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

## TL;DR

This paper introduces the SAT+CAS method, combining SAT solvers and computer algebra systems, to solve combinatorial conjectures and constructs the largest known best matrices, confirming the best matrix conjecture for a new case.

## Contribution

The paper presents a novel SAT+CAS approach applied to construct and verify the existence of large best matrices, extending the conjecture to r=7.

## Key findings

- Confirmed the best matrix conjecture for r=7
- Constructed new skew Hadamard matrices from best matrices
- Validated previous exhaustive search results for r≤6

## 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-\`a-vis best matrices. The SAT+CAS method is a variant of the Davis$\unicode{8211}$Putnam$\unicode{8211}$Logemann$\unicode{8211}$Loveland $\operatorname{DPLL}(T)$ 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 $r^2+r+1$) which was previously known to hold for $r\leq6$ also holds for $r=7$. We also confirmed the results of the exhaustive searches that have been previously completed for $r\leq6$.