The number of the non-full-rank Steiner triple systems

Abstract

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.