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

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 $\mathcal{X}$, define the inclusion number $i(\mathcal{X},\pi )$ to be the number of isomorphic copies of $\mathcal{X}$ in $\pi$. In this paper we prove that if $\mathcal{X}$ has at most ${\log }_{2}p$ lines, then $i(\mathcal{X},\pi )$ can be written as an explicit rational linear combination (depending only on $\mathcal{X}$ 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.