  The dual codes of the ternary linear codes of the residual designs of
biplanes on 56 points are used to prove the nonexistence of quasi-symmetric
2-$(56,12,9)$ and 2-$(57,12,11)$ designs with intersection numbers 0 and 3, and
the nonexistence of a 2-$(267,57,12)$ quasi-3 design. The nonexistence of a
2-$(149,37,9)$ quasi-3 design is also proved.
