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 matrix equation
A^2 + B^2 + C^2 + D^2 = 4m I_m
Then using A,B,C,D in the following Williamson array, we can construct
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.
m=1, 4m=4=1^2+1^2+1^2+1^2, one solution
1
1
1
1
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1
\nu_4 = 1
m=3, 4m=12=3^2+1^2+1^2+1^2, one solution
111
1--
1--
1--
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1
\nu_4 = 1 + 2\omega(-1)
This solution is of Turyn type.
m=5, 4m=20=3^2+3^2+1^2+1^2, one solution
1----
1----
11--1
1-11-
\nu_1 = 1 + 2\omega(-2)
\nu_2 = 1 + 2\omega(-1)
\nu_3 = 1
\nu_4 = 1
This solution is of Yamada type: A, 2A=A, C, 2C.
This solution is also of Turyn type.
m=7, 4m=28=5^2+1^2+1^2+1^2, one solution
1------
11----1
1-1--1-
1--11--
\nu_1 = 1 + 2\omega(-3)
\nu_2 = 1 + 2\omega(-2)
\nu_3 = 1 + 2\omega(-1)
\nu_4 = 1
m=7, 4m=28=3^2+3^2+3^2+1^2, one solution
111--11
111--11
11-11-1
1-1--1-
\nu_1 = 1 + 2\omega(1)
\nu_2 = 1 + 2\omega(2-3)
\nu_3 = 1
\nu_4 = 1
This solution is of Turyn type.
m=9, 4m=36=5^2+3^2+1^2+1^2, two solutions
1111--111
1---11---
11-1--1-1
11-1--1-1
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(1+3-4)
\nu_4 = 1 + 2\omega(-2)
This solution is of Turyn type.
11-1111-1
11------1
11-1--1-1
1-11--11-
\nu_1 = 1 + 2\omega(1-2)
\nu_2 = 1
\nu_3 = 1 + 2\omega(3)
\nu_4 = 1 + 2\omega(-4)
m=9, 4m=36=3^2+3^2+3^2+3^2, one solution
11------1
1-1----1-
1--1--1--
1---11---
\nu_1 = 1 + 2\omega(-4)
\nu_2 = 1 + 2\omega(-3)
\nu_3 = 1 + 2\omega(-2)
\nu_4 = 1 + 2\omega(-1)
m=11, 4m=44=5^2+3^2+3^2+1^2, one solution
11--------1
11-11--11-1
11-1-11-1-1
1-11----11-
\nu_1 = 1 + 2\omega(1-2)
\nu_2 = 1 + 2\omega(-5)
\nu_3 = 1 + 2\omega(-4)
\nu_4 = 1 + 2\omega(3)
m=13, 4m=52=7^2+1^2+1^2+1^2, one solution
1----1--1----
11-1--11--1-1
1---111111---
1-1--1111--1-
\nu_1 = 1 + 2\omega(-2)
\nu_2 = 1 + 2\omega(-4)
\nu_3 = 1 + 2\omega(-1-3+5)
\nu_4 = 1 + 2\omega(6)
m=13, 4m=52=5^2+5^2+1^2+1^2, one solution
111-11--11-11
11-1-1111-1-1
11--1-11-1--1
1---111111---
\nu_1 = 1 + 2\omega(1)
\nu_2 = 1 + 2\omega(5)
\nu_3 = 1 + 2\omega(-3+4)
\nu_4 = 1 + 2\omega(-2+6)
This solution is of Yamada type: A, 5A, C, 5C.
m=13, 4m=52=5^2+3^2+3^2+3^2, two solutions
1111-1--1-111
1---1-11-1---
11---1--1---1
11---1--1---1
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(1-4+5-6)
\nu_4 = 1 + 2\omega(-2-3)
This solution is of Turyn type.
This solution also satisfies A=5A, B=5B, C=5C, D=5D.
1-111-11-111-
11--1----1--1
1----1111----
1---1-11-1---
\nu_1 = 1
\nu_2 = 1 + 2\omega(4-5)
\nu_3 = 1 + 2\omega(-1+6)
\nu_4 = 1 + 2\omega(-2-3)
m=15, 4m=60=7^2+3^2+1^2+1^2, three solutions
1-11-111111-11-
1-11--1111--11-
11-11------11-1
1-1---1111---1-
\nu_1 = 1 + 2\omega(3)
\nu_2 = 1 + 2\omega(-1+2-4+6+7)
\nu_3 = 1
\nu_4 = 1 + 2\omega(-5)
This solution is of Whiteman type.
