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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 \(\mathbf{i}({\mathcal {X}},\pi )\) 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 \(\mathbf{i}({\mathcal {X}},\pi )\) 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.

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