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 M₂₄). 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 M₂₄', 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