A SAT+CAS Approach to Finding Good Matrices: New Examples and Counterexamples

Journal of Symbolic Computation

Heinle

2021

Self Cite

We use techniques from the fields of computer algebra and satisfiability checking to develop a new algorithm to search for complex Golay pairs. We implement this algorithm and use it to perform a complete search for complex Golay pairs of lengths up to 28. In doing so, we find that complex Golay pairs exist in the lengths 24 and 26 but do not exist in the lengths 23, 25, 27, and 28. This independently verifies work done by F. Fiedler in 2013 and confirms the 2002 conjecture of Craigen, Holzmann, and Kharaghani that complex Golay pairs of length 23 don't exist. Our algorithm is based on the recently proposed SAT+CAS paradigm of combining SAT solvers with computer algebra systems to efficiently search large spaces specified by both algebraic and logical constraints. The algorithm has two stages: first, a fine-tuned computer program uses functionality from computer algebra systems and numerical libraries to construct a list containing every sequence that could appear as the first sequence in a complex Golay pair up to equivalence. Second, a programmatic SAT solver constructs every sequence (if any) that pair off with the sequences constructed in the first stage to form a complex Golay pair. This extends work originally presented at the International Symposium on Symbolic and Algebraic Computation (ISSAC) in 2018; we discuss and implement several improvements to our algorithm that enabled us to improve the efficiency of the search and increase the maximum length we search from length 25 to 28.

Section: Stage 2: Construct the Second Sequence B From Amentioning

confidence: 99%

Complex Golay pairs up to length 28: A search via computer algebra and programmatic SAT

Journal of Symbolic Computation

Heinle

2021

Self Cite

“…Our focus was applying the SAT+CAS paradigm to the Williamson conjecture from combinatorial design theory, but we believe the SAT+CAS paradigm shows promise to be applicable to many other problems and conjectures. In fact, the SAT+CAS paradigm has recently been used to enumerate complex Golay pairs (Bright et al, 2018b) and good matrices (Bright et al, 2019). However, the SAT+CAS paradigm is not something that can be effortlessly applied to problems or expected to be effective on all types of problems.…”

Section: Conclusion and Advicementioning

confidence: 99%

Applying computer algebra systems with SAT solvers to the Williamson conjecture

Journal of Symbolic Computation

2020

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We employ tools from the fields of symbolic computation and satisfiability checkingnamely, computer algebra systems and SAT solvers-to study the Williamson conjecture from combinatorial design theory and increase the bounds to which Williamson matrices have been enumerated. In particular, we completely enumerate all Williamson matrices of even order up to and including 70 which gives us deeper insight into the behaviour and distribution of Williamson matrices. We find that, in contrast to the case when the order is odd, Williamson matrices of even order are quite plentiful and exist in every even order up to and including 70. As a consequence of this and a new construction for 8-Williamson matrices we construct 8-Williamson matrices in all odd orders up to and including 35. We additionally enumerate all Williamson matrices whose orders are divisible by 3 and less than 70, finding one previously unknown set of Williamson matrices of order 63.

“…Additionally, the entries of best matrices can be shown to satisfy certain constraints similar to constraints that Williamson matrices [74], good matrices [9], and the coefficients of complex Golay pairs [15] satisfy. In the appendix we show that the entries of best matrices satisfy the relationship a k b k c k d k a 2k b 2k c 2k = −1 for k = 0 with indices reduced mod n. Because of the anti-symmetry of A, B, and C, when k = n/3 the product constraint reduces to d k = 1 and in this case can be encoded as a unit clause.…”

Section: Conquer Phasementioning

confidence: 99%

“…We used MathCheck to show that certain graph theoretic conjectures held up to bounds that had not previously been verified [79]. Later we applied MathCheck to find (and disprove the existence of in certain orders) Williamson matrices [12], complex Golay pairs [14], and good matrices [9].…”

Section: The Sat+cas Paradigmmentioning

confidence: 99%

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

Djoković

2019

EasyChair Preprints

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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 r 2 + 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.