1--1-
111----11-11--11----1-1-11111111-1-1----11--11-11----11
111----11-11--11----1-1-11111111-1-1----11--11-11----11
t=57, q=113,
4t=9^2+7^2+7^2+7^2
1---11-1--1111-111-11---1-111111-1---11-111-1111--1-11---
----11-1--1111-111-11---1-111111-1---11-111-1111--1-11---
--1-1-111--1--1-11---11111-1111-11111---11-1--1--111-1-1-
--1-1-111--1--1-11---11111-1111-11111---11-1--1--111-1-1-
t=61, q=121,
4t=1^2+(-1)^2+11^2+11^2
1-----111--111-1-1--1-11-11-11--11-11-11-1--1-1-111--111-----
------111--111-1-1--1-11-11-11--11-11-11-1--1-1-111--111-----
---11--11-1-1-1111111---1-1111--1111-1---1111111-1-1-11--11--
---11--11-1-1-1111111---1-1111--1111-1---1111111-1-1-11--11--
t=63, q=125,
4t=3^2+1^2+11^2+11^2
11-111--11-11--1--1-11-1-111--------111-1-11-1--1--11-11--111-1
-1-111--11-11--1--1-11-1-111--------111-1-11-1--1--11-11--111-1
1111-11-1-1111-1---1---111---111111---111---1---1-1111-1-11-111
1111-11-1-1111-1---1---111---111111---111---1---1-1111-1-11-111
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