19×19 {-1, +1} matrices of maximal determinant

|Det Rj| = 3411968×218 = 833×46×218 for j=1, 2, 3

Ratio of |Det Rj| to Ehlich bound: 0.975017

M1=R1TR1=R1 R1T:

19  3  3  3  3  3  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 3 19  3  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 3  3 19  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 3  3  3 19 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 3 -1 -1 -1 19  3  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 3 -1 -1 -1  3 19  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 3 -1 -1 -1  3  3 19 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 19  3  3 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1  3 19  3 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1  3  3 19 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 19  3  3 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3 19  3 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3  3 19 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 19  3  3 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3 19  3 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3  3 19 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 19  3  3
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3 19  3
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3  3 19

R1:

+------++++++++++++
-+---------++-++-++
--+-------+-++-++-+
---+------++-++-++-
----+--++++---+---+
-----+-+++-+---++--
------++++--++---+-
++++----+++----+-+-
++++---+-+-+-+----+
++++---++---+-+-+--
++---+++--+--+++---
+-+-+-+-+--+-+++---
+--+++---+--++++---
++--++--+----+--+++
+-+--++--+----+-+++
+--++-++-------++++
++--+-+--++++---+--
+-+-++-+--+++----+-
+--+-++-+-+++-----+
M2=RjTRj=RjRjT for j=2,3:
 19 -1 -1  3 -1 -1  3 -1 -1  3 -1 -1 -1 -1 -1 -1 -1 -1 -1
 -1 19  3  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 -1  3 19  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
  3  3  3 19 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 -1 -1 -1 -1 19  3  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 -1 -1 -1 -1  3 19  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
  3 -1 -1 -1  3  3 19 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1 19  3  3 -1 -1 -1 -1 -1 -1 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1  3 19  3 -1 -1 -1 -1 -1 -1 -1 -1 -1
  3 -1 -1 -1 -1 -1 -1  3  3 19 -1 -1 -1 -1 -1 -1 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 19  3  3 -1 -1 -1 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3 19  3 -1 -1 -1 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3  3 19 -1 -1 -1 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 19  3  3 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3 19  3 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3  3 19 -1 -1 -1
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 19  3  3
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3 19  3
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3  3 19
R2:
-++-++-++----------
+---+-+-++---+-++--
+----+++-+----++-+-
---+++++++---++---+
+-++----+++------++
++-+---+-+-+----+-+
-+++--++++--+---++-
+-++-++----+++-----
++-++-+---+-+-+----
-++++++--+++---+---
-----+--+++++-+-+--
----+--+-+++++---+-
------+++-+++--+--+
--++---+--+--++++--
-+-+----+--+-+++-+-
-++------+--++++--+
-+---++---+--+--+++
--+-+-+----+--+-+++
---+++------+--++++
R3:
-++-++-++----------
+----++-++----+++--
+---+-++-+---+-+-+-
---+++++++---++---+
+-++----+++------++
++-+---+-+-+----+-+
-+++--++++--+---++-
+-++-++----+++-----
++-++-+---+-+-+----
-++++++--+++---+---
----+---++++++--+--
-----+-+-++++-+--+-
------+++-+++--+--+
--++---+--+--++++--
-+-+----+--+-+++-+-
-++------+--++++--+
-+---++---+--+--+++
--+-+-+----+--+-+++
---+++------+--++++

Notes:

  1. Ehlich bound is not achievable as it is not an integer.
  2. This determinant was proved maximal by R. Brent, W. Orrick, J. Osborn, and P. Zimmermann [BOOZ]
  3. It was first reported in [Sm].
  4. Neither M1 nor M2 is of block form.
  5. M1 and M2 have been proved to be the only forms that achieve this determinant value (up to equivalence) [BOOZ].
  6. R1 is equivalent to a matrix found by Smith [Sm].
  7. R2 is equivalent to a matrix found by Cohn [C4].
  8. R3 was constructed by W. Orrick and B. Solomon, 15 May 2003.
  9. R1, R2, and R3 have been proved to be the only matrices (up to equivalence) associated with the given matrices Mj [BOOZ].

Back to maximal determinant main page.
Page created 30 August 2002.
Last modified 20 December 2011.
Comments: maxdet@indiana.edu