In this paper, we provide an overview of the SAT+CAS method that combines satisfiability checkers (SAT solvers) and computer algebra systems (CAS) to resolve combinatorial conjectures, and present new results vis-à-vis best matrices. The SAT+CAS method is a variant of the Davis–Putnam–Logemann–Loveland DPLL(T) architecture, where the T solver is replaced by a CAS. We describe how the SAT+CAS method has been previously used to resolve many open problems from graph theory, combinatorial design theory, and number theory, showing that the method has broad applications across a variety of fields. Additionally, we apply the method to construct the largest best matrices yet known and present new skew Hadamard matrices constructed from best matrices. We show the best matrix conjecture (that best matrices exist in all orders of the form r2 + r + 1) which was previously known to hold for r ≤ 6 also holds for r = 7. We also confirmed the results of the exhaustive searches that have been previously completed for r ≤ 6.

中文翻译：

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用于组合搜索的 SAT+CAS 方法以及对最佳矩阵的应用

在本文中，我们概述了 SAT+CAS 方法，该方法结合了可满足性检查器（SAT 求解器）和计算机代数系统（CAS）来解决组合猜想，并针对最佳矩阵提出新结果。SAT+CAS 方法是 Davis-Putnam-Logemann-Loveland DPLL(T) 架构的变体，其中 T 求解器被 CAS 取代。我们描述了 SAT+CAS 方法以前如何用于解决图论、组合设计理论和数论中的许多开放问题，表明该方法在各个领域都有广泛的应用。此外，我们应用该方法来构建已知的最大的最佳矩阵，并提出由最佳矩阵构建的新偏斜 Hadamard 矩阵。