On Classifying Steiner Triple Systems by Their 3-Rank

Abstract

It was proved recently by Jungnickel and Tonchev (2017) that for every integer $v=3^{m-1}w$, $m\ge 2$, and $w\equiv 1,3 \pmod 6$, there is a ternary linear $[v,v-m]$ code C, such that every Steiner triple system ${{\mathrm{STS}}}(v)$ on v points and having 3-rank $v-m$, is isomorphic to an ${{\mathrm{STS}}}(v)$ supported by codewords of weight 3 in C. In this paper, we consider the ternary $[3^n, 3^n -n]$ code $C_n$ ($n\ge 3$), that supports representatives of all isomorphism classes of ${{\mathrm{STS}}}(3^n)$ of 3-rank $3^n -n$. We prove some structural properties of the triple system supported by the codewords of $C_n$ of weight 3. Using these properties, we compute the exact number of distinct ${{\mathrm{STS}}}(27)$ of 3-rank 24 supported by the code $C_3$. As an application, we prove a lower bound on the number of nonisomorphic ${{\mathrm{STS}}}(27)$ of 3-rank 24, and classify up to isomorphism all ${{\mathrm{STS}}}(27)$ supported by $C_3$ that admit a certain automorphism group of order 3.