  We prove that a certain binary linear code associated with the incidence
matrix of a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3
must be contained in an extremal doubly even self-dual code of length 40. Using
the classification of extremal doubly even self-dual codes of length 40, we
show that a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3
does not exist.
