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

Ratio of |Det R_j| to Ehlich bound: 0.975017

M₁=R₁^TR₁=R₁ R₁^T:

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:

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

 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

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

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

Notes:

1. Ehlich bound is not achievable as it is not an integer.
2. This determinant value has not been proved maximal.
3. It was first reported in [Sm].
4. Neither M₁ nor M₂ is of block form.
5. M₁ and M₂ have not been proved to be the only forms that achieve this determinant value (up to equivalence).
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 not been proved to be the only matrices (up to equivalence) associated with the given matrices M_j.