A Nonexistence Certificate for Projective Planes of Order Ten with Weight 15 Codewords

TL;DR

This paper verifies the nonexistence of projective planes of order ten with codewords of weight fifteen using SAT solvers and symbolic computation, providing a verifiable certificate and improving performance over previous methods.

Contribution

It introduces a SAT+CAS approach to verify a classical combinatorial result, producing a verifiable certificate and enhancing computational efficiency.

Findings

Confirmed no projective planes of order ten with weight 15 codewords exist

Provided a verifiable certificate of unsatisfiability for the result

Achieved faster verification than previous exhaustive searches

Abstract

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.