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Designs, Codes and Cryptography

Erschienen in:

01.12.2015

Maximal arcs and quasi-symmetric designs

verfasst von: Dieter Jungnickel, Vladimir D. Tonchev

Erschienen in: Designs, Codes and Cryptography | Ausgabe 2-3/2015

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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.

Vorheriger Artikel Enumeration of Kerdock codes of length 64

Nächster Artikel Finding shortest lattice vectors faster using quantum search

We assume that the reader is familiar with the basic facts and terminology from Design Theory and Finite Geometry; see, for instance, [1, 5]. For the theory of quasi-symmetric designs, one may consult the monograph [13].

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.

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

Blokhuis A., Haemers W.H.: An infinite family of quasi-symmetric designs. J. Stat. Plan. Inference 95, 117–119 (2001).

de Resmini M.J., Ghinelli D., Jungnickel D.: Arcs and ovals from abelian groups. Des. Codes Cryptogr. 26, 213–228 (2002).

Ghinelli D., Jungnickel D., Metsch K.: Remarks on polarity designs. Des. Codes Cryptogr. 72, 7–19 (2014).

Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998).

Jungnickel D., Tonchev V.D.: Exponential number of quasi-symmetric designs with the symmetric difference property. Des. Codes Cryptogr. 1, 247–253 (1991).

Jungnickel D., Tonchev V.D.: On symmetric and quasi-symmetric designs with the symmetric difference property and their codes. J. Comb. Theory A 59, 40–50 (1992).

Jungnickel D., Tonchev V.D.: Polarities, quasi-symmetric designs, and Hamada’s conjecture. Des. Codes Cryptogr. 51, 131–140 (2009).

Kantor W.M.: Symplectic groups, symmetric designs and line ovals. J. Algebra 33, 43–58 (1975).

10.

Kantor W.M.: 2-Transitive symmetric designs. Graphs Comb. 1, 165–166 (1985).

11.

Rodrigues B.G., Tonchev V.D.: On quasi-symmetric 2-(64, 24, 46) designs derived from codes In: 4th International Castle Meeting, CIM Series in Mathematical Sciences, vol. 3, Portugal, September 15–18, 2014. Springer (2015, to appear).

12.

Shrikhande S.S., Raghavarao D.: A method of construction of incomplete block designs. Sankhyā A 25, 399–402 (1963).

13.

Shrikhande M.S., Sane S.S.: Quasi-symmetric Designs. Cambridge University Press, Cambridge (1991).

Titel: Maximal arcs and quasi-symmetric designs
verfasst von: Dieter Jungnickel
Vladimir D. Tonchev
Publikationsdatum: 01.12.2015
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 2-3/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-015-0065-7

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