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

Bright, Curtis; Kotsireas, Ilias S.; Heinle, Albert

doi:10.1016/j.jsc.2019.10.013

Cited by 8 publications

(5 citation statements)

References 33 publications

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“…Programmatic SAT solvers were introduced by Ganesh et al [11] in order to solve an RNA folding problem. They have since been used to search for various combinatorial objects such as Williamson matrices [5], best matrices [7], and complex Golay sequences [6].…”

Section: Symbolic Computation and Sat+casmentioning

confidence: 99%

Nonexistence Certificates for Ovals in a Projective Plane of Order Ten

Bright

Cheung

Stevens

et al. 2020

Lecture Notes in Computer Science

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In 1983, a computer search was performed for ovals in a projective plane of order ten. The search was exhaustive and negative, implying that such ovals do not exist. However, no nonexistence certificates were produced by this search, and to the best of our knowledge the search has never been independently verified. In this paper, we rerun the search for ovals in a projective plane of order ten and produce a collection of nonexistence certificates that, when taken together, imply that such ovals do not exist. Our search program uses the cube-and-conquer paradigm from the field of satisfiability (SAT) checking, coupled with a programmatic SAT solver and the nauty symbolic computation library for removing symmetries from the search.

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Section: Symbolic Computation and Sat+casmentioning

confidence: 99%

Nonexistence Certificates for Ovals in a Projective Plane of Order Ten

Bright

Cheung

Stevens

et al. 2020

Lecture Notes in Computer Science

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show abstract

“…However, this result had never been independently verified. Using MathCheck we performed the first independent verification of this result [15] by explicitly finding all complex Golay pairs for n ≤ 28, and further provided a complete enumeration of all inequivalent complex Golay pairs up to 28.…”

Section: The Sat+cas Paradigmmentioning

confidence: 99%

“…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%

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

Bright

Djoković

Kotsireas

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.

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“…MathCheck can be used to independently verify the results of Fiedler's searches [8,9]. e rst step is to nd all single polynomials f that could appear as a member of a complex Golay pair.…”

Section: Craigen-holzmann-kharaghani Conjecturementioning

confidence: 99%

SAT Solvers and Computer Algebra Systems: A Powerful Combination for Mathematics

Bright,

Kotsireas,

Ganesh

2019

Preprint

Self Cite

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Over the last few decades, many distinct lines of research aimed at automating mathematics have been developed, including computer algebra systems (CASs) for mathematical modelling, automated theorem provers for rst-order logic, SAT/SMT solvers aimed at program veri cation, and higher-order proof assistants for checking mathematical proofs. More recently, some of these lines of research have started to converge in complementary ways. One success story is the combination of SAT solvers and CASs (SAT+CAS) aimed at resolving mathematical conjectures.Many conjectures in pure and applied mathematics are not amenable to traditional proof methods. Instead, they are best addressed via computational methods that involve very large combinatorial search spaces. SAT solvers are powerful methods to search through such large combinatorial spaces-consequently, many problems from a variety of mathematical domains have been reduced to SAT in an a empt to resolve them. However, solvers traditionally lack deep repositories of mathematical domain knowledge that can be crucial to pruning such large search spaces. By contrast, CASs are deep repositories of mathematical knowledge but lack e cient general search capabilities. By combining the search power of SAT with the deep mathematical knowledge in CASs we can solve many problems in mathematics that no other known methods seem capable of solving.We demonstrate the success of the SAT+CAS paradigm by highlighting many conjectures that have been disproven, veri ed, or partially veri ed using our tool MathCheck. ese successes indicate that the paradigm is positioned to become a standard method for solving problems requiring both a signi cant amount of search and deep mathematical reasoning. For example, the SAT+CAS paradigm has recently been used by Heule, Kauers, and Seidl to nd many new algorithms for 3 × 3 matrix multiplication.

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

Cited by 8 publications

References 33 publications

Nonexistence Certificates for Ovals in a Projective Plane of Order Ten

Nonexistence Certificates for Ovals in a Projective Plane of Order Ten

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

SAT Solvers and Computer Algebra Systems: A Powerful Combination for Mathematics

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