Quasi-symmetric 2-(41,9,9) designs and doubly even self-dual codes of length 40

The existence of a quasi-symmetric 2-(41, 9, 9) design with intersection numbers x = 1, y = 3 is a long-standing open question. Using linear codes and properties of subdesigns, we prove that a cyclic quasi-symmetric 2-(41, 9, 9) design does not exist, and if p < 41 is a prime number being the order of an automorphism of a quasi-symmetric 2-(41, 9, 9) design, then p <= 5. The derived design with respect to a point of a quasi-symmetric 2-(41, 9, 9) design with block intersection numbers 1 and 3 is a quasi-symmetric 1-(40, 8, 9) design with block intersection numbers 0 and 2. The incidence matrix of the latter generates a binary doubly even code of length 40. Using the database of binary doubly even self-dual codes of length 40 classified by Betsumiya et al. (Electron J Combin 19(P18):12, 2012), we prove that there is no quasi-symmetric 2-(41, 9, 9) design with an automorphism phi of order 5 with exactly one fixed point such that the binary code of the derived design is contained in a doubly-even self-dual [40, 20] code invariant under phi.