Computer Science > Discrete Mathematics
[Submitted on 11 Nov 2019 (v1), last revised 25 Mar 2020 (this version, v2)]
Title:A Nonexistence Certificate for Projective Planes of Order Ten with Weight 15 Codewords
View PDFAbstract:Using techniques from the fields of symbolic computation and satisfiability checking we verify one of the cases used in the landmark result that projective planes of order ten do not exist. In particular, we show that there exist no projective planes of order ten that generate codewords of weight fifteen, a result first shown in 1973 via an exhaustive computer search. We provide a simple satisfiability (SAT) instance and a certificate of unsatisfiability that can be used to automatically verify this result for the first time. All previous demonstrations of this result have relied on search programs that are difficult or impossible to verify---in fact, our search found partial projective planes that were missed by previous searches due to previously undiscovered bugs. Furthermore, we show how the performance of the SAT solver can be dramatically increased by employing functionality from a computer algebra system (CAS). Our SAT+CAS search runs significantly faster than all other published searches verifying this result.
Submission history
From: Curtis Bright [view email][v1] Mon, 11 Nov 2019 01:48:58 UTC (22 KB)
[v2] Wed, 25 Mar 2020 22:54:52 UTC (22 KB)
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