A Hamada type characterization of the classical geometric designs View Full Text


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Article Info

DATE

2012-10

AUTHORS

Dieter Jungnickel, Vladimir D. Tonchev

ABSTRACT

The dimension of a combinatorial design over a finite field F = GF(q) was defined in (Tonchev, Des Codes Cryptogr 17:121–128, 1999) as the minimum dimension of a linear code over F that contains the blocks of as supports of nonzero codewords. There it was proved that, for any prime power q and any integer n ≥ 2, the dimension over F of a design that has the same parameters as the complement of a classical point-hyperplane design PGn-1(n, q) or AGn-1(n, q) is greater than or equal to n + 1, with equality if and only if is isomorphic to the complement of the classical design. It is the aim of the present paper to generalize this Hamada type characterization of the classical point-hyperplane designs in terms of associated codes over F = GF(q) to a characterization of all classical geometric designs PGd(n, q), where 1 ≤ d ≤ n − 1, in terms of associated codes defined over some extension field E = GF(qt) of F. In the affine case, we conjecture an analogous result and reduce this to a purely geometric conjecture concerning the embedding of simple designs with the parameters of AGd(n, q) into PG(n, q). We settle this problem in the affirmative and thus obtain a Hamada type characterization of AGd(n, q) for d = 1 and for d > (n − 2)/2. More... »

PAGES

15-28

References to SciGraph publications

  • 1978-10. Ranks of incidence matrices of Steiner triple systems in MATHEMATISCHE ZEITSCHRIFT
  • 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
  • 2005-01. Symmetric (4,4)-Nets and Generalized Hadamard Matrices Over Groups of Order 4 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
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    http://scigraph.springernature.com/pub.10.1007/s10623-011-9580-3

    DOI

    http://dx.doi.org/10.1007/s10623-011-9580-3

    DIMENSIONS

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