Unsatisfiability Proofs for Weight 16 Codewords in Lam's Problem

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 2011, 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.Comment: To appear in Proceedings of the 29th International Joint Conference on Artificial Intelligence (IJCAI 2020

A SAT-based Resolution of Lam's Problem

In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry\unicode{x2014}the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches\unicode{x2014}highlighting the difficulty of relying on special-purpose search code for nonexistence results.Comment: To appear at the Thirty-Fifth AAAI Conference on Artificial Intelligenc

Ternary linear codes

iii+69hlm.;24c

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Proceedings of SAT Competition 2020 : Solver and Benchmark Descriptions

Non peer reviewe