# $\mathbb{Z}_k$-code vertex operator algebras

## TL;DR

This paper constructs a new class of vertex operator algebras associated with $bZ_k$-codes, demonstrating their properties and module structures, and relates them to lattice VOAs and parafermion algebras.

## Contribution

It introduces a novel family of $bZ_k$-code vertex operator algebras with specific properties and describes their module structure and realization as commutants in lattice VOAs.

## Key findings

- The VOAs are simple, self-dual, rational, and $C_2$-cofinite.
- All irreducible modules are constructed within lattice VOA modules.
- The VOAs are realized as commutants in lattice vertex operator algebras.

## Abstract

We introduce a simple, self-dual, rational, and $C_2$-cofinite vertex operator algebra of CFT-type associated with a $\mathbb{Z}_k$-code for $k \ge 2$ based on the $\mathbb{Z}_k$-symmetry among the simple current modules for the parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$. We show that it is naturally realized as the commutant of a certain subalgebra in a lattice vertex operator algebra. Furthermore, we construct all the irreducible modules inside a module for the lattice vertex operator algebra.