Classification of certain genera of codes, lattices and vertex operator algebras

Abstract

We classify the genera of doubly even binary codes, the genera of even lattices, and the genera of rational vertex operator algebras (VOAs) arising from the modular tensor categories (MTCs) of rank up to 4 and central charges up to 16. For the genera of even lattices, there are two types of the genera: code type genera and non code type genera. The number of the code type genera is finite. The genera of the lattices of rank larger than or equal to 17 are non code type. We apply the idea of a vector valued modular form and the representation of the modular group SL[subscript]2(Z) in [Bantay2007] to classify the genera of the VOAs arising from the MTCs of ranks up to 4 and central charges up to 16.