25×25 {-1, +1} matrices of maximal determinant

Ratio of |Det R_j| to Barba bound: 1

M=R_j^TR_j= R_j R_j^T:

25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 25

R₁:

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

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

Notes:

1. A (25, 9, 3)-balanced incomplete block design was first constructed by Bhattacharya [Bh]. That the determinant of its (-1,1)-incidence matrix is maximal is implied by Barba's bound.
2. Raghavarao [R], apparently independently, was the first to discuss maximality of the determinant, and gave the matrix R₁ (with an apparent typo in the elements at positions (10, 9) and (10, 10) which we have corrected.)
3. R₂ was derived using the Brouwer [Br] construction for matrices of order 2q(q+1)+1 with q an odd prime.
4. Denniston [De] showed that there are 78 inequivalent forms. Representatives of each of the 78 equivalence classes have also been computed by Spence.