1-1111-11-1111-
111-11----11-11
1---11-11-11---
1-1---1111---1-
\nu_1 = 1 + 2\omega(4+5-6)
\nu_2 = 1 + 2\omega(2)
\nu_3 = 1 + 2\omega(-1+7)
\nu_4 = 1 + 2\omega(-3)
1111--1111--111
1-1-11-11-11-1-
1-1--11--11--1-
1----111111----
\nu_1 = 1 + 2\omega(2)
\nu_2 = 1 + 2\omega(7)
\nu_3 = 1 + 2\omega(-4+6)
\nu_4 = 1 + 2\omega(-1-3+5)
m=15, 4m=60=5^2+5^2+3^2+1^2, one solution
1-1----11----1-
1-1----11----1-
1--11-1111-11--
111--1----1--11
\nu_1 = 1 + 2\omega(-1-5+7)
\nu_2 = 1 + 2\omega(2-3-4-6)
\nu_3 = 1
\nu_4 = 1
This solution is of Turyn type.
m=17, 4m=68=7^2+3^2+3^2+1^2, three solutions
1--1-1------1-1--
1-1---11--11---1-
1-----111111-----
11--1-11--11-1--1
\nu_1 = 1 + 2\omega(-1-4)
\nu_2 = 1 + 2\omega(-8)
\nu_3 = 1 + 2\omega(-2)
\nu_4 = 1 + 2\omega(-3-5+6+7)
1-----11--11-----
1---11-1--1-11---
11-1--1----1--1-1
1-1---111111---1-
\nu_1 = 1 + 2\omega(-2-8)
\nu_2 = 1 + 2\omega(-1-3+7)
\nu_3 = 1 + 2\omega(-4-5+6)
\nu_4 = 1
1--1-1------1-1--
1-111--------111-
11---1--11--1---1
1--1-111--111-1--
\nu_1 = 1 + 2\omega(-6-7)
\nu_2 = 1 + 2\omega(-1+3-8)
\nu_3 = 1 + 2\omega(-2-4+5)
\nu_4 = 1
m=17, 4m=68=5^2+5^2+3^2+3^2, one solution
1--1-11111111-1--
1-11-111--111-11-
11-1---1--1---1-1
1---111----111---
\nu_1 = 1 + 2\omega(3-4+7)
\nu_2 = 1 + 2\omega(-1+5+6)
\nu_3 = 1 + 2\omega(-2)
\nu_4 = 1 + 2\omega(-8)
This solution is of Yamada type: A, 4A, C, 4C.
m=19, 4m=76=7^2+5^2+1^2+1^2, three solutions
11--11-111111-11--1
1-11--1------1--11-
111---1-1--1-1---11
111---1-1--1-1---11
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(1-3+8)
\nu_4 = 1 + 2\omega(2-4-5+6-7-9)
This solution is of Turyn type.
11-1-1111--1111-1-1
1------111111------
1--11-1-1--1-1-11--
1-11--11----11--11-
\nu_1 = 1 + 2\omega(-2+8)
\nu_2 = 1 + 2\omega(-4+7)
\nu_3 = 1 + 2\omega(3+6-9)
\nu_4 = 1 + 2\omega(-1-5)
11111-1--11--1-1111
1-1-1--1----1--1-1-
11--11-1----1-11--1
111----11--11----11
\nu_1 = 1 + 2\omega(4-8)
\nu_2 = 1 + 2\omega(2-5)
\nu_3 = 1 + 2\omega(1)
\nu_4 = 1 + 2\omega(-3-6+7-9)
m=19, 4m=76=7^2+3^2+3^2+3^2, three solutions
11111-1-1--1-1-1111
11---111-11-111---1
111--1--1111--1--11
1-1--11-1111-11--1-
\nu_1 = 1 + 2\omega(1)
\nu_2 = 1 + 2\omega(6)
\nu_3 = 1 + 2\omega(2-7+8)
\nu_4 = 1 + 2\omega(-3-4+5+9)
11--1-11111111-1--1
1-1---11111111---1-
11-11--1-11-1--11-1
11--111-1--1-111--1
\nu_1 = 1 + 2\omega(-5+7+9)
\nu_2 = 1 + 2\omega(-3+6+8)
\nu_3 = 1 + 2\omega(1-2+4)
\nu_4 = 1
11--1-11111111-1--1
11-11-11----11-11-1
11---11-1111-11---1
1-1-1--111111--1-1-
