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

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.