18×18 {-1, +1} matrices of maximal determinant

Ratio of |Det R_j| to Ehlich/Wojtas bound: 1

M=R_j^TR_j= R_j R_j^T for j=1, 2 , 3:

18  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0
 2 18  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0
 2  2 18  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0
 2  2  2 18  2  2  2  2  2  0  0  0  0  0  0  0  0  0
 2  2  2  2 18  2  2  2  2  0  0  0  0  0  0  0  0  0
 2  2  2  2  2 18  2  2  2  0  0  0  0  0  0  0  0  0
 2  2  2  2  2  2 18  2  2  0  0  0  0  0  0  0  0  0
 2  2  2  2  2  2  2 18  2  0  0  0  0  0  0  0  0  0
 2  2  2  2  2  2  2  2 18  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0 18  2  2  2  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  2 18  2  2  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  2  2 18  2  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  2  2  2 18  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  2  2  2  2 18  2  2  2  2
 0  0  0  0  0  0  0  0  0  2  2  2  2  2 18  2  2  2
 0  0  0  0  0  0  0  0  0  2  2  2  2  2  2 18  2  2
 0  0  0  0  0  0  0  0  0  2  2  2  2  2  2  2 18  2
 0  0  0  0  0  0  0  0  0  2  2  2  2  2  2  2  2 18

R₁:

--+++++++-+++-++-+
+--+++++++-+++-++-
++--+++++-+-+++-++
+++--+++++-+-+++-+
++++--+++++-+-+++-
+++++--++-++-+-+++
++++++--++-++-+-++
+++++++--++-++-+-+
-+++++++-+++-++-+-
+-+--+----+++++++-
-+-+--+----+++++++
--+-+--+-+--++++++
---+-+--+++--+++++
+---+-+--+++--++++
-+---+-+-++++--+++
--+---+-++++++--++
+--+---+-++++++--+
-+--+---++++++++--

--+++++++-+-+++-++
+--+++++++-+++-++-
++--+++++-+++-++-+
+++--+++++++-+--++
++++--+++++-+-+++-
-++++-++++-+-+++-+
++++++--+-++-++++-
+++++++--++-++-+-+
++++++-+-+-++-+-++
+-+---+----+++++++
-+---+--++--++++++
+---+--+-++--+++++
---+-++--+++--++++
--+-+---+++++--+++
-+-+---+--++++-+++
+--+----+++++++--+
--+--+-+-+++++++--
-+--+-+--++++++-+-

--+++++++++-+-++-+
+--+++++++-+-++-++
-+-++++++-++++-++-
+++--+++++-+++-+-+
++++--+++-+++-+-++
+++-+-+++++--++++-
++++++--++-++-+++-
+++++++---++-+++-+
++++++-+-++-++--++
--+-+--+---+++++++
-+-+--+--+--++++++
+----+--+-+-++++++
-+---+-+-+++--++++
+---+-+--++++--+++
--++----++++-+-+++
-+--+---+++++++--+
+--+---+-+++++++--
--+--++--++++++-+-

Notes:

1. R₁ has the form of circulant 9×9 blocks; R₂ and R₃ do not. All three are self-dual.
2. The complete set of three circulant block forms found by Yang [Y3] are all equivalent to R₁. Two of these were given by Ehlich [E1].
3. A matrix equivalent to R₃ was given by Chadjipantelis, Kounias, and Moyssiadis [CKM1]. It is a multi-circulant matrix related to a supplementary difference set construction on the group C₃×C₃.
4. Cohn has proved that {R₁, R₂, R₃} is the complete set of maximal forms up to equivalence [C3].
5. The first reference to R₂ appears to be Cohn [C3].