On Classifying Steiner Triple Systems by Their 3-Rank
| Autor: | Spyros S. Magliveras, Dieter Jungnickel, Vladimir D. Tonchev, Alfred Wassermann |
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| Rok vydání: | 2017 |
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| Zdroj: | Mathematical Aspects of Computer and Information Sciences ISBN: 9783319724522 MACIS |
| DOI: | 10.1007/978-3-319-72453-9_24 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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