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 $mathbb{F}_p := mathbb{Z}/pmathbb{Z}$ 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.