Date: Mon, 1 Nov 1999 22:24:23 +0200 (EET) From: KOUKOUVINOS CHRISTOS

This list contains the so-called good matrices of order m, i.e. one circulant and three back circulant (1,-1) matrices A,B,C,D of order m where A is circulant and skew-type and B,C,D are symmetric (before and after being made) back-circulant and they satisfy the matrix equation

AA^T + BB^T + CC^T + DD^T = 4m I_m

Then using A,B,C,D, where A is circulant and B,C,D are backcirculant, in the following Williamson array, we can construct
skew 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

m=3, 4m=12=3^2+1^2+1^2+1^2, one solution
11-
1--
1--
111

m=5, 4m=20=3^2+3^2+1^2+1^2, one solution
111--
1-11-
1----
1----

m=7, 4m=28=5^2+1^2+1^2+1^2, one solution
1111---
1-1--1-
1--11--
1------

m=7, 4m=28=3^2+3^2+3^2+1^2, two solutions
111-1--
111--11
11-11-1
1-1111-

1111---
11-11-1
11-11-1
1-1111-

m=9, 4m=36=5^2+3^2+1^2+1^2, one solution
1111-1---
11-1--1-1
1---11---
111-11-11

m=11, 4m=44=5^2+3^2+3^2+1^2, three solutions
11-1--11-1-
1111----111
1-111--111-
1---1--1---

11--1-1-11-
111--11--11
11-1-11-1-1
11--------1

11----1111-
111--11--11
1-1-1111-1-
1--1----1--

m=13, 4m=52=7^2+1^2+1^2+1^2, two solutions
11-1---111-1-
1---111111---
11-1--11--1-1
1----1--1----

11--1-1-1-11-
1--111--111--
111-1----1-11
1-----11-----

m=13, 4m=52=5^2+5^2+1^2+1^2, four solutions
11----1-1111-
11-11----11-1
1-1111--1111-
1-111-11-111-

11----1-1111-
11-1-1--1-1-1
111--1111--11
111-11--11-11

11111--11----
1-11--11--11-
1111-1--1-111
11-1-1111-1-1

11-----11111-
1--111--111--
111-1-11-1-11
1-111-11-111-

m=15, 4m=60=7^2+3^2+1^2+1^2, seven solutions
11111-1-1-1----
1---11-11-11---
111--1-11-1--11
111-111--111-11

111111-1-1-----
11---1-11-1---1
1-11-11--11-11-
111-111--111-11

111111-1-1-----
1-1--11--11--1-
111--1-11-1--11
11-1111--1111-1

11-111-1-1---1-
1---11-11-11---
11---111111---1
1-11-111111-11-

11-1111-1----1-
1-1---1111---1-
1--11-1111-11--
11-1111--1111-1

11-1111-1----1-
1-1---1111---1-
1-11--1111--11-
1111-11--11-111

11-1111-1----1-
11-1---11---1-1
111-1--11--1-11
11-1111--1111-1

m=15, 4m=60=5^2+5^2+3^2+1^2, four solutions
111111--11-----
1-11--1111--11-
1----1-11-1----
1-1---1--1---1-

11-11111-----1-
11--1-1111-1--1
1----11--11----
1-1---1--1---1-

1111-111---1---
1-111-1--1-111-
1--11------11--
1----1-11-1----

11---111---111-
1111--1--1--111
1----1-11-1----
1-1---1--1---1-

m=17, 4m=68=7^2+3^2+3^2+1^2, two solutions
11--11-1-1-1--11-
11--1--------1--1
111----1--1----11
11---1-1--1-1---1

111------111111--
1-----11--11-----
1---11-1--1-11---
1-1-1--1--1--1-1-

m=19, 4m=76=7^2+5^2+1^2+1^2, five solutions
1-1-----11--11111-1
11-11111----11111-1
1-1----11--11----1-
1--1-1-11--11-1-1--

