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

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