Counting Steiner triple systems with classical parameters and prescribed rank

By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of a Steiner triple system on $2^n-1$ points is at least $2^n -1 -n$, and equality holds only for the classical point-line design in the projective geometry $PG(n-1,2)$. It follows from results of Assmus \cite{A} that, given any integer $t$ with $1 \leq t \leq n-1$, there is a code $C_{n,t}$ containing representatives of all isomorphism classes of STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$. Using a mixture of coding theoretic, geometric, design theoretic and combinatorial arguments, we prove a general formula for the number of distinct STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$ contained in this code. This generalizes the only previously known cases, $t=1$, proved by Tonchev \cite{T01} in 2001, $t=2$, proved by V. Zinoviev and D. Zinoviev \cite{ZZ12} in 2012, and $t=3$ (V. Zinoviev and D. Zinoviev \cite{ZZ13}, \cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (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$(2^n-1)$ with 2-rank exactly (or at most) $2^n -1 -n + t$. Finally, using our recent systematic study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain analogous results for the ternary case, that is, for STS$(3^n)$ with 3-rank at most (or exactly) $3^n -1 -n + t$. 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.Comment: 27 page

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There are 1239 Steiner triple systems STS(31) of 2-rank 27

A computer search over the words of weight 3 in the code of blocks of a classical Steiner triple system (STS) on 31 points is carried out to classify all STS 31 whose incidence matrix has 2-rank equal to 27, one more than the possible minimum of 26. There is a total of 1239 nonisomorphic STS 31 of 2-rank 27

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Musketeer 1936

Xavier Student Yearbook, published from 1924-2005. Not published: 1943-46, 1972-73, 1988-89, 2006-current.https://www.exhibit.xavier.edu/xavier_yearbook/1012/thumbnail.jp

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum