164/5 and 236/7 (Research on finite groups, algebraic combinatorics and vertex operator algebras)

We show that there does not exist a C_{2}-cofinite and rational vertex operator algebra of CFT type with central charge either 164/5 or 236/7 satisfying conditions (a) the space of characters of simple modules is contained in the space of solutions of a modular linear differential equation of order 3, (b) the graded subspace with weight 1 is trivial. The central charges 164/5 and 236/7 first appeared in the paper by Tuite and Van in the classification of central charges of exceptional vertex operator algebras with lowest primary weight 2, which satisfy the conditions (a) and (b). Later we rediscovered these central charges in the characterization problem of the minimal model. By theorems in this paper our characterization problem is completely proved