Top

Designs, Codes and Cryptography

Published in:

02-08-2023

Block-transitive 3-\((v,4,\lambda )\) designs with sporadic or alternating socle

Authors: Xuan Pang, Xiaoqin Zhan

Published in: Designs, Codes and Cryptography | Issue 12/2023

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper is a contribution to the classification of block-transitive 3-designs. Let \({{\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})}\) be a nontrivial 3-\((v,4,\lambda )\) design and \(G \le Aut({\mathcal {D}})\) acts block-transitively on \({\mathcal {D}}\) with sporadic or alternating socle, then there are exactly 6 incomplete designs as follows:

(i)

\({\mathcal {D}}\) is isomorphic to a 3-\((12,4,\lambda )\) design with \(\lambda \in \{3,6\}\), and \(G \cong M_{11}\);

(ii)

\({\mathcal {D}}\) is isomorphic to a 3-\((22,4,\lambda )\) design with \(\lambda \in \{3,16\}\), and \(Soc(G)=M_{22}\);

(iii)

\({\mathcal {D}}\) is isomorphic to a 3-(10, 4, 1) design, and \(G \cong M_{10}\), \(PGL_2(9)\) or \(P\Gamma L_2(9)\);

(iv)

\({\mathcal {D}}\) is isomorphic to a 3-(10, 4, 6) design, and \(Soc(G)=A_6\).

next article Another look at key randomisation hypotheses

Block R.E.: On the orbits of collineation groups. Math. Z. 96, 33–49 (1967).MathSciNetCrossRefMATH

Bosma W., Cannon J., Playoust C.: The magma algebra system I: the user language. J. Symbolic Comput. 24(3–4), 235–265 (1997).MathSciNetCrossRefMATH

Cameron P.J., Praeger C.E.: Block-transitive \(t\)-designs I: point-imprimitive designs. Discrete Math. 118(1–3), 33–43 (1993).

Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs, 2nd edn Chapman & Hall/CRC, Boca Raton (2007).MATH

Cusack C.A.: Semiregular large sets, University of Nebraska–Lincoln (1998).

Delandtsheer A., Doyen J.: Most block-transitive \(t\)-designs are point-primitive. Geom. Dedicata 29, 307–310 (1989).

Gan Y.S., Liu W.J.: Block-transitive, point-primitive Steiner \(3\)-designs. arXiv preprint (2021). arXiv:2112.00466.

Hanani H., Hartman A., Kramer E.S.: On three-designs of small order. Discrete Math. 45(1), 75–97 (1983).MathSciNetCrossRefMATH

Huber M.: Classification of flag-transitive Steiner quadruple systems. J. Combin. Theory Ser. A 94, 180–190 (2001).MathSciNetCrossRefMATH

10.

Huber M.: The classification of flag-transitive Steiner 3-designs. Adv. Geom. 5(2), 195–221 (2005).MathSciNetCrossRefMATH

11.

Lan T., Liu W.J., Yin F.G.: Block-transitive \(3\)-\((v, k, 1)\) designs associated with alternating groups. Des. Codes Cryptogr. 91, 2791–2807 (2023).MathSciNetCrossRefMATH

12.

Lüneburg H.: Fahnenhomogene quadrupelsysteme. Math. Z. 89(1), 82–90 (1965).MathSciNetCrossRefMATH

13.

Mann A., Tuan N.D.: Block-transitive point-imprimitive \(t\)-designs. Geom. Dedicata 88(1), 81–90 (2001).MathSciNetCrossRefMATH

14.

Muzychuk M., Spiga P.: Finite primitive groups of small rank: symmetric and sporadic groups. J. Algebraic Combin. 52(2), 103–136 (2020).MathSciNetCrossRefMATH

15.

Rahimipour A.R., Moshtagh H.: Janko sporadic group \(J_2\) as automorphism group of 3-designs. Discrete Math. 344(2), 112194 (2021).CrossRefMATH

16.

The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.8.6 (2016). http://www.gap-system.org.

17.

Wielandt H.: Finite Permutation Groups. Acad. Press, New York (1964).MATH

18.

Wilson R., Walsh P., Tripp J., et al.: Atlas of finite group representations. https://brauer.maths.qmul.ac.uk/Atlas/v3/

19.

Zhan X.Q., Pang X., Wang Y.J.: Block-transitive 3-designs with block size at most 6. Graphs Combin. 38(5), 1–14 (2022).MathSciNetCrossRefMATH

20.

Zhou S.L., Wang Y.J.: Flag-transitive non-symmetric 2-designs with \((r,\lambda )= 1\) and alternating socle. Electron. J. Combin. 22(2), P2.6 (2015).

Title: Block-transitive 3- designs with sporadic or alternating socle
Authors: Xuan Pang
Xiaoqin Zhan
Publication date: 02-08-2023
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 12/2023
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-023-01275-9

Springer Professional

Block-transitive 3-\((v,4,\lambda )\) designs with sporadic or alternating socle

Abstract

Premium Partner

Springer Professional

Abstract

Please log in to get access to your license.

Other articles of this Issue 12/2023

Coding and bounds for partially defective memory cells

Skew differential Goppa codes and their application to Mceliece cryptosystem

Optimal Ferrers diagram rank-metric codes from MRD codes

Practical attacks on small private exponent RSA: new records and new insights

On the fourth weight of generalized Reed–Muller codes

Another look at key randomisation hypotheses

Premium Partner