#
Moggie

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## What is Moggie?

Moggie is an applet written in Java for playing with the MOG.

## O.K. then, what's the MOG?

The MOG (or Miracle Octad Generator) is a tool for performing
calculations with the remarkable S(5,8,24) Steiner system and various
interesting finite groups (in particular, the Mathieu group M_{24}).
It was invented by
R. T. Curtis, and full
details can be found in Curtis' original paper [1]. Another account
is given by Conway [2].

Briefly, the points of the system are represented in a 6 × 4 grid,
and it is fairly easy to complete a set of 5 points to a block of the
Steiner system by hand. I wrote this program to help me while I was
still learning how to use the MOG from Conway's book.

## Disclaimer

Moggie is experimental code, and its functionality and interface
might change without notice (or they might not; the applet has been
unchanged for about a year now). It was written
by a mathematics student with not that much experience of Java, so it
is not very slick as applets go. Moreover, while I believe
that bug-free software
exists, I have never seen a constructive proof of the fact. In
other words, it is fairly likely that there are bugs. If
you find any, please send a bug report to me at
simonn@maths.bham.ac.uk.

## Instructions

There are 24 cells in the MOG array, arranged in a 6 x 4 grid.
A cell is red if it contains a 1 and white if
it contains a 0. Other colours have various meanings depending
on which button you've pressed. Here is a (brief)
explanation of what the buttons do.

- Complete - given 5 cells, find the 3 cells that give you
a special octad. (These 3 cells are unique; this is the
special property of an S(5,8,24) Steiner system.)
- Codeword? - say whether or not the cells selected in red
are in the Golay code.
- Clear - turn all cells back to 0.
- Homogeneous - turn all coloured cells to red (i.e. contain 1)
- Sextet - given 4 cells, find the corresponding sextet (i.e.
the other 5 tetrads such that the union of any two is a special
octad).
- EqClass - find the equivalence class of a word modulo the
Golay code. Any arrangement of the MOG array is equivalent to
a word of weight 4 or less. Golay codewords are equivalent to the
empty word. If a word is equivalent to a sextet, only one of the
tetrads will be shown.

## The applet

## Bibliography

[1] 'A new combinatorial approach to M_{24}',
R. T. Curtis,
*Math. Proc. Camb. Phil. Soc.* 1976, **79**, 24

[2] 'The Golay codes and the Mathieu groups', J. H. Conway,
chapter 11 in 'Sphere Packings, Lattices and Groups',
J. H. Conway & N. J. A. Sloane,
Springer-Verlag 1999