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.

中文翻译：

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三元码、双平面以及一些拟对称和拟 3 设计的不存在

56点双平面残差设计的三元线性码的对偶码用于证明拟对称2-$(56,12,9)$和2-$(57,12,11)$不存在交叉点编号为 0 和 3 的设计，并且不存在 2-$(267,57,12)$ 准 3 设计。也证明了不存在 2-$(149,37,9)$ 准 3 设计。