In this paper we demonstrate the first example of a finite translation plane which does not contain a translation hyperoval, disproving a conjecture of Cherowitzo. The counterexample is a semifield plane, specifically a Generalised Twisted Field plane, of order 64. We also relate this non-existence to the covering radius of two associated rank-metric codes, and the non-existence of scattered subspaces of maximum dimension with respect to the associated spread.

中文翻译：

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在半场平面中的平移超椭圆上

在本文中，我们演示了第一个不包含平移超椭圆的有限平移平面的例子，反驳了 Cherowitzo 的猜想。反例是一个半场平面，特别是一个 64 阶的广义扭曲场平面。我们还将这种不存在与两个相关等级度量代码的覆盖半径以及相对于相关散布的最大维度的分散子空间的不存在联系起来。