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Register a new userThe number of the non-full-rank Steiner triple systems
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Minjia Shi Anhui University
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The $p$-rank of a Steiner triple system $B$ is the dimension of the linear span of the set of characteristic vectors of blocks of $B$, over GF$(p)$. We derive a formula for the number of different Steiner triple systems of order $v$ and given $2$-rank $r_2$, $r_2<v$, and a formula for the number of Steiner triple systems of order $v$ and given $3$-rank $r_3$, $r_3<v-1$. Also, we prove that there are no Steiner triple systems of $2$-rank smaller than $v$ and, at the same time, $3$-rank smaller than $v-1$. Our results extend previous work on enumerating Steiner triple systems according to the rank of their codes, mainly by Tonchev, V.A.Zinoviev and D.V.Zinoviev for the binary case and by Jungnickel and Tonchev for the ternary case.
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In a recent work, Jungnickel, Magliveras, Tonchev, and Wassermann derived an overexponential lower bound on the number of nonisomorphic resolvable Steiner triple systems (STS) of order $v$, where $v=3^k$, and $3$-rank $v-k$. We develop an approach to generalize this bound and estimate the number of isomorphism classes of STS$(v)$ of rank $v-k-1$ for an arbitrary $v$ of form $3^kT$.
Let $X$ be a $v$-set, $B$ a set of 3-subsets (triples) of $X$, and $B^+cupB^-$ a partition of $B$ with $|B^-|=s$. The pair $(X,B)$ is called a simple signed Steiner triple system, denoted by ST$(v,s)$, if the number of occurrences of every 2-subset of $X$ in triples $BinB^+$ is one more than the number of occurrences in triples $BinB^-$. In this paper we prove that $st(v,s)$ exists if and only if $vequiv1,3pmod6$, $v e7$, and $sin{0,1,...,s_v-6,s_v-4,s_v}$, where $s_v=v(v-1)(v-3)/12$ and for $v=7$, $sin{0,2,3,5,6,8,14}$.
We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find that there are 2004720 isomorphism classes of STS(21) including a subdesign TD(3,6), or, equivalently, a 6-by-6 latin square.
A Berge-$K_4$ in a triple system is a configuration with four vertices $v_1,v_2,v_3,v_4$ and six distinct triples ${e_{ij}: 1le i< j le 4}$ such that ${v_i,v_j}subset e_{ij}$ for every $1le i<jle 4$. We denote by $cal{B}$ the set of Berge-$K_4$ configurations. A triple system is $cal{B}$-free if it does not contain any member of $cal{B}$. We prove that the maximum number of triples in a $cal{B}$-free triple system on $nge 6$ points is obtained by the balanced complete $3$-partite triple system: all triples ${abc: ain A, bin B, cin C}$ where $A,B,C$ is a partition of $n$ points with $$leftlfloor{nover 3}rightrfloor=|A|le |B|le |C|=leftlceil{nover 3}rightrceil.$$
The Heawood graph is the point-block incidence graph of the Fano plane (the unique Steiner triple system of order 7). We show that the minimum semidefinite rank of this graph is 10. That is, 10 is the smallest number of complex dimensions in which this graph has a faithful orthogonal representation, i.e., an assignment of a vector to each vertex such that the edges occur between precisely those vertices given non-orthogonal pairs. Some of our techniques extend to the incidence graphs of Steiner triple systems of larger order, and we include some observations and questions about the more general case.
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