Witt design

A Witt design is a Steiner system S(t,k,v) with t=5, meaning that every 5-element subset of the v-point set lies in exactly one block of size k. The two classical Witt designs are the 5-(24,8,1) design (often denoted W24) and the 5-(12,6,1) design (W12). These are the only Steiner systems with t=5 that are not trivial, and they are unique up to isomorphism.¹ The parameters satisfy the divisibility conditions of the Fisher inequality, and the designs are self-dual in a sense related to their automorphism groups.

Witt constructed these designs using the extended binary Golay code and the Mathieu groups. The 5-(24,8,1) design can be obtained by taking the 24 coordinate positions of the extended Golay code G24; the blocks are the supports of codewords of weight 8. The automorphism group of W24 is the Mathieu group M24, a sporadic simple group of order 244,823,040. Similarly, W12 arises from the smaller Golay code G12 and has automorphism group M12. These groups act 5-transitively on the points, a property that is extremely rare and characterizes the Witt designs.²

The Witt designs are deeply intertwined with other exceptional objects in mathematics. The 5-(24,8,1) design is used to construct the Leech lattice, a 24-dimensional unimodular lattice with dense sphere packing, and the binary Golay code, which is a perfect error-correcting code used in deep-space communications. The designs also relate to the Steiner system S(5,6,12) and the small Mathieu group M12, which appears in the theory of the exceptional outer automorphism of the symmetric group S6. Furthermore, Witt designs are examples of t-designs with t>2, and they are the only Steiner systems with t=5 that are not trivial.

Beyond the classical W12 and W24, Witt also discovered other designs, such as the 4-(11,5,1) and 3-(10,4,1) designs, which are derived by fixing points. These derived designs are also unique and have connections to the Mathieu groups M11 and M10. The Witt designs are the only Steiner systems with t=5 that are not trivial, and they are the only 5-designs with λ=1. The uniqueness proofs rely on the classification of finite simple groups, and the designs are used in the construction of the Monster group via the Golay code and the Leech lattice. Moreover, the Witt designs have applications in coding theory, particularly in the construction of the Nordstrom-Robinson code and the Kerdock code.

Steiner system

A set system where every t-subset of points appears in exactly one block.

Mathieu group

One of the first discovered sporadic simple groups, acting as automorphisms of Witt designs.

Golay code

A perfect error-correcting binary code used in the construction of Witt designs.

Leech lattice

A 24-dimensional lattice with dense sphere packing, constructed from the Witt design.