The subject of this chapter is a detailed study of the Steiner systems S(4, 5; 11), S(5, 6; 12), S(4, 7; 23) and S(5, 8; 24) which were constructed independently by Carmichael (1937) andWitt (1938a).Witt (1938b) also sketched a proof for the uniqueness of these Steiner systems (up to isomorphism); a detailed uniqueness proof was given by Lüneburg (1969). These Steiner systems are now usually called the Witt designs. Their automorphism groups are the Mathieu groups discovered by Mathieu (1861, 1873), which are the only finite t-transitive permutation groups with t ≥4, except for the symmetric and alternating groups. The (binary respectively ternary) codes of theWitt designs are the Golay codes constructed by Golay (1949), the only perfect t-error correcting codes with t ≥2.

Nowadays there are quite a few existence and uniqueness proofs for these Steiner systems, and many books and papers discuss the relationship between the Witt designs, the Mathieu groups and the Golay codes. We have tried to keep our proofs of the existence and uniqueness of the Witt designs as free from using the methods and results of coding theory as possible; for the opposite (and quite effective) approach, see, for instance, MacWilliams and Sloane (1977).

The Existence of the Witt Designs

Introduction. We have already provided existence proofs for the Witt designs in §III.8, using the Kramer–Mesner approach for the construction of t-designs; see Examples III.8.8 and III.8.9.