Conjectured 22×22 {-1, +1} matrix of maximal determinant

|Det R| = 195312500×221 = 100×59×221

Ratio of |Det R| to Ehlich/Wojtas bound: 0.952381

M=RTR=R RT:

22  2  2  2  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0  0
 2 22  2  2  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0  0
 2  2 22  2  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0  0
 2  2  2 22  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0  0
 2  2  2  2 22  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0  0
 2  2  2  2  2 22  2  2  2  2 -2 -2  0  0  0  0  0  0  0  0  0  0
 2  2  2  2  2  2 22  2  2  2 -2 -2  0  0  0  0  0  0  0  0  0  0
 2  2  2  2  2  2  2 22  2  2 -2 -2  0  0  0  0  0  0  0  0  0  0
 2  2  2  2  2  2  2  2 22  2 -2 -2  0  0  0  0  0  0  0  0  0  0
 2  2  2  2  2  2  2  2  2 22 -2 -2  0  0  0  0  0  0  0  0  0  0
 2  2  2  2  2 -2 -2 -2 -2 -2 22  2  0  0  0  0  0  0  0  0  0  0
 2  2  2  2  2 -2 -2 -2 -2 -2  2 22  0  0  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  0  0  0 22  2  2  2  2  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  0  0  0  2 22  2  2  2  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  0  0  0  2  2 22  2  2  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  0  0  0  2  2  2 22  2  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  0  0  0  2  2  2  2 22  2  2  2  2  2
 0  0  0  0  0  0  0  0  0  0  0  0  2  2  2  2  2 22  2  2  2  2
 0  0  0  0  0  0  0  0  0  0  0  0  2  2  2  2  2  2 22  2  2  2
 0  0  0  0  0  0  0  0  0  0  0  0  2  2  2  2  2  2  2 22  2  2
 0  0  0  0  0  0  0  0  0  0  0  0  2  2  2  2  2  2  2  2 22  2
 0  0  0  0  0  0  0  0  0  0  0  0  2  2  2  2  2  2  2  2  2 22

R:

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

Notes:

  1. Cannot achieve Ehlich/Wojtas bound since 21=22-1 is not the sum of two squares.
  2. This form has not been proved to be optimal.
  3. This lower bound was reported by Dowdeswell, Neubauer, Solomon and Tumer [DNST].
  4. Are M and R unique (up to equivalence)?

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Page created 26 January 2002.
Last modified 11 April 2003.
Comments: maxdet@indiana.edu