\nu_1 = 1 + 2\omega(1-2+6)
\nu_2 = 1 + 2\omega(4-5+7)
\nu_3 = 1 + 2\omega(-3+8+9)
\nu_4 = 1
m=19, 4m=76=5^2+5^2+5^2+1^2, no solution
m=21, 4m=84=9^2+1^2+1^2+1^2, one solution
11-11111-1--1-11111-1
11--1-1-11--11-1-1--1
1111-1---1--1---1-111
1--1111---11---1111--
\nu_1 = 1 + 2\omega(1+9-10)
\nu_2 = 1 + 2\omega(-2+4+6)
\nu_3 = 1 + 2\omega(3+5-8)
\nu_4 = 1 + 2\omega(-7)
m=21, 4m=84=7^2+5^2+3^2+1^2, three solutions
1-1--1----11----1--1-
1111-1-1--11--1-1-111
111-1-----11-----1-11
11---11-11--11-11---1
\nu_1 = 1 + 2\omega(2-6-8-9+10)
\nu_2 = 1 + 2\omega(-4+5)
\nu_3 = 1 + 2\omega(-3-7)
\nu_4 = 1 + 2\omega(1)
1-1-----11--11-----1-
1111--11-1--1-11--111
1-1--111------111--1-
1---1-11-1111-11-1---
\nu_1 = 1 + 2\omega(2-4-10)
\nu_2 = 1 + 2\omega(-5+9)
\nu_3 = 1 + 2\omega(-1-3)
\nu_4 = 1 + 2\omega(6+7-8)
1-1-----11--11-----1-
1111-1-11----11-1-111
1-1--111------111--1-
1---11-11-11-11-11---
\nu_1 = 1 + 2\omega(2-4-10)
\nu_2 = 1 + 2\omega(-6+8)
\nu_3 = 1 + 2\omega(-1-3)
\nu_4 = 1 + 2\omega(5+7-9)
m=21, 4m=84=5^2+5^2+5^2+3^2, three solutions
111-1--111--111--1-11
111-1--111--111--1-11
111-111-1----1-111-11
1--1---1-1111-1---1--
\nu_1 = 1 + 2\omega(1+2-3+4+8-10)
\nu_2 = 1 + 2\omega(-5-6+7+9)
\nu_3 = 1
\nu_4 = 1
This solution is of Turyn type.
11111--11----11--1111
111-1--1-1111-1--1-11
111--1-111--111-1--11
1---1-1-11--11-1-1---
\nu_1 = 1 + 2\omega(1+2-6+7)
\nu_2 = 1 + 2\omega(4-5)
\nu_3 = 1 + 2\omega(8-10)
\nu_4 = 1 + 2\omega(-3+9)
11-111--1-11-1--111-1
111-111-1----1-111-11
111-11---1111---11-11
1-1----11-11-11----1-
\nu_1 = 1 + 2\omega(1+4+5-7)
\nu_2 = 1 + 2\omega(8-9)
\nu_3 = 1 + 2\omega(-6+10)
\nu_4 = 1 + 2\omega(2-3)
m=23, 4m=92=9^2+3^2+1^2+1^2, no solution
m=23, 4m=92=7^2+5^2+3^2+3^2, one solution
1-11-11--111111--11-11-
11---1---1-11-1---1---1
111---11-1-11-1-11---11
111-111-1------1-111-11
\nu_1 = 1 + 2\omega(-4-8+9+11)
\nu_2 = 1 + 2\omega(5-7)
\nu_3 = 1 + 2\omega(2+6)
\nu_4 = 1 + 2\omega(1-3-10)
m=25, 4m=100=9^2+3^2+3^2+1^2, one solution
11-1111--1-1111-1--1111-1
1----1111--1--1--1111----
1-1-11---1--11--1---11-1-
1---1-1--11111111--1-1---
\nu_1 = 1 + 2\omega(5-10)
\nu_2 = 1 + 2\omega(-2+6+11)
\nu_3 = 1 + 2\omega(4-7-8+9+12)
\nu_4 = 1 + 2\omega(-1-3)
m=25, 4m=100=7^2+7^2+1^2+1^2, three solutions
1-----1-11--11--11-1-----
1-----1-11--11--11-1-----
1-11-1-111------111-1-11-
11--1-1---111111---1-1--1
\nu_1 = 1 + 2\omega(-1-4+8+9-10-11)
\nu_2 = 1 + 2\omega(-2-3-5+6-7+12)
\nu_3 = 1
\nu_4 = 1
This solution is of Yamada type: A, 7A=A, C, 7C.
