Self-dual Codes over the Kleinian Four Group
Gerald Höhn
TL;DR
Self-dual Kleinian codes over $K= {\bf Z}_2\times{\bf Z}_2$ are developed as a natural fourth link in the binary codes–lattices–VOAs paradigm. The paper establishes a comprehensive foundational framework, including definitions, duality, weight enumerators, and MacWilliams identities, and performs a complete classification up to length $8$, alongside extremal-code analysis, generalized $2$-designs, and lexicographic constructions. It then situates Kleinian codes within the broader algebraic landscape by connecting them to binary codes, lattices, and vertex operator algebras through two constructions $\rho_A$ and $\rho_B$, showing, for example, that $\rho_B({\cal C}_6)$ yields the Golay code and that even/odd Kleinian codes correspond to VOA/SVOA extensions with a sub-VOA $V_{D_4}^{\otimes n}$. This framework advances the analogy among codes, lattices, and VOAs and opens avenues for quantum codes and VOA-theoretic insights grounded in Kleinian codes.
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.