Crnković, Dean; Rodrigues, B. G.; Rukavina, Sanja; Tonchev, Vladimir D. Quasi-symmetric \(2\)-\((64, 24, 46)\) designs derived from \(\mathrm{AG}(3, 4)\). (English) Zbl 1367.05028 Discrete Math. 340, No. 10, 2472-2478 (2017). Summary: This paper completes the enumeration of quasi-symmetric \(2\)-\((64, 24, 46)\) designs supported by the dual code \(C^\bot\) of the binary linear code \(C\) spanned by the lines of \(\mathrm{AG}(3, 4)\), initiated in [B. G. Rodrigues and V. D. Tonchev, in: Coding theory and applications. 4th international Castle Meeting, ICMCTA, Palmela Castle, Portugal, September 15–18, 2014. Cham: Springer. 327–333 (2015; Zbl 1328.05021)]. It is shown that \(C^\bot\) supports exactly 30,264 nonisomorphic quasi-symmetric \(2\)-\((64, 24, 46)\) designs. The automorphism groups of the related strongly regular graphs are computed. Cited in 1 Document MSC: 05B30 Other designs, configurations 94B05 Linear codes (general theory) 05E30 Association schemes, strongly regular graphs 05A15 Exact enumeration problems, generating functions 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) Keywords:quasi-symmetric design; linear code; automorphism group; strongly regular graph Citations:Zbl 1328.05021 Software:Cliquer; Magma PDFBibTeX XMLCite \textit{D. Crnković} et al., Discrete Math. 340, No. 10, 2472--2478 (2017; Zbl 1367.05028) Full Text: DOI References: [1] Assmus Jr., E. F.; Key, J. D., Designs and their Codes (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.05001 [2] Beth, T.; Jungnickel, D.; Lenz, H., Design Theory (1999), Cambridge University Press: Cambridge University Press Cambridge [3] Blokhuis, A.; Haemers, W. H., An infinite family of quasi-symmetric designs, J. Statist. Plann. Inference, 95, 117-119 (2001) · Zbl 0978.05008 [4] Bosma, W.; Cannon, J., Handbook of Magma Functions (November 1994), Department of Mathematics, University of Sydney, http://magma.maths.usyd.edu.au/magma [7] Colbourn, C. J.; Dinitz, J. F., Handbook of Combinatorial Designs (2007), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton · Zbl 1101.05001 [8] Haemers, W.; Tonchev, V. D., Spreads in strongly regular graphs, Des. Codes Cryptogr., 8, 145-157 (1996) · Zbl 0870.05078 [9] Hamada, N., On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error correcting codes, Hiroshima Math. J., 3, 153-226 (1973) · Zbl 0271.62104 [10] Hirschfeld, J. W.P., Projective Geometries Over Finite Fields (1998), Oxford University Press · Zbl 0899.51002 [12] Jungnickel, D.; Tonchev, V. D., Maximal arcs and quasi-symmetric designs, Des. Codes Cryptogr., 77, 365-374 (2015) · Zbl 1323.05026 [13] Niskanen, S.; Östergård, P. R.J., Cliquer User’S Guide, Version 10 Tech. Rep. T48 (2003), Communications Laboratory, Helsinki University of Technology: Communications Laboratory, Helsinki University of Technology Espoo, Finland [14] Rodrigues, B. G.; Tonchev, V. D., On quasi-symmetric 2-(64,24, 46) designs derived from codes, (Pinto, R.; etal., Coding Theory and Applications. Coding Theory and Applications, CIM Series in Mathematical Sciences, vol. 3 (2015), Springer International Publishing: Springer International Publishing Switzerland), 327-333 · Zbl 1328.05021 [15] Rose, H. E., A Course on Finite Groups (2009), Springer-Verlag: Springer-Verlag London · Zbl 1200.20001 [16] Shrikhande, S. S.; Raghavarao, D., A method of construction of incomplete block designs, Sankhyā Ser. A, 25, 399-402 (1963) · Zbl 0126.35601 [17] Shrikhande, M. S.; Sane, S. S., Quasi-Symmetric Designs (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0767.05019 [18] Tonchev, V. D., (Combinatorial Configurations: Designs, Codes, Graphs. Combinatorial Configurations: Designs, Codes, Graphs, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 40 (1988), Wiley: Wiley New York) 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.