Frame Stabilizers for framed vertex operator algebras associated to lattices having 4-frames

In this paper, we study certain Virasoro frames for lattice vertex operator algebras and their ${\mathbb{Z}}_2$-orbifolds using linear codes over ${\mathbb{Z}}_4$. We also compute the corresponding frame stabilizer from the view point of binary codes and ${\mathbb{Z}}_4$-codes. As an application, we determine the frame stabilizers of several Virasoro frames of the vertex operator algebra $V_{E_8}$ and the moonshine vertex operator algebra $V^{\mathrm{♮}}$.