41×41 {-1, +1} matrices of maximal determinant

|Det R| = 900000000000000000000×240 = 90×1019×240

Ratio of |Det R| to Barba bound: 1

M=RTR=R RT=40 I + J
where I is the 41×41 identity matrix, and J is the matrix with all entries 1.

R:

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

Notes:

  1. This matrix comes from the symmetric block design with parameters (41, 16, 6) found by Bridges, Hall and Hayden [BHH].
  2. A design was also found by van Trung [vT].
  3. It has been established by Ted Spence [Sp2] that the number of inequivalent designs with an automorphism of order 5 (of which the above design is one) is 419, with an automorphism of order 3 fixing 11 points is 3076, and with an automorphism of order 3 fixing 5 points is greater than 112,000. In addition, there are at least 10 designs with trivial automorphism group [Sp3].

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Page created 26 January 2002.
Last modified 3 June 2003.
Comments: maxdet@indiana.edu