Vertex Operator Algebras with central charge 8 and 16

We will partially classify spaces of characters of vertex operator algebras $V$ with central charges 8 and 16, such that the spaces of characters is 3-dimensional and the characters forms a basis of the solution space of a third order monic modular linear differential equation with rational indicial roots. Assuming a mild arithmetic condition, we show that the space of characters of $V$ coincides with the space of characters of lattice vertex operators associated with integral lattices $\sqrt{2}E_8$ or the affine vertex operator algebra of type $D_{20}^{(1)}$ for $c=8$, and the Barnes--Wall lattice $\Lambda_{16}$, the affine vertex operator algebras of type $D_{16}^{(1)}$ with level 1 and type $D_{28}^{(1)}$ with level 1 for $c=16$. (The central charge of the affine vertex operator algebra of type $D_{28}^{(1)}$ with level 1 is 28, but the space of characters satisfies the differential equations for $c=16$.) Supposing a mild condition on characters of $V$, then it uniquely determines (up to isomorphism) the spaces of characters of the lattice $\sqrt{2}E_8$ and the Barnes--Wall lattice $\Lambda_{16}$, respectively. The reason why vertex operator algebras with central charges 8 and 16 are intensively studied is that there are solutions which do not depend on extra parameters (which represent conformal weights). This fact is well understood using the hypergeometric function $3F2$. Hence we cannot apply our standard method to classify vertex operator algebras in which we are interested. In appendix we classify $c=4$ vertex operator algebras with the same conditions mentioned above.