20×20 {-1, +1} matrices of maximal determinant

|Det Rj| = 19531250×219 = 10×59×219 for j=1, 2, 3

Ratio of |Det Rj| to Hadamard bound: 1

M=RjTRj= Rj RjT for j=1, 2, 3: 

20  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0 20  0  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0 20  0  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0 20  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0 20  0  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0 20  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0 20  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0 20  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0 20  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0 20  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0 20  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 20  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 20  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 20  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 20  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 20

R1: 

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

Notes:

  1. These matrices taken from N. J. A. Sloane's Library of Hadamard matrices.
  2. Maximal matrix first reported by Hadamard [Had].
  3. The number of inequivalent Hadamard matrices of order 20 was shown to be three by Hall [Hal].
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Page created 26 January 2002.
Last modified 26 January 2003.
Comments: maxdet@indiana.edu