The Witt Design

In the 1860s and 70s, French mathematician Émile Mathieu discovered five remarkable groups of permutations, now known as the Mathieu groups, which play a vital role in the twentieth century discovery of all the basic building blocks for symmetry, described in my book Symmetry and the Monster. The largest, known as M_24, permutes 24 symbols in such a way that any sequence of five symbols can be sent to any other. No other group — except Mathieu’s group M₁₂, which permutes 12 symbols — can do this unless it contains all even permutations. In 1934/35, Ernst Witt constructed a remarkable design using 24 symbols and having M₂₄ as its group of symmetries; it is called the Witt design.

Witt’s design has been used to obtain the Leech Lattice and the Golay Code. It is a collection of subsets, called octads, each having 8 symbols, with the property that each set of 5 symbols lies in exactly one octad. The number of octads must be 759, as we now demonstrate.

First count the number of sequences of five symbols. We are choosing from 24 symbols, so there are 24 choices for the first symbol, 23 for the second one, 22 for the third, 21 for the fourth, and 20 for the fifth. The number of such quintuples is therefore:

24×23×22×21×20

Now count them in a different way. The number of quintuples in each octad is 8×7×6×5×4 (8 choices for the first member of the quintuple, 7 for the second, etc.). Each quintuple lies in exactly one octad, so if N denotes the number of octads, then the number of quintuples must be N×8×7×6×5×4. Hence:

N×8×7×6×5×4 = 24×23×22×21×20

Dividing both sides by 8×7×6×5×4 yields N = 759.