New invariants for incidence structures View Full Text


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Article Info

DATE

2013-07

AUTHORS

Dieter Jungnickel, Vladimir D. Tonchev

ABSTRACT

We exhibit a new, surprisingly tight, connection between incidence structures, linear codes, and Galois geometry. To this end, we introduce new invariants for finite simple incidence structures , which admit both an algebraic and a geometric description. More precisely, we will associate one invariant for the isomorphism class of with each prime power q. On the one hand, we consider incidence matrices M with entries from GF(qt) for the complementary incidence structure , where t may be any positive integer; the associated codes C = C(M) spanned by M over GF(qt); and the corresponding trace codes Tr(C(M)) over GF(q). The new invariant, namely the q-dimension of , is defined to be the smallest dimension over all trace codes which may be obtained in this manner. This modifies and generalizes the q-dimension of a design as introduced in Tonchev (Des Codes Cryptogr 17:121–128, 1999). On the other hand, we consider embeddings of into projective geometries , where an embedding means identifying the points of with a point set V in in such a way that every block of is induced as the intersection of V with a suitable subspace of . Our main result shows that the q-dimension of always coincides with the smallest value of N for which may be embedded into the (N − 1)-dimensional projective geometry PG(N − 1, q). We also give a necessary and sufficient condition when actually an embedding into the affine geometry AG(N − 1, q) is possible. Several examples and applications will be discussed: designs with classical parameters, some Steiner designs, and some configurations. More... »

PAGES

163-177

References to SciGraph publications

  • 1978-10. Ranks of incidence matrices of Steiner triple systems in MATHEMATISCHE ZEITSCHRIFT
  • 1968. Finite Geometries, Reprint of the 1968 Edition in NONE
  • 1987-08. 83 inPG(2,q) in ARCHIV DER MATHEMATIK
  • 2014-07. Remarks on polarity designs in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2005-01. Symmetric (4,4)-Nets and Generalized Hadamard Matrices Over Groups of Order 4 in DESIGNS, CODES AND CRYPTOGRAPHY
  • 1999-09. Linear Perfect Codes and a Characterization of the Classical Designs in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2003-05. A Note on MDS Codes, n-Arcs and Complete Designs in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2011-08. Recent results on designs with classical parameters in JOURNAL OF GEOMETRY
  • 2010-05. The number of designs with geometric parameters grows exponentially in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2012-06. Weight enumeration of codes from finite spaces in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2009-05. Polarities, quasi-symmetric designs, and Hamada’s conjecture in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2012-10. A Hamada type characterization of the classical geometric designs in DESIGNS, CODES AND CRYPTOGRAPHY
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    http://scigraph.springernature.com/pub.10.1007/s10623-012-9636-z

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    http://dx.doi.org/10.1007/s10623-012-9636-z

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1011702149


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