This solution is also of Turyn type.
1---1-111--------111-1---
11-1--1-1--------1-1--1-1
1--11-1--111--111--1-11--
1-1111--1---11---1--1111-
\nu_1 = 1 + 2\omega(-2-5+6-12)
\nu_2 = 1 + 2\omega(8-9-10-11)
\nu_3 = 1 + 2\omega(-1+4)
\nu_4 = 1 + 2\omega(3-7)
This solution is of Yamada type: A, 7A, C, 7C.
1---1-1-11------11-1-1---
1--1--1-1---11---1-1--1--
11-11-111--------111-11-1
1-1111----11--11----1111-
\nu_1 = 1 + 2\omega(-2-5+6+8-10-11)
\nu_2 = 1 + 2\omega(-1-7)
\nu_3 = 1 + 2\omega(4-12)
\nu_4 = 1 + 2\omega(3-9)
m=25, 4m=100=7^2+5^2+5^2+1^2, three solutions
1--11-1-1--------1-1-11--
111-111---1-11-1---111-11
1-11-111-11----11-111-11-
1-1111----11--11----1111-
\nu_1 = 1 + 2\omega(6-11)
\nu_2 = 1 + 2\omega(4-7-9)
\nu_3 = 1 + 2\omega(-1+3-12)
\nu_4 = 1 + 2\omega(2+5-8+10)
1-1------111--111------1-
1-111-11--1-11-1--11-111-
1-1--11--11111111--11--1-
11----1-11-1111-11-1----1
\nu_1 = 1 + 2\omega(-1+2-8+10)
\nu_2 = 1 + 2\omega(-5)
\nu_3 = 1 + 2\omega(-3-4-7+9+11)
\nu_4 = 1 + 2\omega(6+12)
111----1---1--1---1----11
111-111---1-11-1---111-11
1--1-1-11111--11111-1-1--
111--1-1--11--11--1-1--11
\nu_1 = 1
\nu_2 = 1 + 2\omega(1+2-3-8-9)
\nu_3 = 1 + 2\omega(-4-6+7+11-12)
\nu_4 = 1 + 2\omega(5+10)
m=25, 4m=100=5^2+5^2+5^2+5^2, three solutions
1-1--111--111111--111--1-
111--1--1111--1111--1--11
11--1--1-11111111-1--1--1
1111-1-1-1--11--1-1-1-111
\nu_1 = 1 + 2\omega(-3+10+11)
\nu_2 = 1 + 2\omega(2-4+5)
\nu_3 = 1 + 2\omega(7-8+12)
\nu_4 = 1 + 2\omega(1-6+9)
This solution is of Yamada type: A, 7A, C, 7C.
1--1--1111-1111-1111--1--
111----11-111111-11----11
111-1-1-11--11--11-1-1-11
111--111-1-1--1-1-111--11
\nu_1 = 1 + 2\omega(-5+8+12)
\nu_2 = 1 + 2\omega(-4+7+11)
\nu_3 = 1 + 2\omega(6+9-10)
\nu_4 = 1 + 2\omega(1+2-3)
This solution is of Yamada type: A, 7A, C, 7C.
1--11111--1-11-1--11111--
11-111--111----111--111-1
11--1--1-11111111-1--1--1
1111-1-1-1--11--1-1-1-111
\nu_1 = 1 + 2\omega(-2+4+10)
\nu_2 = 1 + 2\omega(3+5-11)
\nu_3 = 1 + 2\omega(7-8+12)
\nu_4 = 1 + 2\omega(1-6+9)
This solution is of Yamada type: A, 7A, C, 7C.
m=27, 4m=108=9^2+5^2+1^2+1^2, two solutions
1--1---111--------111---1--
1-11-1----11----11----1-11-
1-1-111--1---11---1--111-1-
1111-1---1--1--1--1---1-111
\nu_1 = 1 + 2\omega(-1-12)
\nu_2 = 1 + 2\omega(3-4-6-13)
\nu_3 = 1 + 2\omega(9-10-11)
\nu_4 = 1 + 2\omega(2+5-7-8)
1---1---1--1-11-1--1---1---
1----1-111---11---111-1----
11-1--1--111----111--1--1-1
1-1-----11-111111-11-----1-
\nu_1 = 1 + 2\omega(-2-12)
\nu_2 = 1 + 2\omega(-1-3-6+8-10+13)
\nu_3 = 1 + 2\omega(-5-7+11)
\nu_4 = 1 + 2\omega(-4+9)
m=27, 4m=108=9^2+3^2+3^2+3^2, no solution
m=27, 4m=108=7^2+7^2+3^2+1^2, three solutions
1---11111-1-1111-1-11111---
1---11111-1-1111-1-11111---
111-11-1---11--11---1-11-11
1--1--1-111--11--111-1--1--
\nu_1 = 1 + 2\omega(-3+4+5+7-9+12)
\nu_2 = 1 + 2\omega(-1-2+6+8+10-11+13)
\nu_3 = 1
\nu_4 = 1
This solution is of Turyn type.
