Using techniques from the fields of symbolic computation and satisfiability checking we verify one of the cases used in the landmark result that projective planes of order ten do not exist. In particular, we show that there exist no projective planes of order ten that generate codewords of weight fifteen, a result first shown in 1973 via an exhaustive computer search. We provide a simple satisfiability (SAT) instance and a certificate of unsatisfiability that can be used to automatically verify this result for the first time. All previous demonstrations of this result have relied on search programs that are difficult or impossible to verify---in fact, our search found partial projective planes that were missed by previous searches due to previously undiscovered bugs. Furthermore, we show how the performance of the SAT solver can be dramatically increased by employing functionality from a computer algebra system (CAS). Our SAT+CAS search runs significantly faster than all other published searches verifying this result.

中文翻译：

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权重为 15 码字的 10 阶射影平面的不存在证明

使用符号计算和可满足性检查领域的技术，我们验证了标志性结果中使用的一种情况，即十阶射影平面不存在。特别是，我们表明不存在生成权重为 15 的码字的 10 阶射影平面，该结果于 1973 年通过详尽的计算机搜索首次显示。我们提供了一个简单的可满足性 (SAT) 实例和一个可用于首次自动验证此结果的不可满足性证书。此结果的所有先前演示都依赖于难以或不可能验证的搜索程序——事实上，我们的搜索发现了由于先前未发现的错误而被先前搜索遗漏的部分投影平面。此外，我们展示了如何通过使用计算机代数系统 (CAS) 的功能来显着提高 SAT 求解器的性能。我们的 SAT+CAS 搜索运行速度明显快于验证此结果的所有其他已发布搜索。