38×38 {-1, +1} matrices of maximal determinant

|Det Rj| = 5553501505988967477×237 = 333×917×237 for j=1, 2 , ..., 8

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

M=RjTRj= Rj RjT for j=1, 2 , ..., 8:

    | S   0 |
M = |       |
    | 0   S |
with S = 36 I + 2 J where I is the 19×19 identity matrix and J is the 19×19 matrix with all entries 1.

There are eight inequivalent matrices composed of circulant blocks, A and B:

 | A   B |
 |       |
 |  T   T|
 |-B   A |
The first rows of A and B are:
R1:
+++++---+-++-++++-+      +++--+--+++-+-+++-+
R2:
+++++---+-++-++++-+      ++++--+++-+-+--++-+
R3:
++++++---+-+-++++-+      +++--+++-++-++--+-+
R4:
++++++-++-++--+-+-+      +++++---++-+-+++--+
R5:
++++++---++-++-++-+      +++++--+++--+-+-+-+
R6:
+++++++-+--+++--+-+      +++--+--+++-+-+++-+
R7:
+++++++-+--+++--+-+      ++++--+++-+-+--++-+
R8:
++++++++--++-+-+--+      +++--+-+++-+--+++-+

Notes:

  1. A maximal determinant was first reported by Ehlich [E1].
  2. The complete set of circulant block forms was found by Yang [Y3]. That there are eight inequivalent ones was shown by Kounias, Koukouvinos, Nikolaou and Kakos [KKNK1] who gave the forms listed here.
  3. Are there other inequivalent matrices not in the form of circulant blocks?

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This page created 10 March 2002.