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Designs, Codes and Cryptography

, Volume 68, Issue 1–3, pp 105–126 | Cite as

The existence of maximal (q 2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

  • Thomas Honold
  • Michael Kiermaier
Article

Abstract

We prove that (q 2, 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q 2 and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q 2 + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear [q 3, 6, q 3q 2q] codes over \({\mathbb {F}_q}\) for every prime power q > 1 and linear [q 3 + q, 6,q 3q 2 −1] codes over \({\mathbb {F}_q}\) for the special case q = 2 r . Furthermore, we construct 2-arcs of size (q + 1)2/4 in the planes PHG(2, R) over Galois rings R of length 2 and odd characteristic p 2.

Keywords

Hjelmslev geometry Projective Hjelmslev plane Arc Maximal finite chain ring Galois ring Quadratic map Dembowski–Ostrom polynomial 

Mathematics Subject Classification

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

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

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

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