Vol. 12, No. 1, 2011

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New quotients of the $d$-dimensional Veronesean dual hyperoval in $\mathrm{PG}(2d+1,2)$

Hiroaki Taniguchi and Satoshi Yoshiara

Vol. 12 (2011), No. 1, 151–165
Abstract

Let d 3. For each e 1, Thas and Van Maldeghem constructed a d-dimensional dual hyperoval in PG(d(d + 3)2,q) with q = 2e, called the Veronesean dual hyperoval. A quotient of the Veronesean dual hyperoval with ambient space PG(2d + 1,q), denoted Sσ, is constructed by Taniguchi, using a generator σ of the Galois group Gal(GF(qd+1)GF(q)).

In this note, using the above generator σ for q = 2 and a d-dimensional vector subspace H of GF(2d+1) over GF(2), we construct a quotient Sσ,H of the Veronesean dual hyperoval in PG(2d + 1,2) in case d is even. Moreover, we prove the following: for generators σ and τ of the Galois group Gal(GF(2d+1)GF(2)),

  • Sσ above (for q = 2) is not isomorphic to Sτ,H,
  • Sσ,H is isomorphic to Sσ,H for any d-dimensional vector subspaces H and H of GF(2d+1), and
  • Sσ,H is isomorphic to Sτ,H if and only if σ = τ or σ = τ1.

Hence, we construct many new non-isomorphic quotients of the Veronesean dual hyperoval in PG(2d + 1,2).

Keywords
dual hyperoval, Veronesean, quotient
Mathematical Subject Classification 2010
Primary: 05BXX, 05EXX, 51EXX
Milestones
Received: 5 January 2011
Authors
Hiroaki Taniguchi
Satoshi Yoshiara