For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1 2 , two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge 1 2 are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.Proposition 2.14 (1) The code C is even, i.e. the weight wt(c) = r i=1 c i ∈ Z + of every codeword c ∈ C is divisible by 2.

We provide a novel and simple description of Schellekens' seventy-one affine Kac-Moody structures of self-dual vertex operator algebras of central charge 24 by utilizing cyclic subgroups of the glue codes of the Niemeier lattices with roots. We also discuss a possible uniform construction procedure of the self-dual vertex operator algebras of central charge 24 starting from the Leech lattice. This also allows us to consider the uniqueness question for all non-trivial affine Kac-Moody structures. We finally discuss our description from a Lorentzian viewpoint.

We determine the orbits of fixed-point sublattices of the Leech lattice with respect to the action of the Conway group Co 0 . There are 290 such orbits. Detailed information about these lattices, the corresponding coinvariant lattices, and the stabilizing subgroups, is tabulated in several tables.

We introduce the notion of a conformal design based on a vertex operator algebra. This notation is a natural analog of the notion of block designs or spherical designs when the elements of the design are based on self-orthogonal binary codes or integral lattices, respectively. It is shown that the subspaces of fixed degree of an extremal self-dual vertex operator algebra form conformal 11-, 7-, or 3-designs, generalizing similar results of Assmus-Mattson and Venkov for extremal doubly-even codes and extremal even lattices. Other examples are coming from group actions on vertex operator algebras, the case studied first by Matsuo. The classification of conformal 6and 8-designs is investigated. Again, our results are analogous to similar results for codes and lattices.

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