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

|Det Rj| = 1114112×217 = 68×47×217 for j=1, 2 , 3

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

M=RjTRj= Rj RjT 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:

--+++++++-+++-++-+
+--+++++++-+++-++-
++--+++++-+-+++-++
+++--+++++-+-+++-+
++++--+++++-+-+++-
+++++--++-++-+-+++
++++++--++-++-+-++
+++++++--++-++-+-+
-+++++++-+++-++-+-
+-+--+----+++++++-
-+-+--+----+++++++
--+-+--+-+--++++++
---+-+--+++--+++++
+---+-+--+++--++++
-+---+-+-++++--+++
--+---+-++++++--++
+--+---+-++++++--+
-+--+---++++++++--
R2:
--+++++++-+-+++-++
+--+++++++-+++-++-
++--+++++-+++-++-+
+++--+++++++-+--++
++++--+++++-+-+++-
-++++-++++-+-+++-+
++++++--+-++-++++-
+++++++--++-++-+-+
++++++-+-+-++-+-++
+-+---+----+++++++
-+---+--++--++++++
+---+--+-++--+++++
---+-++--+++--++++
--+-+---+++++--+++
-+-+---+--++++-+++
+--+----+++++++--+
--+--+-+-+++++++--
-+--+-+--++++++-+-
R3:
--+++++++++-+-++-+
+--+++++++-+-++-++
-+-++++++-++++-++-
+++--+++++-+++-+-+
++++--+++-+++-+-++
+++-+-+++++--++++-
++++++--++-++-+++-
+++++++---++-+++-+
++++++-+-++-++--++
--+-+--+---+++++++
-+-+--+--+--++++++
+----+--+-+-++++++
-+---+-+-+++--++++
+---+-+--++++--+++
--++----++++-+-+++
-+--+---+++++++--+
+--+---+-+++++++--
--+--++--++++++-+-

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].

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Page created 26 January 2002.
Last modified 13 November 2005.
Comments: maxdet@indiana.edu