Annals of Combinatorics

, Volume 5, Issue 2, pp 141–152 | Cite as

On Generalized k-Arcs in $ \Bbb {PG} $(2n, q)

  • B.N. Cooperstein
  • J.A. Thas


The notion of a generalized k-arc in \( \Bbb {PG} \)(2n, q) is introduced. When \( k = {{q^{n+1}-1} \over {q-1}} +1 \) it is demonstrated that the existence of a generalized k-arc in \( \Bbb {PG} \)(2n, q) leads to a construction of a partial geometry, a strongly regular graph and a two-weight code. Such k-arcs are called generalized hyperovals. It is proved that no such generalized hyperovals exist when q is odd. For each \( n \ge 2 \) and q = 2 it is shown that each generalized hyperoval of \( \Bbb {PG} \)(2n, q) is a partition of \( \Bbb {PG} \)(2n, 2)\ \( \Bbb {PG} \)(n, 2). Related structures are also discussed.

Keywords: projective space, k-arc, oval, hyperoval, strongly regular graph, partial geometry, two-weight code 


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Copyright information

© Birkhäuser Verlag Basel, 2001

Authors and Affiliations

  • B.N. Cooperstein
    • 1
  • J.A. Thas
    • 2
  1. 1.Department of Mathematics, University of California, Santa Cruz, CA 95064, USAUS
  2. 2.Department of Pure Mathematics and Computer Algebra, Ghent University, BelgiumBE

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