Science China Mathematics

, Volume 55, Issue 1, pp 73–92 | Cite as

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.

Keywords

Hjelmslev geometry projective Hjelmslev plane arc finite chain ring Galois ring subplane affine subplane 

MSC(2010)

05B25 51C05 51E15 51E21 51E26 16P10 94B05 94B27 

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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Information and Electronic EngineeringZhejiang UniversityHangzhouChina
  2. 2.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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