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 P G(2, F p ) over the field F p := Z/pZ is the unique projective plane of order p. Let π be any projective plane of order p. For any partial linear space X , define the inclusion number i(X , π) to be the number of isomorphic copies of X in π. In this paper we prove that if X has at most log 2 p lines, then i(X , π) can be written as an explicit rational linear combination (depending only on X and p) of the coefficients of the complete weight enumerator (c.w.e.) of the p-ary code of π. Thus, the c.w.e. of this code carries an enormous amount of structural information about π. In consequence, it is shown that if p > 2 9 = 512, and π has the same c.w.e. as P G(2, F p ), then π must be isomorphic to P G(2, 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.