2017/09/18 by Jungnickel, Dieter, Tonchev, Vladimir D.
#05B05 #51E10 #94B27 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1709.06044
By a famous result of Doyen, Hubaut and Vandensavel \citeDHV, 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 \citeA 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 \citeT01 in 2001, t=2, proved by V. Zinoviev and D. Zinoviev \citeZZ12 in 2012, and t=3 (V. Zinoviev and D. Zinoviev \citeZZ13, \citeZZ13a (2013), D. Zinoviev \citeZ16 (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 \citeJT, 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.