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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 807–816 | Cite as

Maximal arcs and extended cyclic codes

  • Stefaan De Winter
  • Cunsheng Ding
  • Vladimir D. TonchevEmail author
Article
  • 108 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

It is proved that for every \(d\ge 2\) such that \(d-1\) divides \(q-1\), where q is a power of 2, there exists a Denniston maximal arc A of degree d in \({\mathrm {PG}}(2,q)\), being invariant under a cyclic linear group that fixes one point of A and acts regularly on the set of the remaining points of A. Two alternative proofs are given, one geometric proof based on Abatangelo–Larato’s characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes.

Keywords

Maximal arc 2-Design Two-weight code Cyclic code 

Mathematics Subject Classification

05B05 05B25 51E15 94B15 

Notes

Acknowledgements

This material is based upon work that was done while the first author was serving at the National Science Foundation. The research of Cunsheng Ding was supported by the Hong Kong Grants Council, Proj. No. 16300415. Vladimir Tonchev acknowledges support by NSA Grant H98230-16-1-0011. The authors wish to thank the reviewers for their helpful remarks and suggestions that improved the manuscript.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Stefaan De Winter
    • 1
  • Cunsheng Ding
    • 2
  • Vladimir D. Tonchev
    • 3
    Email author
  1. 1.Michigan Technological UniversityHoughtonUSA
  2. 2.The Hong Kong University of Science and TechnologyHong KongHong Kong
  3. 3.Michigan Technological UniversityHoughtonUSA

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