The SAT+CAS Method for Combinatorial Search with Applications to Best Matrices

Abstract

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 DPLL(<em>T</em>) architecture, where the <em>T</em> solver is replaced by a CAS. We describe how the SAT+CAS method has been 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 find new skew Hadamard matrices constructed from best matrices. As a consequence of this we show that a conjecture on the existence of best matrices that was previously known to hold for <em>r</em> ≤ 6 also holds for <em>r</em> = 7.