I have a maybe simple question about even positive-definite lattices and lattices coming from binary codes. They seemed to be used in framed vertex operator algebras.

What is known about even lattices, which have $(2\mathbb{Z})^n$ (with standard scalar product) as a sublattice. I am particular interested in lattices, which are sublattices of the lattice $(1/2\mathbb{Z})^n$. I guess non-trivial examples are just possible for $n=8,16,\ldots$ and that these come from binary codes. Is this true? I have as examples $E_8$ and the Leech lattice in mind.

Is there maybe any correspondence between such lattices and binary codes?