1-1----1--11-1111-1
11-111-1-11-1-111-1
1--11----11----11--
1-1----111111----1-

1-1----111---1111-1
11-1111-1--1-1111-1
1---11--1--1--11---
1-111-1------1-111-

1-1---1111----111-1
1-1111-11--11-1111-
11-1-1--------1-1-1
1--1--111--111--1--

1-1---1111----111-1
1-11-1111--1111-11-
111--1--------1--11
1-1-1--11--11--1-1-

m=19, 4m=76=5^2+5^2+5^2+1^2, three solutions
111------1-111111--
1---1--1-11-1--1---
1-1----11--11----1-
1---1-11----11-1---

111-1-1-11--1-1-1--
1-11--1------1--11-
11----1--11--1----1
111----1----1----11

11--1111-1-1----11-
1---1-1--11--1-1---
1-11-1--------1-11-
11-----11--11-----1

m=21, 4m=84=9^2+1^2+1^2+1^2, four solutions
11--1111111-------11-
111--1111-11-1111--11
1--11-1-1-11-1-1-11--
1---111-1-11-1-111---

11------111---111111-
1-111--11111111--111-
11-11--1-1--1-1--11-1
1-1-11-11----11-11-1-

111--1--1-1-1-11-11--
111111-1--11--1-11111
1---111-1-11-1-111---
11-11----1111----11-1

11-1--1---1-111-11-1-
11-1--1111111111--1-1
1---111-1-11-1-111---
1--11---111111---11--

m=21, 4m=84=7^2+5^2+3^2+1^2, six solutions
111111-11-1-1--1-----
111--1-111--111-1--11
11-1-----1--1-----1-1
1-1--11---11---11--1-

111111--1-1-1-11-----
111--111--11--111--11
11-1--1--------1--1-1
1-1--11-1----1-11--1-

11111-11--1-11--1----
1-11111---11---11111-
1-----1-1-11-1-1-----
1-11---1--11--1---11-

1111-111--1-11---1---
11--1--11111111--1--1
1---1-1--1--1--1-1---
111---1-1----1-1---11

1111-11-11-1--1--1---
11-1111-1----1-1111-1
1-1-----11--11-----1-
11-1---11----11---1-1

1111-1-111-1---1-1---
11111---1-11-1---1111
11--1----1--1----1--1
1---11-1--11--1-11---

m=23, 4m=92=9^2+3^2+1^2+1^2, six solutions
111-1-------1111111-1--
11----1--1----1--1----1
11-1--11---11---11--1-1
1--11-1-1-1111-1-1-11--

11---1-1----1111-1-111-
11--1------11------1--1
111---1-1--11--1-1---11
111-11--1-1--1-1--11-11

11-1--11----1111--11-1-
1-11---1--------1---11-
1--11-111------111-11--
111-1-1-11----11-1-1-11

111-1----1--11-1111-1--
1--1-----11--11-----1--
1--1-1---111111---1-1--
1---111-1-1111-1-111---

111-1-----1-1-11111-1--
11---1----1--1----1---1
1--1---111-11-111---1--
11-11-1---1111---1-11-1

11-1--1---1-1-111-11-1-
1--1-----11--11-----1--
1-1----1111--1111----1-
1111-1---11--11---1-111

m=25, 4m=100=9^2+3^2+3^2+1^2, three solutions
11-----1-1---111-1-11111-
11-1111-1-11--11-1-1111-1
1---1--1111----1111--1---
1--1-111--1----1--111-1--

111--1--1----1111-11-11--
11-111---11111111---111-1
111--1-1-1------1-1-1--11
1--1-1---11-11-11---1-1--

11---1-111---111---1-111-
1-1-11-11-111111-11-11-1-
111-11------11------11-11
1---11-1-11----11-1-11---

m=25, 4m=100=7^2+5^2+5^2+1^2, six solutions
11--1---1----1111-111-11-
11-1----1---11---1----1-1
111--1111-1----1-1111--11
11-1-11--1-1111-1--11-1-1

