{"type":"ScholarlyArticle","title":"Unsatisfiability Proofs for Weight 16 Codewords in Lam's Problem","authors":[{"name":"Curtis Bright","slug":"curtis-bright","given":"Curtis","family":"Bright","affiliation":"University of Waterloo , Department of Electrical and Computer Engineering; Carleton University , School of Mathematics and Statistics"},{"name":"Kevin K.H. Cheung","slug":"kevin-k-h-cheung","given":"Kevin K.H.","family":"Cheung","affiliation":"Carleton University , School of Mathematics and Statistics"},{"name":"Brett Stevens","slug":"brett-stevens","given":"Brett","family":"Stevens","affiliation":"Carleton University , School of Mathematics and Statistics"},{"name":"Ilias Kotsireas","slug":"ilias-kotsireas","given":"Ilias","family":"Kotsireas","affiliation":"Wilfrid Laurier University , Department of Physics and Computer Science"},{"name":"Vijay Ganesh","slug":"vijay-ganesh","given":"Vijay","family":"Ganesh","affiliation":"University of Waterloo , Department of Electrical and Computer Engineering"}],"abstract":"In the 1970s and 1980s, searches performed by L. Carter, C. Lam, L. Thiel, and S. Swiercz showed that projective planes of order ten with weight 16 codewords do not exist. These searches required highly specialized and optimized computer programs and required about 2,000 hours of computing time on mainframe and supermini computers. In 2010, these searches were verified by D. Roy using an optimized C program and 16,000 hours on a cluster of desktop machines. We performed a verification of these searches by reducing the problem to the Boolean satisfiability problem (SAT). Our verification uses the cube-and-conquer SAT solving paradigm, symmetry breaking techniques using the computer algebra system Maple, and a result of Carter that there are ten nonisomorphic cases to check. Our searches completed in about 30 hours on a desktop machine and produced nonexistence proofs of about 1 terabyte in the DRAT (deletion resolution asymmetric tautology) format.","doi":"10.24963/ijcai.2020/203","url":"https://ijcai.org/proceedings/2020/203","pdf":"https://ijcai.org/proceedings/2020/0203.pdf","markdown":"https://ijcai.org/proceedings/2020/203.md","bibtex":"https://ijcai.org/proceedings/2020/bibtex/203","volume":{"slug":"2020","title":"Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2020","url":"https://ijcai.org/proceedings/2020/","year":2020,"location":"Yokohama","dates":"January 2021 (postponed from 11-17 July 2020)","editors":"Christian Bessiere","isbn":"978-0-9992411-6-5","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2020"},"section":"Heuristic Search and Game Playing","track":"Main track","number":203,"pages":{"first":1460,"last":1466},"keywords":["Constraints and SAT: Constraint Satisfaction","Constraints and SAT: SAT: Algorithms and Techniques","Heuristic Search and Game Playing: Combinatorial Search and Optimisation","Constraints and SAT: SAT: : Solvers and Applications"],"videos":[{"label":"Video","url":"https://videos.ijcai20.org/24880/file_24880.m3u8","length_sec":297,"provider":"hls"}],"references":[{"position":1,"raw":"[Aloul et al., 2003] Fadi A. 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