Hyperovals in projective planes turn out to have a link with t‐designs. Motivated by an unpublished work of Lonz and Vanstone, we present a construction for t‐designs and s‐resolvable t‐designs from hyperovals in projective planes of order $2^n$. We prove that the construction works for $t\le 5$. In particular, for $t=5$ the construction yields a family of 5‐ $\left(2^n+2,8,70\left(2^{n-2}-1\right)\right)$ designs. For $t=4$ numerous infinite families of 4‐designs on $2^n+2$ points with block size $2k$ can be constructed for any $k\ge 4$. The construction assumes the existence of a 4‐ $\left(2^{n-1}+1,k,\lambda \right)$ design, called the indexing design, including the complete 4‐ $\left(2^{n-1}+1,k,\left(\genfrac{}{}{0ex}{}{2^{n-1}-3}{k-4}\right)\right)$ design. Moreover, we prove that if the indexing design is s‐resolvable, then so is the constructed design. As a result, many of the constructed designs are s‐resolvable for $s=2,3$. We include a short discussion on the simplicity or non‐simplicity of the designs from hyperovals.

中文翻译：

---

关于超卵形的t设计和s可解析t设计

投影平面上的超卵形体与t设计有联系。由龙泽和范斯通的未发表的作品的启发，我们提出了一个建设牛逼-designs和小号-resolvable牛逼-designs从顺序射影平面超卵形$2^{ñ}$。我们证明该建筑工程适用于$Ť\le 5$。特别是对于$Ť=5$ 该建筑产生了5个家庭 $（2^{ñ}+2，8，70（2^{ñ-2}-\mathrm{1个}））$设计。对于$Ť=4$ 上的4设计的无穷系列 $2^{ñ}+2$ 点与块大小 $2ķ$ 可以为任何构造 $ķ\ge 4$。该构造假设存在4个$（2^{ñ-\mathrm{1个}}+\mathrm{1个}，ķ，\lambda ）$ 设计，称为索引设计，包括完整的4‐ $（2^{ñ-\mathrm{1个}}+\mathrm{1个}，ķ，（\genfrac{}{}{0ex}{}{2^{ñ-\mathrm{1个}}-3}{ķ-4}））$设计。此外，我们证明了如果索引设计s ^ -resolvable的，那么所构造的设计。其结果是，许多构造设计都是小号-resolvable为$s=2，3$。我们将简短讨论超级椭圆形设计的简单性或非简单性。