Existence of Projective Planes

TL;DR

This paper reviews the history and properties of finite projective planes, and confirms through independent verification that a projective plane of order 10 does not exist, supporting previous computational results.

Contribution

It provides an overview of finite projective planes and independently verifies the non-existence of a projective plane of order 10.

Findings

No projective plane of order 10 exists.

Confirmed previous computational results from 1970.

Validated the absence of a specific code vector in the related code.

Abstract

This report gives an overview of the history of finite projective planes and their properties before going on to outline the proof that no projective plane of order 10 exists. The report also investigates the search carried out by MacWilliams, Sloane and Thompson in 1970 [12] and confirms their result by providing independent verification that there is no vector of weight 15 in the code generated by the projective plane of order 10.