Quantum dimensions and fusion rules of the VOA $V^τ_{L_{C \times D}}$

In this article, we determine quantum dimensions and fusion rules for the orbifold code VOA $V^τ_{L_{C \times D}}$. As an application, we also construct certain $3$-local subgroups inside the automorphism group of the VOA $V^\sharp$, where $V^\sharp$ is a holomorphic VOA obtained by the $\mathbb{Z}_3$-orbifold construction on the Leech lattice VOA.

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