Counting Steiner triple systems with classical parameters and prescribed rank

Abstract

By a famous result of Doyen, Hubaut and Vandensavel [6], the 2-rank of a Steiner triple system on 2n−1 points is at least 2n−1−n, and equality holds only for the classical point-line design in the projective geometry PG(n−1,2). It follows from results of Assmus [1] that, given any integer t with 1≤t≤n−1, there is a code Cn,t containing representatives of all isomorphism classes of STS(2n−1) with 2-rank at most 2n−1−n+t. Using a mixture of coding theoretic, geometric, design theoretic and combinatorial arguments, we prove a general formula for the number of distinct STS(2n−1) with 2-rank at most 2n−1−n+t contained in this code. This generalizes the only previously known cases, t=1, proved by Tonchev [13] in 2001, t=2, proved by V. Zinoviev and D. Zinoviev [16] in 2012, and t=3 (V. Zinoviev and D. Zinoviev [17], [18] (2013), D. Zinoviev [15] (2016)), while also unifying and simplifying the proofs.This enumeration result allows us to prove lower and upper bounds for the number of isomorphism classes of STS(2n−1) with 2-rank exactly (or at most) 2n−1−n+t. Finally, using our recent systematic study of the ternary block codes of Steiner triple systems [10], we obtain analogous results for the ternary case, that is, for STS(3n) with 3-rank at most (or exactly) 3n−1−n+t.We note that this work provides the first two infinite families of 2-designs for which one has non-trivial lower and upper bounds for the number of non-isomorphic examples with a prescribed p-rank in almost the entire range of possible ranks.