Title

On classifying Steiner triple systems by their 3-rank

Authors

Dieter Jungnickel, University of Augsburg
Spyros Magliveras, Florida Atlantic University
Vladimir Tonchev, Michigan Technological UniversityFollow
Alfred Wassermann, University of Bayreuth

Document Type

Conference Proceeding

Publication Date

12-21-2017

Abstract

It was proved recently by Jungnickel and Tonchev (2017) that for every integer v=3^m−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 C_n ( 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 C_n 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.

Publisher's Statement

© Springer International Publishing AG 2017. Publisher’s version of record: https://doi.org/10.1007/978-3-319-72453-9_24

Publication Title

International Conference on Mathematical Aspects of Computer and Information Sciences

ISBN

978-3-319-72453-9

Recommended Citation

Jungnickel, D., Magliveras, S., Tonchev, V., & Wassermann, A. (2017). On classifying Steiner triple systems by their 3-rank. International Conference on Mathematical Aspects of Computer and Information Sciences, 295-305. http://doi.org/10.1007/978-3-319-72453-9_24
Retrieved from: https://digitalcommons.mtu.edu/math-fp/75