1-1-11-111--1111--111-11-1-
11-1-1-11-11-11-11-11-1-1-1
1111-11-----1111-----11-111
1111---11---1--1---11---111
\nu_1 = 1 + 2\omega(5+13)
\nu_2 = 1 + 2\omega(-6+7+8)
\nu_3 = 1 + 2\omega(2-10-11+12)
\nu_4 = 1 + 2\omega(1+3-4-9)
1111-1111--1----1--1111-111
111---11-111-11-111-11---11
11-1--11-1-1-11-1-1-11--1-1
1-111----1--1111--1----111-
\nu_1 = 1 + 2\omega(1-4+6+7+11-12)
\nu_2 = 1 + 2\omega(2)
\nu_3 = 1 + 2\omega(3-10)
\nu_4 = 1 + 2\omega(-5-8+9+13)
m=27, 4m=108=7^2+5^2+5^2+3^2, one solution
1111-1--111--11--111--1-111
11-----1--1-1111-1--1-----1
111--1-1---1----1---1-1--11
1-1-1111--1--11--1--1111-1-
\nu_1 = 1 + 2\omega(1-4-6)
\nu_2 = 1 + 2\omega(10-11+13)
\nu_3 = 1 + 2\omega(2+5-12)
\nu_4 = 1 + 2\omega(-3+7-8-9)
m=29, 4m=116=9^2+5^2+3^2+1^2, one solution
1111-11-1---111111---1-11-111
11--1--1-111-1111-111-1--1--1
111---11--1-1----1-1--11---11
1-1---11--1-111111-1--11---1-
\nu_1 = 1 + 2\omega(1)
\nu_2 = 1 + 2\omega(13+14)
\nu_3 = 1 + 2\omega(2-4+6-9-11+12)
\nu_4 = 1 + 2\omega(-3-5+7-8+10)
m=29, 4m=116=7^2+7^2+3^2+3^2, no solution
(in particular, no solution of Yamada type)
m=31, 4m=124=11^2+1^2+1^2+1^2, no solution
m=31, 4m=124=9^2+5^2+3^2+3^2, no solution
m=31, 4m=124=7^2+7^2+5^2+1^2, one solution
11-1---11111-1-11-1-11111---1-1
11111--111-1--1--1--1-111--1111
11-11--1----11----11----1--11-1
1---111-11-1-1----1-1-11-111---
\nu_1 = 1 + 2\omega(1+3-5-6+7)
\nu_2 = 1 + 2\omega(8+9+11-12)
\nu_3 = 1 + 2\omega(-2+13-14)
\nu_4 = 1 + 2\omega(4-10-15)
m=31, 4m=124=7^2+5^2+5^2+5^2, one solution
11--1-1111--11-11-11--1111-1--1
1-11-1----11--1--1--11----1-11-
1111---1---1-1----1-1---1---111
1111---1---1-1----1-1---1---111
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(1-5+7-10+13-14)
\nu_4 = 1 + 2\omega(2+3-4-6-8-9+11-12-15)
This solution is of Turyn type.
