Conference Matrices and Unimodular Lattices

We use conference matrices to define an action of the complex numbers on the real Euclidean vector space R. In certain cases, the lattice D n becomes a module over a ring of quadratic integers. We can then obtain new unimodular lattices, essentially by multiplying the lattice D n by a non-principal ideal in this ring. We show that lattices constructed via quadratic residue codes, including the Leech lattice, can be constructed in this way. Recall that a lattice Λ is a discrete subgroup of a finite dimensional real vector space V . We suppose that V has a given Euclidean inner product (u,v) 7→ u · v and the rank of Λ equals the dimension of V . In this case Λ has a bounded fundamental region in V . We call the volume of such a fundamental region (measured with respect to the Euclidean structure on V ) the volume of the lattice Λ. The lattice Λ is integral if u · v ∈ Z for all u, v ∈ Λ. It is even if |u| = u · u ∈ 2Z for all u ∈ Λ. Even lattices are necessarily integral. The lattice Λ is unimodular if Λ is integral and has volume 1. It is well known [9, Chapter VIII, Theorem 8] that if Λ is an even unimodular then the rank of Λ is divisible by 8. For convenience we call the square of the length of a vector its norm. The minimum norm of a lattice is the smallest non-zero norm of its vectors.