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

|Det Rj| = 15237476352×224 = 42×611×224

Ratio of |Det Rj| to Barba bound: 1

M=RjTRj= Rj RjT:

```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:

```+++++++++----------------
++----+--++++++----------
++----+--------++++++----
----+++------++++------++
------+++++------++----++
--++--+----++------++--++
+-+-+-----+---+---++--+-+
+----+-+-+-+----+---+-+-+
+--+----+---++-+-+----+-+
-+-+-+---+----+--+-+--++-
-+--+--+--+-+--+----+-++-
-++-----+--+-+--+-+---++-
+-+--+----+--+---+--++-+-
+---+---++--+---+--+-+-+-
+--+---+---+--++--+--+-+-
-+-++----+---+----+-++--+
-+---+--+-++---+---+-+--+
-++----+----+-+-++---+--+
--++--+--++----++----++--
--+-+--+-+-+-+-+-+-+-----
--+--+--++--+-++--+-+----
----+++----++----++--++--
---+-+-+--+-++--+-++-----
---++---+-++--+-++--+----
------+++----++----++++--
```
R2:
```+--++----++----++----++--
+--++--++----++----++----
+--++------++----++----++
+++------++------++++----
+++----++------++------++
+++--------++++------++--
+----++--++--++--------++
+----++++--------++--++--
+----++----++--++--++----
--------++--+-++--+-++--+
-------+-+-+-+-+-+-+-+-+-
--------+-++--+-++--+-++-
-------+--+-++--+-++--+-+
--++--+------+--++--++--+
-+-+-+--------+-+-++-+-+-
--+-++-------+-+--+-+-++-
-+--+-+-------++-+-+--+-+
--++--+-++--+------+--++-
-+-+-+-+-+-+--------+-+-+
--+-++--+-++-------+-+--+
-+--+-++--+-+-------++-+-
--++--++--++--++--+------
-+-+-+--+-+-++-+-+-------
--+-++-+-+--+-+-++-------
-+--+-+-++-+-+--+-+------
```

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.

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Page created 26 January 2002.