# 34×34 {-1, +1} matrices of largest known determinant

|Det Rj| = 9007199254740992×233 = 256×815×233

Ratio of |Det Rj| to Ehlich/Wojtas bound: 0.969697

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 2×16 matrix of 1s, K is a 2×16 matrix of the form
``` ++++++...+
------...-
```
and A and B are 16×16 circulant matrices.

There are 2 distinct choices for the first rows of A and B:
a1, b1:

```+++++--+-+--+---      +++-+-+---+-++--
```
a2, b2:
```+++++--+---++---      +++-+-+-+--+--+-
```

Notes:

1. This determininant surpasses a previous record.
2. Cannot achieve Ehlich/Wojtas bound since 33=34-1 is not the sum of two squares.
3. This form has not been proved to be optimal.
4. This determinant was discovered by Will Orrick on 13 March 2005.
5. From the 2 row-pairs above, 40 inequivalent matrices can be obtained by modifying the corner, X, and by transposition.

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Page created 19 April 2005.
Last modified 19 April 2005.
Comments: maxdet@indiana.edu