# 15×15 {-1, +1} matrix of maximal determinant

|Det R| = 25515×214 = 105×35×214

Ratio of |Det R| to Ehlich bound: 0.970725

M=RTR=R RT:

```15  3  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
3 15  3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
3  3 15 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 15  3  3  3 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1  3 15  3  3 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1  3  3 15  3 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1  3  3  3 15 -1 -1 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 15  3  3  3 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1  3 15  3  3 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1  3  3 15  3 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1  3  3  3 15 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 15  3  3  3
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3 15  3  3
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3  3 15  3
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  3  3  3 15
```

R:

```----+++++-++++-
---+-++-++++-++
---++-++-++-+++
-++--+++-++----
++-+--+++-+----
+-+-+-+-+++----
++++++-++++--+-
-++-----+-+-+++
+++---++++-++++
+-+----+--++-++
++-------+++++-
++--+++------++
+-++-++-----++-
-++++-+----+-+-
+++++++---+++-+
```

Notes:

1. Ehlich bound is not achievable as it is not an integer.
2. This determinant was independently found by Smith [Sm] and Cohn [C1,C4] who conjectured it to be maximal. A proof of optimality was given by Orrick [O1].
3. M has block form.
4. R is unique up to equivalence [O1].

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Page created 26 January 2002.