# 17×17 {-1, +1} matrices of maximal determinant

|Det Rj| = 327680×216 = 20×47×216 for j=1, 2, 3

Ratio of |Det Rj| to Barba bound: 0.870388

M=RjTRj=Rj RjT:

```17 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3
-3 17  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
-3  1 17  1  1  1  1  1  1  1  1  1  1  1  1  1  1
-3  1  1 17  1  1  1  1  1  1  1  1  1  1  1  1  1
-3  1  1  1 17  1  1  1  1  1  1  1  1  1  1  1  1
-3  1  1  1  1 17  1  1  1  1  1  1  1  1  1  1  1
-3  1  1  1  1  1 17  1  1  1  1  1  1  1  1  1  1
-3  1  1  1  1  1  1 17  1  1  1  1  1  1  1  1  1
-3  1  1  1  1  1  1  1 17  1  1  1  1  1  1  1  1
-3  1  1  1  1  1  1  1  1 17  1  1  1  1  1  1  1
-3  1  1  1  1  1  1  1  1  1 17  1  1  1  1  1  1
-3  1  1  1  1  1  1  1  1  1  1 17  1  1  1  1  1
-3  1  1  1  1  1  1  1  1  1  1  1 17  1  1  1  1
-3  1  1  1  1  1  1  1  1  1  1  1  1 17  1  1  1
-3  1  1  1  1  1  1  1  1  1  1  1  1  1 17  1  1
-3  1  1  1  1  1  1  1  1  1  1  1  1  1  1 17  1
-3  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 17
```

R1:

```+++++++++++++++++
+----------++++++
+------++++----++
+++----+--++--+--
++--+---+-+-++---
+--+-+---++++----
+-+---+-++--+-+--
+---+-++-+-+-+---
+--+-+-++----++--
+---+++---+---+-+
+-++--+---+--+-+-
+++--+---+---+--+
++---+++----+--+-
++-+--+-+--+----+
++-++----+----++-
+-+++--+----+---+
+-+-++--+--+---+-
```
R2:
```+++++++++++++++++
+----------++++++
+------++++----++
+++----+--++--+--
++--+---+-+-++---
+--+-+---++++----
+-+---+-++--+-+--
+---+-++-+-+-+---
+--+-+-++----++--
+---+++---+---+-+
+-++--+---+--+-+-
+++--+---+---+--+
++-+--++----+---+
++---++-+--+---+-
++-++----+----++-
+-+-++-+----+--+-
+-+++---+--+----+
```
R3:
```+++++++++++++++++
+----------++++++
+------++++----++
+--+--+--++-++---
+-+--+--+-++-+---
++--+--+--+++----
+--+--+++--+--+--
++--+---++---++--
+-+--+-+-+--+-+--
+---+++---+---+-+
++++------+---++-
+--+++---+-+---+-
++---+++-----+-+-
+-+-+-+-+---+--+-
++-+-+--+---+---+
+-+++--+-----+--+
+++---+--+-+----+
```

Notes:

1. Cannot achieve Barba bound as 33=2×17-1 is not a perfect square.
2. A construction that achieves the maximal determinant was discovered by Schmidt [Sc] and conjectured to be optimal. A proof of optimality was given by Moyssiadis and Kounias [MK].
3. The set of three inequivalent matrices Rj is complete [MK][Hu]. (See also [O3].)

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Page created 26 January 2002.