# 106×106 {-1, +1} matrices of largest known determinant

|Det Rj| = 3941372061724234462448657884773553789208582127386093522798996947985764450304×2105 = 2704×2651×2105

Ratio of |Det Rj| to Ehlich/Wojtas bound: 0.990476

M=RjTRj= Rj RjT:

Rj:
The matrices of have the form

``` | X   J   K |
|           |
|  T        |
| J   A   B |
|           |
|  T   T   T|
| K  -B   A |
```
where X is an arbitrary 2×2 matrix, J is a 2×52 matrix of 1s, K is a 2×52 matrix of the form
``` ++++++...+
------...-
```
and A and B are 52×52 circulant matrices.

There are 3 known choices for the first rows of A and B:
a1, b1:

```+++++++-+-+-+-+-++-++-+--+-+-++-+---++-----+----++--
+--++--+-++-+++---+-+++-----+++++-+++-----++--++--+-
```
a2, b2:
```+++++++-+++-+-+-+---++--+----+-+--++--+-++--++-+----
+-+++--+----+-+++--++-+--+--+-+-+++------++++--++++-
```
a3, b3:
```+++++++-+-+-+-+--+--+++--+++--++------+----++++--+--
+----+-++-+----+--+++++-++--+-+-+-+-++-++--++-+---++
```

Notes:

1. Cannot achieve Ehlich/Wojtas bound since 105=106-1 is not the sum of two squares.
2. This form has not been proved to be optimal.
3. This determinant was discovered by Will Orrick on 24 April 2005.
4. From the row-pairs above, 60 inequivalent matrices can be obtained by modifying the corner, X, and by transposition.

Back to maximal determinant main page.
Page created 24 April 2005.