We give a brief introduction to sphere-packing in n-dimensions and, in particular, to lattice packings. The notions of density and kissing number are explained. The 24-dimensional Leech lattice Λ is defined following Conway’s approach in his Three lectures on exceptional groups, see Conway (1971), with vectors in the lattice invariably displayed as they appear in the MOG. We explore the factor space Λ/2Λ, a 24-dimensional vector space over ${\mathbb{Z}}_2$, and show that its non-zero elements may be taken to be vectors of type 2 and 3 together with sets of 24 mutually orthogonal vectors of type 4 (and their negatives). These last elements are known as frames of reference or crosses; the six orbits on crosses under permutations of M24 and sign changes on $\mathcal{C}$-sets are described explicitly in the text, as are the orbits on vectors of types 2, 3 and 4. Finally, we explain how Λ can be defined in terms of a single Lorentz-type vector with 25 space-like coordinates and one time-like coordinate.

In this chapter we construct the Golay codes and the Witt designs, both in several ways. The uniqueness is proved in a self-contained way for the binary case; in the ternary case some details are left out. We then study the associated Witt designs, which are remarkable Steiner systems on 12 and 24 points. We show uniqueness of these, and of the (multiply) derived designs. We define the two standard near polygonsfrom the ternary Golay code and the large Witt design. We discuss the geometry of the projective plane of order 4 providing an alternative construction and uniqueness proof of the Witt designs. Finally, we introduce the Leech lattice and its binary and complex variants.