Previous 45×45 {-1, +1} matrix of largest known determinant

|Det R| = 620336654418268743029613×244 = 10143×1119×244

Ratio of |Det R| to Barba bound: 0.807781

M=RTR=R RT:

 45 -3  1  1  1  1  1  1  1  1 -3 -3 -3 -3 -3  5  5 -3  1  1  1  1  1  1  1  5  1  1  1  1  1  1  1  1  1  5  1  1  1  1  1  1  1  1  1 
 -3 45 -3 -3 -3 -3 -3 -3 -3 -3  1  1  1  1  1 -7 -7  1 -3 -3 -3 -3 -3 -3 -3 -3  1  1  1  1  1  1  1  1  1 -3  1  1  1  1  1  1  1  1  1 
  1 -3 45  1  5  5  5  5  5  5  1  1  1  1  1  5  5 -3  1  1  1  1  1  1  1  5  1  1  1  1  1  1  1  1  1  5  1  1  1  1  1  1  1  1  1 
  1 -3  1 45  5  5  5  5  5  5  1  1  1  1  1  5  5 -3  1  1  1  1  1  1  1  5  1  1  1  1  1  1  1  1  1  5  1  1  1  1  1  1  1  1  1 
  1 -3  5  5 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  5  5  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  5  5  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  5  5  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  5  5  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  5  5  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
 -3  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
 -3  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
 -3  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
 -3  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
 -3  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  5 -7  5  5  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  5 -7  5  5  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
 -3  1 -3 -3  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1 -3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  5 -3  5  5  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1  1 
  5 -3  5  5  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45  1 
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 45 

R:

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

Notes:

  1. This determinant has been surpassed by a new record.
  2. This matrix was constructed in August 2009 by [BOOZ] from a doubly-three-normalized Hadamard matrix of size 44 and excess 236 by the method used in size 29 in [OS]. The resulting matrix is not a local maximum under flips of single matrix elements since the diagonal elements of the 3×3 intersection of the normalizing rows and columns do not have the same sign as their cofactors. Negating a single one of these elements gives this matrix, which surpasses the old record based on the three-normalized maximal excess construction.
  3. Cannot achieve Barba bound as 89=2×45-1 is not a perfect square.

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Page created 5 October 2012.
Last modified 6 October 2012.
Comments: maxdet@indiana.edu