11---1-----1-1-11111-111-
11----1-1--1--1--1-1----1
11-111---11-11-11---111-1
11-11-1---111111---1-11-1

11-1-11------111111--1-1-
11--1-1---1----1---1-1--1
111-111-1---11---1-111-11
111---11-11-11-11-11---11

11--1--1-1---111-1-11-11-
11-----1-1-1--1-1-1-----1
1-11----1111111111----11-
1--11--111-1111-111--11--

111----1---1-1-111-1111--
111------1-1--1-1------11
1--1-111--111111--111-1--
11-1-11-11--11--11-11-1-1

11--11-1-1---111-1-1--11-
11-1--1----1--1----1--1-1
1---1111--111111--1111---
1-11---1-11111111-1---11-

m=27, 4m=108=9^2+5^2+1^2+1^2, six solutions
1111-------1--11-1111111---
1-1---1---11----11---1---1-
1-11-1-----11--11-----1-11-
1-1--111-1--1--1--1-111--1-

111------11-1-1-1--111111--
1----1--11-1----1-11--1----
11-1--1--1---11---1--1--1-1
1111---1-1---11---1-1---111

1111-1-1-1----1111-1-1-1---
1--11---1----11----1---11--
11------11-1-11-1-11------1
111-11--1--1----1--1--11-11

11-1-111-1----1111-1---1-1-
11-1---1----1--1----1---1-1
1----111--1--11--1--111----
1--1----111-1111-111----1--

1111---1---1--11-111-111---
1-----1-11-1----1-11-1-----
1-1-11--11--------11--11-1-
1--1111--1-1----1-1--1111--

11-1---1--1111----11-111-1-
1----11-1-1------1-1-11----
11-11----11------11----11-1
1---1--11-1-1111-1-11--1---

m=27, 4m=108=7^2+7^2+3^2+1^2, six solutions
11---1-1--11-1-1--11-1-111-
1111---11-11-11-11-11---111
1111-111-11------11-111-111
11---1111-1-1--1-1-1111---1

11---1---1111-1----111-111-
1-11-11--1111--1111--11-11-
1-1--1--111111111111--1--1-
11-1---111-1-11-1-111---1-1

11-1-11-----11--11111--1-1-
11-11-11-1--1111--1-11-11-1
111111-1---11--11---1-11111
111---1-1-111--111-1-1---11

11-11-1---1---111-111-1--1-
1111---111-11--11-111---111
11111-11-1---11---1-11-1111
111--1-1-1--1111--1-1-1--11

11-11-1-----11--11111-1--1-
11-1---1111-1111-1111---1-1
11-11111-1---11---1-11111-1
111--11--1-1-11-1-1--11--11

111---1--1-11-1--1-11-111--
111--1-1111--11--1111-1--11
111111---1-11--11-1---11111
1-1-1---11-111111-11---1-1-

m=29, 4m=116=9^2+5^2+3^2+1^2, five solutions
11--1----1-1-11--1-1-1111-11-
11---1111-11--11--11-1111---1
11-1----11-1--11--1-11----1-1
1111111---1-1-11-1-1---111111

11--11----1-1--11-1-1111--11-
1-111-1---11-1111-11---1-111-
111----11--1-1--1-1--11----11
1-11---1-111111111111-1---11-

11--1111-----1-1-11111----11-
11--1-11-111--11--111-11-1--1
11--1-1-1--11----11--1-1-1--1
1111111-1---1-11-1---1-111111

111--1----11--1-11--1111-11--
11-11-11--1-1-11-1-1--11-11-1
11---1-1----111111----1-1---1
1-1111111-1---11---1-1111111-

11------1--11-1-1--11-111111-
11-1---1111-11--11-1111---1-1
11--11-1--1-1----1-1--1-11--1
11-1-111111---11---111111-1-1

m=31, 4m=124=11^2+1^2+1^2+1^2, two solutions
11--11----1---1-1-111-1111--11-
1-11-1---1-111----111-1---1-11-
111---1--1--111--111--1--1---11
1-11111111--1-1--1-1--11111111-

