Date: Sun, 31 Oct 1999 10:38:09 +0200 (EET)
From: KOUKOUVINOS CHRISTOS <ckoukouv@cc.uoa.gr>
<p>
In the following list we present some D-optimal designs of order n=2mod4
which are of circulant type.
If n=2m, m odd, and A, B are mxm commuting matrices, with elements 1,-1,
such that
<p>
 AA^T + BB^T = (2m-2) I_m + 2J_m   (1)
<p>
where J_m is an mxm matrix of 1's, and I_m is the identity matrix of
order m, then the nxn matrix
<br>
        ~~~~~A ~~~~~     B<br>
  R =<br>
      ~~~-B^T~~~~    A^T
<br>
has the maximum determinant among all nxn, 1,-1 matrices. Such matrices
are called D-optimal designs of order n.
<br>If the matrices A, B are circulant, then the corresponding D-optimal
designs are called circulant. In this case we can form two sets
P={p_1,p_2,...,p_r} and Q={q_1,q_2,...,q_s} where p_i, q_j denote the
positions of -1's in the first row of A, B respectively. Since the
circulant matrices A, B satisfy the matrix equation (1), the corresponding
sets P, Q are supplementary difference sets 2-{m;r,s;\lambda}, where
$\lambda=r+s-(n-2)/4$ and $s \geq r \geq 0$ are found from
(m-2r)^2 + (m-2s)^2 = 4m-2.
<br>Hence the construction of the two circulant matrices A,B satisfying (1)
is equivalent to the construction of the corresponding supplementary
difference sets. For n=22,34,58,70,78,94  D-optimal designs satisfying (1)
do not exist because n-1 is not the sum of two squares.
<p>
In the following Table we give the non-equivalent supplementary difference
sets, i.e. the non-equivalent circulant D-optimal designs for n=2m,
m=1,3,5,7,9,13,15,19,21.
<br><p>
m=1, n=2; 2-{1;0,0;0} (this is a trivial case)
<br>
A_1={ }, B_1={ }
<br>
The corresponding circulant matrices A, B have first row (+) and (+)
respectively, where + stands for 1.
<br><p>

m=3, n=6; 2-{3;0,1;0} (this is also a trivial case)
<br>
A_1={ }, B_1={1}
<br>
The corresponding circulant matrices A, B have first row (+++) and (+-+)
respectively, where + stands for 1 and - stands for -1.
    1 1-1 1 1 1<br>
   -1 1 1 1 1 1<br>
    1-1 1 1 1 1<br>
   -1-1-1 1-1 1<br>
   -1-1-1 1 1-1<br>
   -1-1-1-1 1 1<br>

<br><p>
m=5, n=10; 2-{5;1,1;0} 
<br>
A_1={4}, B_1={4}
<br>
The corresponding circulant matrices A, B have first row (++++-) and
(++++-) respectively.
<br><p>
m=7, n=14; 2-{7;1,3;1}
<br>
A_1={6}, B_1={3,5,6}
<br>
The corresponding circulant matrices A, B have first row (++++++-) and
(+++-+--) respectively.
<br><p>
m=9, n=18; 2-{9;2,3;1}
<br>
A_1={7,8}, B_1={3,6,8}
<br>
The corresponding circulant matrices A, B have first row (+++++++--) and
(+++-++-+-) respectively.
<br><p>
m=13, n=26; 2-{13;3,6,3}
<br>
A_1={8,11,12}, B_1={3,4,5,7,10,12}
<br>A_2={8,11,12}, B_2={4,5,7,9,11,12}
<br>
The corresponding circulant matrices A, B have first row (++++++++-++--) and
(+++---+-++-+-) respectively for the first solution, and (++++++++-++--) and
(++++--+-+-+--) respectively for the second solution.
<br><p>
m=13, n=26; 2-{13;4,4;2}
<br>
A_1={5,7,8,12}, B_1={5,7,8,12}
<br>
The corresponding circulant matrices A, B have first row (+++++-+--+++-) and
(+++++-+--+++-) respectively.
<br><p>
m=15, n=30; 2-{15;4,6;3}
<br>
A_1={6,9,12,14}, B_1={3,7,8,9,10,14}
<br>A_2={8,9,12,14}, B_2={4,5,7,8,12,14}
<br>A_3={8,9,12,14}, B_3={5,6,9,11,13,14}
<br><p>
m=19, n=38; 2-{19;6,7;4}
<br>
A_1={6,7,8,10,13,18},   B_1={4,5,7,8,12,14,18}
<br>A_2={6,7,8,10,13,18},   B_2={5,6,10,12,14,15,18}
<br>A_3={7,8,9,11,13,18},   B_3={4,5,9,12,15,16,18}
<br>A_4={7,10,13,14,16,18}, B_4={6,7,8,11,13,17,18}
<br>A_5={7,8,9,12,15,18},   B_5={6,7,11,12,14,16,18}
<br>A_6={8,10,11,15,16,18}, B_6={4,5,7,8,12,14,18}
<br>A_7={8,10,11,15,16,18}, B_7={5,6,10,12,14,15,18}
<br>A_8={9,10,13,15,17,18}, B_8={4,5,7,11,13,14,18}
<br>
<br><p>
This page created 1st November 1999 with matrices supplied by Dr Christos
Koukouvinos.