有限射影平面即指数为1($\lambda=1$)的对称设计，是一类最特殊，研究最深入的对称设计。在组合设计里面有着重要的地位，也是目前所知道的唯一的无穷类（在$\lambda$ 固定的情况下）。射影平面具有非常强的几何背景，悠久的历史和丰富的结果,而且它也是组合设计构造的丰富源泉。半域是构造射影平面的重要手段，半域的构造不仅直接丰富了组合设计的理论，而且半域编码密码学上有着广泛的应用。本项目主要从研究某些半域的子半域开始，尝试利用"扩张"的方法得到更多的半域、射影平面，并讨论半域这种非结合代数和射影平面在组合设计、编码密码学等方向上的相关应用。

Finite projective planes, i.e. the symmetric designs with $\lambda=1$, are one of the most special and well studied symmetric designs. It is the only infinite class of known symmetric designs (with fixed $\lambda$), and very important in the research of combinatorial designs. Projective planes have strong background on geometry. The research of projective planes has long history and plentiful results, which are also the sources of the other combinatorial designs. Semifield is one of the main methods to construct projective planes. The construction of semifields not noly enriches combinatorial designs theory, but also have wide applications in codes and cryptographies. This project is trying to find more semifields and projective planes by the extension of semifields, which is based on the study of the subsemifields of some semifields. Their applications on designs,codes and crypotographies are also discussed.