A $k$-semiframe of type $g^u$ is a $k$-GDD of type $g^u\left(\mathcal{X},\mathcal{G},\mathcal{ℬ}\right)$, in which the collection of blocks $\mathcal{ℬ}$ can be written as a disjoint union $\mathcal{ℬ}=\mathcal{P}\cup \mathcal{Q}$, where $\mathcal{P}$ is partitioned into parallel classes of $\mathcal{X}$ and $\mathcal{Q}$ is partitioned into holey parallel classes, each holey parallel class being a partition of $\mathcal{X}\backslash G$ for some $G\in \mathcal{G}$. In this paper, we will introduce a new concept of $t$-perfect semiframe and use it to prove the existence of a 3-semiframe of type $g^u$ with even group size. This completes the proof of the existence of 3-semiframes with uniform group size.

中文翻译：

---

完成块大小为 3 的半帧频谱

一个$ķ$- 类型的半帧$G^{你}$是一个$ķ$-GDD 类型$G^{你}\left(\mathcal{X},\mathcal{G},\mathcal{ℬ}\right)$, 其中块的集合$\mathcal{ℬ}$可以写成不相交的并集$\mathcal{ℬ}=\mathcal{磷}\cup \mathcal{问}$， 在哪里$\mathcal{磷}$被划分为并行的类$\mathcal{X}$和$\mathcal{问}$被划分成孔平行类，每个孔平行类是$\mathcal{X}\backslash G$对于一些$G\in \mathcal{G}$. 在本文中，我们将介绍一个新的概念$吨$-完美的半帧并用它来证明类型的 3-semiframe 的存在$G^{你}$组大小均匀。这样就完成了具有统一组大小的 3-semiframes 存在的证明。