Self-dual Codes over the Kleinian Four Group

Abstract

We introduce self-dual codes over the Kleinian four group $K={\mathbb{Z}}_2\times {\mathbb{Z}}_2$ for a natural quadratic form on $K^n$ and develop the theory. Topics studied are: weight enumerators, mass formulas, classification up to length 8, neighbourhood graphs, extremal codes, shadows, generalized t-designs, lexicographic codes, the Hexacode and its odd and shorter cousin, automorphism groups, marked codes. Kleinian codes form a new and natural fourth step in a series of analogies between binary codes, lattices and vertex operator algebras. This analogy will be emphasized and explained in detail.

26 pages with 5 tables and 1 figure, LaTeX

References in corpus (3)

• Meromorphic c=24 Conformal Field Theories
• Framed vertex operator algebras, codes and the moonshine module
• The Leech lattice and other lattices