Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs.

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.