m=33, 4m=132=11^2+3^2+1^2+1^2, one solution
111--1---1--1--------1--1---1--11
111----1-1-1--11--11--1-1-1----11
1-11--1-1---1111--1111---1-1--11-
11--1----11-1-111111-1-11----1--1
\nu_1 = 1 + 2\omega(2-4-10-16)
\nu_2 = 1 + 2\omega(1-3-6-8+9-13)
\nu_3 = 1 + 2\omega(-7-11+12)
\nu_4 = 1 + 2\omega(-5+14+15)
m=33, 4m=132=9^2+7^2+1^2+1^2, one solution
1--111---1-111111111111-1---111--
11-1---1---1--1-11-1--1---1---1-1
11-11-1-11---11----11---11-1-11-1
111-1----111-1--11--1-111----1-11
\nu_1 = 1 + 2\omega(-2+3-10+14)
\nu_2 = 1 + 2\omega(-6-8+11+16)
\nu_3 = 1 + 2\omega(4-7+9+13)
\nu_4 = 1 + 2\omega(1-5-12-15)
m=33, 4m=132=9^2+5^2+5^2+1^2, two solutions
1--11-111111---1111---111111-11--
1-11111-11-11--------11-11-11111-
11-1-1--1-1111--11--1111-1--1-1-1
1---111--11-1-1-11-1-1-11--111---
\nu_1 = 1 + 2\omega(3+8+11-14)
\nu_2 = 1 + 2\omega(-1+4+6+9-13)
\nu_3 = 1 + 2\omega(-2+10+16)
\nu_4 = 1 + 2\omega(5-7+12-15)
1--111--1--111111111111--1--111--
111111----11-1--11--1-11----11111
11-1-1-1--111-1-11-1-111--1-1-1-1
11--1-1--11-111----111-11--1-1--1
\nu_1 = 1 + 2\omega(3+5-6-9+11+16)
\nu_2 = 1 + 2\omega(4-7+13)
\nu_3 = 1 + 2\omega(-2+12+14)
\nu_4 = 1 + 2\omega(1-8+10-15)
m=33, 4m=132=7^2+7^2+5^2+3^2, one solution
11----1-11--1-1----1-1--11-1----1
1-1--1-----111--11--111-----1--1-
111-11----111-11--11-111----11-11
1111-1-1---1--1----1--1---1-1-111
\nu_1 = 1 + 2\omega(-3-7+12)
\nu_2 = 1 + 2\omega(-4-10-15)
\nu_3 = 1 + 2\omega(1-13+14-16)
\nu_4 = 1 + 2\omega(2+5-6-8-9+11)
m=35, 4m=140=11^2+3^2+3^2+1^2, no solution
m=35, 4m=140=9^2+7^2+3^2+1^2, no solution
m=35, 4m=140=9^2+5^2+5^2+3^2, no solution
m=37, 4m=148=11^2+5^2+1^2+1^2, no solution
m=37, 4m=148=11^2+3^2+3^2+3^2, one solution
1--111-1-----1----11----1-----1-111--
11111-1-----11----11----11-----1-1111
1---11-11--1-1-11----11-1-1--11-11---
1--1-1-1-11---1--1111--1---11-1-1-1--
\nu_1 = 1 + 2\omega(4-9-10+13-14-17)
\nu_2 = 1 + 2\omega(3-8-11-15-16+18)
\nu_3 = 1 + 2\omega(-1-2+5-6+7-12)
\nu_4 = 1
This solution satisfies A=6A, B=6B, C=6C, D=6D.
m=37, 4m=148=9^2+7^2+3^2+3^2, one solution
1111-1111-1--1---1111---1--1-1111-111
1---1----1-11-111----111-11-1----1---
11-1-1----11--1--1111--1--11----1-1-1
11-1-1----11--1--1111--1--11----1-1-1
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(1+3-4+5-9+10-12-15-16+17+18)
\nu_4 = 1 + 2\omega(-2-6-7-8+11-13+14)
This solution is of Turyn type.
m=37, 4m=148=7^2+7^2+7^2+1^2, no solution
m=37, 4m=148=7^2+7^2+5^2+5^2, two solutions
11---1-----1-1-11-11-11-1-1-----1---1
1--11-111------11----11------111-11--
1-1111-1-11--1-1--11--1-1--11-1-1111-
1--111-1----111-111111-111----1-111--
\nu_1 = 1 + 2\omega(-12-14+15-17)
\nu_2 = 1 + 2\omega(-2-9-10+16)
\nu_3 = 1 + 2\omega(5-6-8+13+18)
\nu_4 = 1 + 2\omega(-1+3+4+7-11)
This solution is of Yamada type: A, 6A, C, 6C.
