A coding theoretic approach to the uniqueness conjecture for projective planes of prime order

TL;DR

This paper links the structure of projective planes of prime order to their associated codes' weight enumerators, providing a new approach to the uniqueness conjecture for these planes.

Contribution

It establishes that the inclusion numbers of small partial linear spaces are explicitly determined by the codes' weight enumerators, advancing the understanding of the uniqueness conjecture.

Findings

The complete weight enumerator encodes extensive structural information about the plane.

If two planes have the same c.w.e. and order > 512, they are isomorphic.

The approach reduces the conjecture to analyzing possible c.w.e. of such planes.

Abstract

An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\mathbb{F}_p)$ over the field $\mathbb{F}_p := \mathbb{Z}/p\mathbb{Z}$ is the unique projective plane of order $p$. Let $\pi$ be any projective plane of order $p$. For any partial linear space ${\cal X}$, define the inclusion number $i({\cal X},\pi)$ to be the number of isomorphic copies of ${\cal X}$ in $\pi$. In this paper we prove that if ${\cal X}$ has at most $\log_2 p$ lines, then $i({\cal X},\pi)$ can be written as an explicit rational linear combination (depending only on ${\cal X}$ and $p$) of the coefficients of the complete weight enumerator (c.w.e.) of the $p$-ary code of $\pi$. Thus, the c.w.e. of this code carries an enormous amount of structural information about $\pi$. In consequence, it is shown that if $p > 2^ 9=512$, and $\pi$ has the same c.w.e. as $PG(2,\mathbb{F}_p)$, then $\pi$ must be isomorphic to $PG(2,\mathbb{F}_p)$. Thus, the uniqueness conjecture can be approached via a thorough study of the possible c.w.e. of the codes of putative projective planes of prime order.