11×11 {-1, +1} matrices of maximal determinant

Ratio of |Det R| to Ehlich bound: 0.944911

M₁=R₁^TR₁=R₁ R₁^T:

11  3  3 -1 -1 -1 -1 -1 -1 -1 -1
 3 11  3 -1 -1 -1 -1 -1 -1 -1 -1
 3  3 11  3 -1 -1 -1 -1 -1 -1 -1
-1 -1  3 11  3 -1 -1 -1 -1 -1 -1
-1 -1 -1  3 11 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 11  3 -1 -1 -1 -1
-1 -1 -1 -1 -1  3 11 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 11  3 -1 -1
-1 -1 -1 -1 -1 -1 -1  3 11 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 11  3
-1 -1 -1 -1 -1 -1 -1 -1 -1  3 11

R₁:

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

M₂=R₂^TR₂=R₂ R₂^T:

11  3  3 -1 -1 -1 -1 -1 -1 -1 -1
 3 11 -1 -1 -1 -1 -1 -1 -1 -1 -1
 3 -1 11  3 -1 -1 -1 -1 -1 -1 -1
-1 -1  3 11  3 -1 -1 -1 -1 -1 -1
-1 -1 -1  3 11 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 11  3 -1 -1 -1 -1
-1 -1 -1 -1 -1  3 11 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 11  3 -1 -1
-1 -1 -1 -1 -1 -1 -1  3 11 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 11  3
-1 -1 -1 -1 -1 -1 -1 -1 -1  3 11

R₂:

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

M₃=R₃^TR₃=R₃ R₃^T:

11  3  3  3  3 -1 -1 -1 -1 -1 -1
 3 11  3  3  3 -1 -1 -1 -1 -1 -1
 3  3 11  3  3 -1 -1 -1 -1 -1 -1
 3  3  3 11  3 -1 -1 -1 -1 -1 -1
 3  3  3  3 11 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 11  3 -1 -1 -1 -1
-1 -1 -1 -1 -1  3 11 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 11  3 -1 -1
-1 -1 -1 -1 -1 -1 -1  3 11 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 11  3
-1 -1 -1 -1 -1 -1 -1 -1 -1  3 11

R₃:

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

Notes:

1. Ehlich bound is not achievable as it is not an integer.
2. M₂ was found by Ehlich and Zeller [EZ].
3. M₁ and M₃ were found by Ehlich and, along with M₂, proved by him to be the complete set of optimal forms. The first published description appears in Galil and Kiefer [GK].
4. M₃ has block form. M₁ and M₂ do not.
5. The decompositions R₁, R₂ and R₃ are unique up to equivalence. (See [O3].)