On classifying Steiner triple systems by their 3-rank
It was proved recently by Jungnickel and Tonchev (2017) that for every integer v=3m−1w , m≥2 , and w≡1,3 (mod 6) , there is a ternary linear [v,v−m] code C, such that every Steiner triple system STS(v) on v points and having 3-rank v−m , is isomorphic to an STS(v) supported by codewords of weight 3 in C. In this paper, we consider the ternary [3n,3n−n] code Cn ( n≥3 ), that supports representatives of all isomorphism classes of STS(3n) of 3-rank 3n−n . We prove some structural properties of the triple system supported by the codewords of Cn of weight 3. Using these properties, we compute the exact number of distinct STS(27) of 3-rank 24 supported by the code C3 . As an application, we prove a lower bound on the number of nonisomorphic STS(27) of 3-rank 24, and classify up to isomorphism all STS(27) supported by C3 that admit a certain automorphism group of order 3.
International Conference on Mathematical Aspects of Computer and Information Sciences
On classifying Steiner triple systems by their 3-rank.
International Conference on Mathematical Aspects of Computer and Information Sciences, 295-305.
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