On some d-dimensional dual hyperovals in PG(d(d+3)/2,2)

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• 作者：Hiroaki Taniguchi
• 刊名：European Journal of Combinatorics
• 出版年：2010
• 出版时间：January 2010
• 年：2010
• 卷：31
• 期：1
• 页码：401-410
• 全文大小：656 K

文摘

Let d≥3. Let H be a d+1-dimensional vector space over GF(2) and {e₀,…,e_d} be a specified basis of H. We define Supp(t){e_t1,…,e_tl}, a subset of a specified base for a non-zero vector t=e_t1++e_tl of H, and Supp(0)0/. We also define J(t)Supp(t) if Supp(t) is odd, and J(t)Supp(t){0} if Supp(t) is even.

For s,tH, let {a(s,t)} be elements of H(HH) which satisfy the following conditions: (1) a(s,s)=(0,0), (2) a(s,t)=a(t,s), (3) a(s,t)≠(0,0) if s≠t, (4) a(s,t)=a(s^′,t^′) if and only if {s,t}={s^′,t^′}, (5) {a(s,t)tH} is a vector space over GF(2), (6) {a(s,t)s,tH} generate H(HH). Then, it is known that S{X(s)sH}, where X(s){a(s,t)tH{s}}, is a dual hyperoval in PG(d(d+3)/2,2)=(H(HH)){(0,0)}.

In this note, we assume that, for s,tH, there exists some x_s,t in GF(2) such that a(s,t) satisfies the following equation: Then, we prove that the dual hyperoval constructed by {a(s,t)} is isomorphic to either the Huybrechts’ dual hyperoval, or the Buratti and Del Fra’s dual hyperoval.