×

Enumeration of symmetric \((45,12,3)\) designs with nontrivial automorphisms. (English) Zbl 1427.05017

Summary: We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence [J. Comb. Des. 4, No. 3, 155–175 (1996; Zbl 0912.05006)] have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric (45, 12, 3) designs. We prove that \(k\)-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric (45, 12, 3) designs.

MSC:

05A15 Exact enumeration problems, generating functions
05B05 Combinatorial aspects of block designs
20D45 Automorphisms of abstract finite groups
94B05 Linear codes (general theory)
05C38 Paths and cycles

Citations:

Zbl 0912.05006

Software:

Code Tables; Magma; GAP; MinT
PDFBibTeX XMLCite

References:

[1] R. W. Ahrens, G. Szekeres, On a combinatorial generalization of the 27 lines associated with a cubic surface, J. Austral. Math. Soc. 10 (1969) 485-492. · Zbl 0183.52203
[2] E. F. Assmus Jr., J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992. · Zbl 0762.05001
[3] T. Beth, D. Jungnickel, H. Lenz, Design Theory, 2nd Edition, Cambridge University Press, Cambridge, 1999. · Zbl 0945.05005
[4] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3-4) (1997) 235-265. · Zbl 0898.68039
[5] V. Ćepulić, On symmetric block designs (45,12,3) with automorphisms of order 5, Ars Combin. 37 (1994) 33-48. · Zbl 0811.05007
[6] C. J. Colbourn, J. F. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, 2nd Edition, CRC Press, Boca Raton, 2007. · Zbl 1101.05001
[7] K. Coolsaet, J. Degraer, E. Spence, The strongly regular (45, 12, 3, 3) graphs, Electron. J. Combin. 13(1) (2006) Research Paper 32, 1-9. · Zbl 1098.05081
[8] D. Crnković, B. G. Rodrigues, S. Rukavina, L. Simčić, Ternary codes from the strongly regular (45,12,3,3) graphs and orbit matrices of 2-(45,12,3) designs, Discrete Math. 312(20) (2012) 3000- 3010. · Zbl 1251.94047
[9] D. Crnković, S. Rukavina, Construction of block designs admitting an abelian automorphism group, Metrika 62(2-3) (2005) 175-183. · Zbl 1080.05011
[10] D. Crnković, S. Rukavina, On some symmetric (45, 12, 3) and (40, 13, 4) designs, J. Comput. Math. Optim. 1(1) (2005) 55-63. · Zbl 1064.05027
[11] D. Crnković, S. Rukavina, L. Simčić, On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes, Util. Math., to appear. · Zbl 1373.05028
[12] U. Dempwolff, Primitive rank 3 groups on symmetric designs, Des. Codes Cryptogr. 22(2) (2001) 191-207. · Zbl 0972.05008
[13] C. E. Frasser, k-geodetic Graphs and Their Application to the Topological Design of Computer Networks, Argentinian Workshop in Theoretical Computer Science, 28 JAIIO-WAIT ’99 (1999) 187- 203.
[14] The GAP Group,GAP - Groups,Algorithms,and Programming,Version 4.7.2;2013. (http://www.gap-system.org)
[15] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at http://www.codetables.de, Accessed on 2014-03-11. · Zbl 0666.35020
[16] Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Combinatorics ’90 (Gaeta, 1990), Ann. Discrete Math. 52 (1992) 275-277. · Zbl 0773.05010
[17] P. Kaski, P. R. J. Östergård, Classification Algorithms for Codes and Designs, Springer, Berlin, 2006. · Zbl 1089.05001
[18] P. Kaski and P. R. J. Östergård, There are exactly five biplanes with k = 11, J. Combin. Des. 16(2) (2008) 117-127. · Zbl 1140.05012
[19] T. Kölmel, Einbettbarkeit symmetrischer (45,12,3) Blockplaene mit fixpunktfrei operierenden Automorphismen, Heidelberg, 1991. · Zbl 0784.05013
[20] C. W. H. Lam, G. Kolesova, L. H. Thiel, A computer search for finite projective planes of order 9, Discrete Math. 92(1-3) (1991) 187-195. · Zbl 0749.05002
[21] E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press, Cambridge, 1983. · Zbl 0502.05010
[22] R. Mathon, A. Rosa, 2-(v, k, λ) Designs of Small Order, in: Handbook of Combinatorial Designs, 2nd ed., C.J. Colbourn, J.H. Dinitz (Editors), Chapman & Hall/CRC, Boca Raton (2007) 25-58. · Zbl 1113.05016
[23] R. L. MacFarland, A family of difference sets in non-cyclic groups, J. Combin. Theory Ser. A 15 (1973) 1-10. · Zbl 0268.05011
[24] V. Mandekić-Botteri, On symmetric block designs (45,12,3) with involutory automorphism fixing 15 points, Glas. Mat. Ser. III 36(56)(2) (2001) 193-222. · Zbl 1002.05004
[25] R. Mathon, E. Spence, On 2-(45,12,3) designs, J. Combin. Des. 4(3) (1996) 155-175. · Zbl 0912.05006
[26] MinT, Dept. of Mathematics, University of Salzburg, The online database for optimal parameters of (t,m,s)-nets, (t,s)-sequences, orthogonal arrays, linear codes, and OOAs, Online available at http://mint.sbg.ac.at/index.php, Accessed on 2014-03-11.
[27] R. M. Ramos, J. Sicilia, M. T. Ramos, A generalization of geodetic graphs: K-geodetic graphs, Investigacion Operativa 6 (1998) 85-101.
[28] L. Rudolph, A class of majority logic decodable codes, IEEE Trans. Inform. Theory 13(2) (1967) 305-307. · Zbl 0152.15410
[29] N. Srinivasan, J. Opatrný, V. S. Alagar, Construction of geodetic and bigeodetic blocks of connectivity k ≥ 3 and their relation to block designs, Ars Combin. 24 (1987) 101-114. · Zbl 0645.05015
[30] L.H.Soicher,DESIGN-aGAPpackage,Version1.6,23/11/2011.(http://www.gapsystem.org/Packages/design.html)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.