Designs, Codes and Cryptography

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

The existence of maximal (q2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

Article

Abstract

We prove that (q2, 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q2 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 q2 + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear [q3, 6, q3q2q] codes over \({\mathbb {F}_q}\) for every prime power q > 1 and linear [q3 + q, 6,q3q2 −1] codes over \({\mathbb {F}_q}\) for the special case q = 2r. 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 p2.

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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