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The maximal size of 6- and 7-arcs in projective Hjelmslev planes over chain rings of order 9

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Abstract

We complete the determination of the maximum sizes of (k, n)-arcs, n ≤ 12, in the projective Hjelmslev planes over the two (proper) chain rings ℤ9 = ℤ/9ℤ and \(\mathbb{S}_3 = \mathbb{F}_3 {{[X]} \mathord{\left/ {\vphantom {{[X]} {(X^2 )}}} \right. \kern-\nulldelimiterspace} {(X^2 )}}\) of order 9 by resolving the hitherto open cases n = 6 and n = 7. Parts of our proofs rely on decidedly geometric properties of the planes such as Desargues’ theorem and the existence of certain subplanes.

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Authors and Affiliations

  1. Department of Information and Electronic Engineering, Zhejiang University, Hangzhou, 310027, China

    Thomas Honold

  2. Mathematisches Institut, Universität Bayreuth, D-95440, Bayreuth, Germany

    Michael Kiermaier

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  1. Thomas Honold
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  2. Michael Kiermaier
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Correspondence to Thomas Honold.

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Dedicated to Professor Feng Keqin on the Occasion of his 70th Birthday

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Honold, T., Kiermaier, M. The maximal size of 6- and 7-arcs in projective Hjelmslev planes over chain rings of order 9. Sci. China Math. 55, 73–92 (2012). https://doi.org/10.1007/s11425-011-4296-4

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  • DOI: https://doi.org/10.1007/s11425-011-4296-4

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