11--1--1---1----1111-111-11-11-
11-11-1-1-----1111-----1-1-11-1
11-11---11--1-1--1-1--11---11-1
111-1-111---11111111---111-1-11

m=31, 4m=124=7^2+7^2+5^2+1^2, one solution
11--------111--1-11---11111111-
11---1-1111-11-11-11-1111-1---1
1-1-1111---11-1111-11---1111-1-
1-1-11--1--1---11---1--1--11-1-

m=33, 4m=132=11^2+3^2+1^2+1^2, two solutions
111-111111-11----1111--1------1--
1-1111--1-1---1-11-1---1-1--1111-
1-1----111-11--1--1--11-111----1-
1---1---11----1-11-1----11---1---

1-11111-1-1-11---111--1-1-1-----1
111--1-1--1-11--11--11-1--1-1--11
11-----1-1-111--11--111-1-1-----1
111--1--1-------11-------1--1--11

m=33, 4m=132=9^2+7^2+1^2+1^2, one solution
1-1--1-11-111-111---1---1--1-11-1
1-1-1--11--111111111111--11--1-1-
111-----11--1111--1111--11-----11
1-1---11-1----1-11-1----1-11---1-

m=33, 4m=132=9^2+5^2+5^2+1^2, one solution
111-1---1-1-11---111--1-1-111-1--
11--1-11-1--1111111111--1-11-1--1
1--111-111--1-11--11-1--111-111--
11-1-----111-11111111-111-----1-1

m=35, 4m=140=11^2+3^2+3^2+1^2, one solution
1111-1-11----11-11--1--1111----1---
1-1111111---11-11--11-11---1111111-
1-1-11---11-1-11-11-11-1-11---11-1-
1-1-111111--11--------11--111111-1-

m=127, 4m=508=1^2+13^2+13^2+13^2, at least one solution
The following sets are 4-{127;63,70,70,70;146} sds say S_0, S_1, S_2, S_3 with the extra properties
i belongs S_0 iff m-1 doesn't belongs S_0
i belongs S_k iff m-1 belongs S_k, k=1,2,3

The sets are:

S_0 = {1,2,4,8,16,32,64,3,6,12,24,48,65,96,47,61,87,94,107,117,122, 7,14,28,56,67,97,112,9,17,18,34,36,68,72,11,22,44,49,69,88,98, 23,46,57,75,92,101,114,19,25,38,50,73,76,100,21,37,41,42,74,81,82}

S_1 = {1,2,4,8,16,32,64,63,95,111,119,123,125,126,3,6,12,24,48,65,96, 31,62,79,103,115,121,124,5,10,20,33,40,66,80,47,61,87,94,107,117,122, 11,22,44,49,69,88,98,29,39,58,78,83,105,116,19,25,38,50,73,76,100, 27,51,54,77,89,102,108}

S_2 = {1,2,4,8,16,32,64,63,95,111,119,123,125,126,5,10,20,33,40,66,80, 47,61,87,94,107,117,122,7,14,28,56,67,97,112,15,30,60,71,99,113,120, 9,17,18,34,36,68,72,55,59,91,93,109,110,118,19,25,38,50,73,76,100, 27,51,54,77,89,102,108}

S_3 = {1,2,4,8,16,32,64,63,95,111,119,123,125,126,5,10,20,33,40,66,80, 47,61,87,94,107,117,122,13,26,35,52,70,81,104,23,46,57,75,92,101,114, 19,25,38,50,73,76,100,27,51,54,77,89,102,108,21,37,41,42,74,81,82, 45,46,53,85,86,90,106}

Created 2nd November 1999. Matrices used supplied by Dr Christos Koukouvinos. Koukouvinos has verified the list is complete upto and including m=31. Partial results for m=33, m=35 were found by Djokovic (1993). The result for m=127, found by Djokovic (1993), was added 3rd November 1999.