An Introduction to S(5,8,24)

Abstract

S(5,8,24) is one of the largest known Steiner systems and connects combinatorial designs, error-correcting codes, finite simple groups, and sphere packings in a truly remarkable way. This thesis discusses the underlying structure of S(5,8,24), its construction via the (24,12) Golay code, as well its automorphism group, which is the Mathieu group M24, a member of the sporadic simple groups. Particular attention is paid to the calculation of the size of automorphism groups of Steiner systems using the Orbit-Stabilizer Theorem. We conclude with a section on the sphere packing problem and elaborate on how the 8-sets of S(5,8,24) can be used to form Leech's Lattice, which Leech used to create the densest known sphere packing in 24-dimensions. The appendix contains code written for Matlab which has the ability to construct the octads of S(5,8,24), permute the elements to obtain isomorphic S(5,8,24) systems, and search for certain subsets of elements within the octads.