# 11×11 {-1, +1} matrices of maximal determinant

|Det Rj| = 320×210 = 40×23×210 for j = 1, 2, 3

Ratio of |Det R| to Ehlich bound: 0.944911

M1=R1TR1=R1 R1T:

```11  3  3 -1 -1 -1 -1 -1 -1 -1 -1
3 11  3 -1 -1 -1 -1 -1 -1 -1 -1
3  3 11  3 -1 -1 -1 -1 -1 -1 -1
-1 -1  3 11  3 -1 -1 -1 -1 -1 -1
-1 -1 -1  3 11 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 11  3 -1 -1 -1 -1
-1 -1 -1 -1 -1  3 11 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 11  3 -1 -1
-1 -1 -1 -1 -1 -1 -1  3 11 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 11  3
-1 -1 -1 -1 -1 -1 -1 -1 -1  3 11
```

R1:

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

M2=R2TR2=R2 R2T:

```11  3  3 -1 -1 -1 -1 -1 -1 -1 -1
3 11 -1 -1 -1 -1 -1 -1 -1 -1 -1
3 -1 11  3 -1 -1 -1 -1 -1 -1 -1
-1 -1  3 11  3 -1 -1 -1 -1 -1 -1
-1 -1 -1  3 11 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 11  3 -1 -1 -1 -1
-1 -1 -1 -1 -1  3 11 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 11  3 -1 -1
-1 -1 -1 -1 -1 -1 -1  3 11 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 11  3
-1 -1 -1 -1 -1 -1 -1 -1 -1  3 11
```

R2:

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

M3=R3TR3=R3 R3T:

```11  3  3  3  3 -1 -1 -1 -1 -1 -1
3 11  3  3  3 -1 -1 -1 -1 -1 -1
3  3 11  3  3 -1 -1 -1 -1 -1 -1
3  3  3 11  3 -1 -1 -1 -1 -1 -1
3  3  3  3 11 -1 -1 -1 -1 -1 -1
-1 -1 -1 -1 -1 11  3 -1 -1 -1 -1
-1 -1 -1 -1 -1  3 11 -1 -1 -1 -1
-1 -1 -1 -1 -1 -1 -1 11  3 -1 -1
-1 -1 -1 -1 -1 -1 -1  3 11 -1 -1
-1 -1 -1 -1 -1 -1 -1 -1 -1 11  3
-1 -1 -1 -1 -1 -1 -1 -1 -1  3 11
```

R3:

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

Notes:

1. Ehlich bound is not achievable as it is not an integer.
2. M2 was found by Ehlich and Zeller [EZ].
3. M1 and M3 were found by Ehlich and, along with M2, proved by him to be the complete set of optimal forms. The first published description appears in Galil and Kiefer [GK].
4. M3 has block form. M1 and M2 do not.
5. The decompositions R1, R2 and R3 are unique up to equivalence. (See [O3].)

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Page created 26 January 2002.