Abstract
We show that the construction of quasi-symmetric designs with parameters 2-\((q^3, q^{2}(q-1)/2, q(q^3 -q^2 -2)/4)\) and block intersection numbers \(q^{2}(q-2)/4\) and \(q^{2}(q-1)/4\) (where \(q \ge 4\) is a power of 2) given by Blokhuis and Haemers (J Stat Plan Inference 95:117–119, 2001) leads to exponential numbers of such designs. For \(q=4\), there are already at least 28,844 isomorphism classes.
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Notes
This result is essentially a special case of a more general construction due to Kantor [9]; we will discuss Kantor’s work at the end of this section.
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Acknowledgments
The second author wishes to thank the University of Augsburg, Germany, for the hospitality during his visit. He is also grateful for the support provided by a grant of the Alexander von Humboldt Foundation. Research partially supported by a research grant of the Alexander von Humboldt Foundation, and NSA Grant H98230-15-1-0042.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Cryptography, Codes, Designs and Finite Fields: In Memory of Scott A. Vanstone”.
This paper is dedicated to the memory of our esteemed colleague and collaborator Scott A. Vanstone. In particular, the first author published 17 joint papers with Scott during the years from 1986 to 1999, who hosted him regularly at the University of Waterloo. He wishes to thank him not only for this fruitful research collaboration but even more for the close personal friendship which developed over the years.
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Jungnickel, D., Tonchev, V.D. Maximal arcs and quasi-symmetric designs. Des. Codes Cryptogr. 77, 365–374 (2015). https://doi.org/10.1007/s10623-015-0065-7
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DOI: https://doi.org/10.1007/s10623-015-0065-7