  In this paper, we study certain Virasoro frames for lattice vertex operator
algebras and their $\Z_2$-orbifolds using linear codes over $\Z_4$. We also
compute the corresponding frame stabilizer from the view point of binary codes
and $\Z_4$-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 $V^\natural$.
