Designs, Codes and Cryptography

, Volume 50, Issue 3, pp 359–372 | Cite as

Complete (q 2 + q + 8)/2-caps in the spaces PG(3, q), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG(3, 5)

  • Alexander A. DavydovEmail author
  • Stefano Marcugini
  • Fernanda Pambianco


An infinite family of complete (q 2 + q + 8)/2-caps is constructed in PG(3, q) where q is an odd prime ≡ 2 (mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q 2 + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails for q = 17.


Complete caps Projective spaces of the dimension three Projective planes 

Mathematics Subject Classification (2000)

51E21 51E22 94B05 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Alexander A. Davydov
    • 1
    Email author
  • Stefano Marcugini
    • 2
  • Fernanda Pambianco
    • 2
  1. 1.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussian Federation
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly

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