In 1983, a computer search was performed for ovals in a projective plane of order ten. The search was exhaustive and negative, implying that such ovals do not exist. However, no nonexistence certificates were produced by this search, and to the best of our knowledge the search has never been independently verified. In this paper, we rerun the search for ovals in a projective plane of order ten and produce a collection of nonexistence certificates that, when taken together, imply that such ovals do not exist. Our search program uses the cube-and-conquer paradigm from the field of satisfiability (SAT) checking, coupled with a programmatic SAT solver and the nauty symbolic computation library for removing symmetries from the search.

中文翻译：

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十阶投影平面中椭圆的不存在证明

1983 年，计算机搜索了 10 阶投影平面中的椭圆。搜索是详尽的和否定的，这意味着这样的椭圆不存在。然而，此次搜索并未产生不存在证书，据我们所知，该搜索从未经过独立验证。在本文中，我们在 10 阶投影平面中重新搜索椭圆，并生成不存在证明的集合，将这些证明放在一起时，意味着此类椭圆不存在。我们的搜索程序使用来自可满足性 (SAT) 检查领域的立方体和征服范式，结合编程 SAT 求解器和用于从搜索中删除对称性的顽皮符号计算库。