1--1-1111-----1-1----1-1-----1111-1--
11---1-1--11---1--11--1---11--1-1---1
111--1111-111-1--------1-111-1111--11
11--1-1-1111-11--1--1--11-1111-1-1--1
\nu_1 = 1 + 2\omega(-4+5+7-9-13-17)
\nu_2 = 1 + 2\omega(-2-12)
\nu_3 = 1 + 2\omega(6+8+14-15-18)
\nu_4 = 1 + 2\omega(1-3+10+11-16)
m=39, 4m=156=11^2+5^2+3^2+1^2, no solution
m=39, 4m=156=9^2+7^2+5^2+1^2, no solution
m=39, 4m=156=9^2+5^2+5^2+5^2, one solution
111--1-1-----1--11----11--1-----1-1--11
111--11-1---1-1--1----1--1-1---1-11--11
1111---1--11----1-1--1-1----11--1---111
1---11-1-1-----111-11-111-----1-1-11---
\nu_1 = 1 + 2\omega(1+2-4-9-15-19)
\nu_2 = 1 + 2\omega(-3+5-10-11+17-18)
\nu_3 = 1 + 2\omega(-6+7-8-12-14+16)
\nu_4 = 1 + 2\omega(-13)
m=39, 4m=156=7^2+7^2+7^2+3^2, no solution
The results are complete upto this point.
m=41, 4m=164=9^2+9^2+1^2+1^2, one solution (no solution of Yamada type)
1-1---111-11-1--1111111111--1-11-111---1-
1-1---111-11-1--1111111111--1-11-111---1-
11----1-111-11-11---11---11-11-111-1----1
1-1111-1---1--1--111--111--1--1---1-1111-
\nu_1 = 1 + 2\omega(-3-4-5+6+8+10+13-14+16+20)
\nu_2 = 1 + 2\omega(-1+2+7-9+11-12-15+17+18+19)
\nu_3 = 1
\nu_4 = 1
This solution is of Turyn type.
This solution also satisfies A=9A, B=9B, C=9C, D=9D.
m=43, 4m=172=7^2+7^2+7^2+5^2, one solution
1---11--1111-1-111-11--11-111-1-1111--11---
11-111111----1-1--11-11-11--1-1----111111-1
111-1-11--1-1-1111-1----1-1111-1-1--11-1-11
11---1111-1--1--11--------11--1--1-1111---1
\nu_1 = 1 + 2\omega(4+15+19)
\nu_2 = 1 + 2\omega(-2+5+8-12+13-14)
\nu_3 = 1 + 2\omega(-3+10+16+17-18-21)
\nu_4 = 1 + 2\omega(1+6+7-9-11-20)
This solution satisfies A=6A, B=6B, C=6C, D=6D.
m=45, 4m=180=13^2+3^2+1^2+1^2, no solution
m=45, 4m=180=11^2+7^2+3^2+1^2, no solution
m=45, 4m=180=11^2+5^2+5^2+3^2, no solution
m=45, 4m=180=9^2+9^2+3^2+3^2, no solution
m=45, 4m=180=9^2+7^2+7^2+1^2, no solution
m=45, 4m=180=9^2+7^2+5^2+5^2, one solution
111--111-11-1-11111--------11111-1-11-111--11
1--11---1--1-1-----11111111-----1-1--1---11--
11--1111-1-1-111--1--1--1--1--111-1-1-1111--1
11--1111-1-1-111--1--1--1--1--111-1-1-1111--1
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(1-3+5+6+7-8+9+14+15+18-19-20-22)
\nu_4 = 1 + 2\omega(-2+4-10+11-12+13-16-17+21)
This solution is of Turyn type.
Djokovic and Van Vliet believe the results for m=45 are complete.
m=49, 4m=196=9^2+9^2+5^2+3^2, one solution
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
1----1-1111--1-111-1-111--111-1-111-1--1111-1----
11111-1----11-1---1-1---11---1-1---1-11----1-1111
\nu_1 = 1 + 2\omega(-4+5+8-11+13-14+15+16+17-18-20+21+22)
\nu_2 = 1 + 2\omega(1+2+3+6-7-9-10+12-19-23+24)
\nu_3 = 1
\nu_4 = 1
This solution is of Turyn type.
m=51, 4m=204=13^2+5^2+3^2+1^2, no solution
m=51, 4m=204=11^2+9^2+1^2+1^2, one solution
1---111-11-1-111--11111--11--11111--111-1-11-111---
1111---1--1-1---11-----11--11-----11---1-1--1---111
1-11-1111-1---1-1-----11-11-11-----1-1---1-1111-11-
1-11-1111-1---1-1-----11-11-11-----1-1---1-1111-11-
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(-1+5+6+8-12+14-17+22-24+25)
\nu_4 = 1 + 2\omega(2+3-4+7-9+10-11-13-15+16-18-19-20-21+23)
This solution is of Turyn type.
