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Designs, Codes and Cryptography

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15.06.2018

The classification of Steiner triple systems on 27 points with 3-rank 24

verfasst von: Dieter Jungnickel, Spyros S. Magliveras, Vladimir D. Tonchev, Alfred Wassermann

Erschienen in: Designs, Codes and Cryptography | Ausgabe 4/2019

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Abstract

We show that there are exactly 2624 isomorphism classes of Steiner triple systems on 27 points having 3-rank 24, all of which are actually resolvable. More generally, all Steiner triple systems on \(3^n\) points having 3-rank at most \(3^n-n\) are resolvable. Combining this observation with the lower bound on the number of such \({\mathrm {STS}}(3^n)\) recently established by two of the present authors, we obtain a strong lower bound on the number of Kirkman triple systems on \(3^n\) points. For instance, there are more than \(10^{99}\) isomorphism classes of \({\mathrm {KTS}}(81)\).

Vorheriger Artikel The weight spectrum of certain affine Grassmann codes

Nächster Artikel Linear codes close to the Griesmer bound and the related geometric structures

Assmus Jr. E.F.: On \(2\)-ranks of Steiner triple systems. Electron. J. Comb. 2, Research paper 9 (1995).

Assmus Jr. E.F., Key J.D.: Designs and their codes. Cambridge University Press, Cambridge (1992).CrossRefMATH

Babai L.: Almost all Steiner triple systems are asymmetric. Ann. Discr. Math. 7, 37–39 (1980).MathSciNetCrossRefMATH

Beth T., Jungnickel D., Lenz H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999).CrossRefMATH

Colbourn C.J., Dinitz J.F. (eds.): Handbook of Combinatorial Designs, vol. 2. Chapman & Hall/, Boca Raton (2007).MATH

Colbourn C.J., Magliveras S.S., Mathon R.A.: Transitive Steiner and Kirkman triple systems of order 27. Math. Comp. 58(197), 441–450 (1992).MathSciNetMATH

Doyen J., Hubaut X., Vandensavel M.: Ranks of incidence matrices of Steiner triple systems. Math. Z. 163, 251–259 (1978).MathSciNetCrossRefMATH

GAP user manual. https://www.gap-system.org/Manuals/doc/ref/chap39.html.

Jungnickel D., Magliveras S.S., Tonchev V.D., Wassermann A.: On classifying Steiner triple systems by their 3-rank. In: Blömer J. (ed.) MACIS 2017, pp. 295–305. Lecture Notes in Computer ScienceSpringer, New York (2018).

10.

Jungnickel D., Tonchev V.D.: On Bonisoli’s theorem and the block codes of Steiner triple systems. Des. Codes Cryptogr. 86, 449–462 (2018).MathSciNetCrossRefMATH

11.

Jungnickel, D., Tonchev, V.D.: Counting Steiner triple systems with classical parameters and prescribed rank. ArXiv e-prints 1709.06044 (2017).

12.

Key J.D., Shobe F.D.: Some transitive Steiner triple systems of Bagchi and Bagchi. J. Stat. Plann. Inference 58, 79–86 (1997).MathSciNetCrossRefMATH

13.

McKay, B., Piperino, A.: Nauty and Traces. http://pallini.di.uniroma1.it/.

14.

Osuna O.P.: There are 1239 Steiner triple systems \(\text{ STS }(31)\) of 2-rank 27. Des. Codes Cryptogr. 40, 187–190 (2006).MathSciNetCrossRefMATH

15.

Soicher, L.H.: DESIGN Package (Version 1.6) for GAP. http://www.maths.qmul.ac.uk/~leonard/designtheory.org/software/.

16.

Teirlinck L.: On projective and affine hyperplanes. J. Combin. Th. Ser. A 28, 290–306 (1980).MathSciNetCrossRefMATH

17.

Tonchev V.D.: A mass formula for Steiner triple systems STS\((2^{n}-1)\) of 2-rank \(2^n -n\). J. Combin. Theory Ser. A 95, 197–208 (2001).MathSciNetCrossRefMATH

Titel: The classification of Steiner triple systems on 27 points with 3-rank 24
verfasst von: Dieter Jungnickel
Spyros S. Magliveras
Vladimir D. Tonchev
Alfred Wassermann
Publikationsdatum: 15.06.2018
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 4/2019
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-018-0502-5

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