Remarks on polarity designs View Full Text


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Article Info

DATE

2012-09-25

AUTHORS

Dina Ghinelli, Dieter Jungnickel, Klaus Metsch

ABSTRACT

Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140, [11]) used polarities of PG(2d − 1, q) to construct non-classical designs with a hyperplane and the same parameters and same intersection numbers as the classical designs PGd(2d, q), for every prime power q and every integer d ≥ 2. Our main result shows that these properties already characterize their polarity designs. Recently, Jungnickel and Tonchev (Des. Codes Cryptogr. [14] introduced new invariants for simple incidence structures \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document}, which admit both a coding theoretic and a geometric description. Geometrically, one considers embeddings of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} into projective geometries Π = PG(n, q), where an embedding means identifying the points of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} with a point set V in Π in such a way that every block of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} is induced as the intersection of V with a suitable subspace of Π. Then the new invariant—which we shall call the geometric dimension geomdimq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} —is the smallest value of n for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} may be embedded into the n-dimensional projective geometry PG(n, q). The classical designs PGd(n, q) always have the smallest possible geometric dimension among all designs with the same parameters, namely n, and are actually characterized by this property. We give general bounds for geomdimq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \mathcal{D}}$$\end{document} whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} is one of the (exponentially many) “distorted” designs constructed in Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140, [11]; Des. Codes Cryptogr. 55:131–140, [12]—a class of designs with classical parameters which includes the polarity designs as a very special case. We also show that this class contains designs with the same parameters as PGd(n, q) and geomdimq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D} = n + 1}$$\end{document}, for every prime power q and for all values of d and n with 2 ≤ d ≤ n−1. Regarding the polarity designs, we conjecture that their geometric dimension always satisfies our general upper bound with equality, that is, geomdimq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D} = 4d}$$\end{document} for the polarity design \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} with the parameters of PGd(2d, q), but we are only able to establish this result if we restrict ourselves to the special case of “natural” embeddings. More... »

PAGES

7-19

References to SciGraph publications

  • 2010-09-14. Designs having the parameters of projective and affine spaces in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2012-02-29. New invariants for incidence structures in DESIGNS, CODES AND CRYPTOGRAPHY
  • 1999-09. Linear Perfect Codes and a Characterization of the Classical Designs in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2005-04-07. A new family of distance-regular graphs with unbounded diameter in INVENTIONES MATHEMATICAE
  • 2006-08. Independent Sets In Association Schemes in COMBINATORICA
  • 2011-08. Recent results on designs with classical parameters in JOURNAL OF GEOMETRY
  • 2009-05-23. The number of designs with geometric parameters grows exponentially in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2008-11-26. Polarities, quasi-symmetric designs, and Hamada’s conjecture in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2011-11-11. 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/s10623-012-9748-5

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    http://dx.doi.org/10.1007/s10623-012-9748-5

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The classical designs PGd(n, q) always have the smallest possible geometric dimension among all designs with the same parameters, namely n, and are actually characterized by this property. We give general bounds for geomdimq\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${ \\mathcal{D}}$$\\end{document} whenever \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal{D}}$$\\end{document} is one of the (exponentially many) \u201cdistorted\u201d designs constructed in Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131\u2013140, [11]; Des. Codes Cryptogr. 55:131\u2013140, [12]\u2014a class of designs with classical parameters which includes the polarity designs as a very special case. We also show that this class contains designs with the same parameters as PGd(n, q) and geomdimq\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal{D} = n + 1}$$\\end{document}, for every prime power q and for all values of d and n with 2\u00a0\u2264 d\u00a0\u2264 n\u22121. 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    15 schema:description Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140, [11]) used polarities of PG(2d − 1, q) to construct non-classical designs with a hyperplane and the same parameters and same intersection numbers as the classical designs PGd(2d, q), for every prime power q and every integer d ≥ 2. Our main result shows that these properties already characterize their polarity designs. Recently, Jungnickel and Tonchev (Des. Codes Cryptogr. [14] introduced new invariants for simple incidence structures \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document}, which admit both a coding theoretic and a geometric description. Geometrically, one considers embeddings of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} into projective geometries Π = PG(n, q), where an embedding means identifying the points of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} with a point set V in Π in such a way that every block of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} is induced as the intersection of V with a suitable subspace of Π. Then the new invariant—which we shall call the geometric dimension geomdimq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} —is the smallest value of n for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} may be embedded into the n-dimensional projective geometry PG(n, q). The classical designs PGd(n, q) always have the smallest possible geometric dimension among all designs with the same parameters, namely n, and are actually characterized by this property. We give general bounds for geomdimq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \mathcal{D}}$$\end{document} whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} is one of the (exponentially many) “distorted” designs constructed in Jungnickel and Tonchev (Des. Codes Cryptogr. 51:131–140, [11]; Des. Codes Cryptogr. 55:131–140, [12]—a class of designs with classical parameters which includes the polarity designs as a very special case. We also show that this class contains designs with the same parameters as PGd(n, q) and geomdimq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D} = n + 1}$$\end{document}, for every prime power q and for all values of d and n with 2 ≤ d ≤ n−1. Regarding the polarity designs, we conjecture that their geometric dimension always satisfies our general upper bound with equality, that is, geomdimq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D} = 4d}$$\end{document} for the polarity design \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document} with the parameters of PGd(2d, q), but we are only able to establish this result if we restrict ourselves to the special case of “natural” embeddings.
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