On the Embedding of (k, p)-Arcs in Maximal Arcs

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Abstract

Our main result is that a (k, p)-arc in PG(2, q), q = ph, with k ≥ qp - q + p - 1/2 4√q can be extended to a maximal arc. Combining this result with the recent Ball, Blokhuis, Mazzocca theorem about the non-existence of maximal arcs for p > 2, it gives an upper bound for the size of a (k, p)-arc. The method can be regarded as a generalization of B. Segre's method for proving similar embeddability theorems for k-arcs (that is when n = 2). It is based on associating an algebraic envelope containing the short lines to the (k, p)-arc. However, the construction of the envelope is independent of Segre's method using the generalization of Menelaus' theorem.

Original languageEnglish
Pages (from-to)235-246
Number of pages12
JournalDesigns, Codes, and Cryptography
Volume18
Issue number1-3
Publication statusPublished - Dec 1 1999

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Keywords

  • (k, n)-arc
  • Arc

ASJC Scopus subject areas

  • Computer Science Applications
  • Applied Mathematics

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