Finite projective planes and the Delsarte LP-bound

TL;DR

This paper uses an improved Delsarte LP-bound, aided by computer, to prove the non-existence of finite projective planes of order 6 and the uniqueness of those of order 7, with potential for higher orders.

Contribution

It introduces a new, computer-aided proof technique using an improved Delsarte LP-bound for finite projective planes.

Findings

No finite projective plane of order 6 exists.

Unique finite projective plane of order 7.

Potential applicability to higher orders like 8, 9, 10, and 12.

Abstract

We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12.