m=51, 4m=204=11^2+7^2+5^2+3^2, one solution
1111--1--111-11-1---11111--11111---1-11-111--1--111
1--1-11--11-111----1-1111111111-1----111-11--11-1--
11-1-1-1----11--1-1--1-11--11-1--1-1--11----1-1-1-1
11--111-111-1-1111----1------1----1111-1-111-111--1
\nu_1 = 1 + 2\omega(3-4-8+13-15-17+21+23+24)
\nu_2 = 1 + 2\omega(6-7+9+10+14-18+22)
\nu_3 = 1 + 2\omega(1+16-19-25)
\nu_4 = 1 + 2\omega(-2+5-11+12-20)
m=51, 4m=204=9^2+7^2+7^2+5^2, no solution
Van Vliet believes the results for m=51 are complete.
m=53, 4m=212=9^2+9^2+5^2+5^2, no solution of Yamada type
m=55, 4m=220=11^2+9^2+3^2+3^2, one solution
1-1--1-1-11--1-11111-111--1111--111-11111-1--11-1-1--1-
11-11-1-1--11-1-----1---11----11---1-----1-11--1-1-11-1
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
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(2-3-4-6+7+10-12+15+22+26+27)
\nu_4 = 1 + 2\omega(1-5+8-9+11-13+14-16-17-18-19+20-21-23+24+25)
This solution is of Turyn type.
m=57, 4m=228=9^2+7^2+7^2+7^2, one solution
1---11-1--1111-111-11---1-111111-1---11-111-1111--1-11---
1111--1-11----1---1--111-1------1-111--1---1----11-1--111
11-1-1---11-11-1--111-----1----1-----111--1-11-11---1-1-1
11-1-1---11-11-1--111-----1----1-----111--1-11-11---1-1-1
\nu_1 = 1
\nu_2 = 1
\nu_3 = 1 + 2\omega(-2+5-6-8+10+12+13-14+15+19+20-21-22-23-25+26)
\nu_4 = 1 + 2\omega(1+3-4-7+9-11-16-17+18-24-27-28)
This solution is of Turyn type.
m=61, 4m=244=11^2+11^2+1^2+1^2, one solution
11--1--11--1-1-1111--1-----1------1-----1--1111-1-1--11--1--1
11--1--11--1-1-1111--1-----1------1-----1--1111-1-1--11--1--1
1---1-1-1111---11--1-11-1---111111---1-11-1--11---1111-1-1---
1111-1-1----111--11-1--1-111------111-1--1-11--111----1-1-111
\nu_1 = 1 + 2\omega(-2-3+4-5+8+11-12-14+15+16-20+21-23-25-26)
\nu_2 = 1 + 2\omega(1-6+7-9-10+13+17+18-19-22-24+27-28-29-30)
\nu_3 = 1
\nu_4 = 1
This solution is of Yamada type: A, 11A=A, C, 11C.
m=63, 4m=252=11^2+11^2+3^2+1^2, one solution
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
11-111--11-11--1--1-11-1-111--------111-1-11-1--1--11-11--111-1
1-1---11--1--11-11-1--1-1---11111111---1-1--1-11-11--1--11---1-
\nu_1 = 1 + 2\omega(1+3+5-7+8+11+12-14+15-16-17-22+23+25-28)
\nu_2 = 1 + 2\omega(2-4+6-9+10+13-18+19-20-21+24-26-27+29+30+31)
\nu_3 = 1
\nu_4 = 1
This solution is of Turyn type.
Notes: The decomposition m=31, 4m=124=7^2+5^2+5^2+5^2, was found by
Turyn (1972) and the decomposition m=31, 4m=124=7^2+7^2+5^2+1^2, found
by Djokovic (1992).
The exhaustive search for the orders 29 and 31 is confirmed in:
D.Z.Djokovic, Williamson matrices of orders 4.29 and 4.31, J.Combin.
Theory, Ser. A, 59 (1992), 309-311.
D.Z.Djokovic, Williamson matrices of order 4n for n=33,35,39, Discrete
Math. 115 (1993), 267-271.
D.Z.Djokovic, Good matrices of orders 33,35 and 127 exist, JCMCC, 14
(1993), 145-152.
Updated 4th August 2001.
The material for m=51, 4m=204=11^2+7^2+5^2+3^2
was provided by Rudy van Vliet. Van Vliet states "I am confident that also
the results for m=37, m=39, m=45 and m=51 (given here plus my new solution) are
complete. My confidence is based on a comparison of the results from my
computer program, and those from the program that Dragomir Djokovic used. Because
the (intermediate and final) results of our programs were equal, we concluded
that both our programs were correct."
Modified version herewith provided by Rudy van Vliet 11 August 2001.
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