On Ovoids of the Generalized Quadrangle \(H(3,q^2)\)

Abstract

We construct examples and families of locally Hermitian ovoids of the generalized quadrangle \(H(3,q^2)\). We also obtain a computer classification of all locally Hermitian ovoids of \(H(3,q^2)\) for \(q \le 4\), and compare the obtained classification for \(q=3\) with the classification of all ovoids of H(3, 9) which is also obtained by computer.

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Notes

  1. 1.

    If \(\gcd (a,q^2-1)\), then \(a^{-1}\) denotes the inverse of a modulo \(q^2-1\).

  2. 2.

    Note that the group generated by the automorphisms mentioned under (G1), (G5), (G6), (G7) induces the full group of automorphisms of L stabilizing B.

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Communicated by Marni Mishna

Determination of Some \(E_l(q^2)\)’s

Determination of Some \(E_l(q^2)\)’s

If \(\lambda x^e\) is an I-monomial of \({{\mathbb {F}}}_{q^2}[x]\), then the map \(x \mapsto \lambda x^e\) is bijective and hence \(gcd(e,q^2-1)=1\). Also, as \(\frac{\lambda x^e- \lambda 0^e}{x-0} = \lambda x^{e-1} \not \in {{\mathbb {F}}}_q\) for all \(x \in {{\mathbb {F}}}_{q^2}^*\), we have \(\lambda \not \in {{\mathbb {F}}}_q\) and \(gcd(e-1,q^2-1) \not = 1\). For each prime power \(q \le 32\), we computed all I-monomials \(\lambda x^e\), see [7]. If we denote by \(E(q^2)\) the set of all \(e \in \{ 1,2,\ldots ,q^2-1 \}\) for which there exists an I-monomial of \({{\mathbb {F}}}_{q^2}[x]\) of the form \(\lambda x^e\), then we have

  • \(E(2^2)=\{ 1 \}\),

  • \(E(3^2)=\{ 1,3,5 \}\),

  • \(E(4^2)=\{ 1 \}\),

  • \(E(5^2)=\{ 1,5,7,13 \}\),

  • \(E(7^2)=\{ 1,7,17,25 \}\),

  • \(E(8^2)=\{ 1,4,10,16,19 \}\),

  • \(E(9^2)=\{ 1,3,9,11,27,41,51 \}\),

  • \(E(11^2)=\{ 1,11,13,37,61 \}\),

  • \(E(13^2)=\{ 1,13,29,85 \}\),

  • \(E(16^2) = \{ 1 \}\),

  • \(E(17^2)=\{ 1,17,19,37,91,109,145 \}\),

  • \(E(19^2)=\{ 1,19,181 \}\),

  • \(E(23^2)=\{ 1,23,25,49,97,169,265 \}\),

  • \(E(25^2)=\{ 1,5,25,53,125,313,365 \}\),

  • \(E(27^2)=\{1,3,9,27,29,57,81,99,113,243,281,365,393,477,603 \}\),

  • \(E(29^2)=\{ 1,29,31,271,421 \}\),

  • \(E(31^2)=\{ 1,31,481 \}\),

  • \(E(32^2) = \{ 1,4,16,34,64,67,256,331,397 \}\).

More generally, if \(l \in {{\mathbb {N}}}^*\), then \(E_l(q^2)\) denotes the set of all subsets \(\{ e_1,e_2,\ldots ,e_l \} \subseteq {{\mathbb {N}}}^*\) such that \(1 \le e_1< e_2< \cdots < e_l \le q^2-1\) and there exist \(\lambda _1,\lambda _2,\ldots ,\lambda _l \in {{\mathbb {F}}}_{q^2}^*\) such that \(x \mapsto \sum _{i=1}^l \lambda _i x^{e_i}\) is an I-permutation of \({{\mathbb {F}}}_{q^2}\). We denote by \(E_l'(q^2)\) the subset of \(E_l(q^2)\) consisting of all \(\{ e_1,e_2,\ldots ,e_l \} \subseteq {{\mathbb {N}}}^*\) satisfying the above conditions but with the extra requirement that \(\lambda _1 \not \in {{\mathbb {F}}}_q\) if \(e_1=1\). If \(I \in E_l(q^2) {\setminus } E_l'(q^2)\) for some \(l \in {{\mathbb {N}}}^*\), then \(l \ge 2\), \(1 \in I\) and \(I {\setminus } \{ 1 \} \in E_{l-1}'(q^2)\) by construction (C5). Conversely, by construction (C5) we know that if \(l \ge 2\) and \(I \in E_{l-1}'(q^2)\) with \(1 \not \in I\), then \(I \cup \{ 1 \} \in E_l(q^2)\). If one is interested in finding only one representative for each equivalence class of I-polynomials, it suffices to consider the sets \(E_l'(q^2)\). With the aid of a computer, we found all sets \(E_2'(q^2)\) for \(q \le 9\) and all sets \(E_3'(q^2)\) for \(q \le 5\) [7]:

  • \(E_2'(2^2) = E_2'(3^2) = \{ \}\),

  • \(E_2'(4^2) = \{ \{ 1,4 \} , \{ 2,8 \} \}\),

  • \(E_2'(5^2) = \{ \{ 1,5 \} , \{ 7,19 \} \}\),

  • \(E_2'(7^2) = \{ \{ 1,7 \} , \{ 5,17 \} , \{ 5,29 \} , \{17,41 \} \}\),

  • \(E_2'(8^2) = \{ \{ 1,8 \} , \{ 2,16 \} , \{ 4,32 \} , \{ 16,37 \} \}\),

  • \(E_2'(9^2) = \{ \{ 1,9 \} , \{ 3,27 \} , \{ 11,27 \}, \{ 21,61 \} , \{ 29,61 \} , \{ 31,71 \} \}\),

  • \(E_3'(2^2) = E_3'(3^2) = \{ \}\),

  • \(E_3'(4^2) = \{ \{ 1,6,11 \} \}\),

  • \(E_3'(5^2) = \{ \{ 1,7,13 \},\{ 1,9,17 \},\{ 3,11,19 \}\}\).

For additive I-polynomials, the corresponding ovoids of \(H(3,q^2)\) are translation ovoids related to so-called semifield spreads of \(\mathrm{PG}(3,q)\), see [5]. These spreads and their related semifields have extensively been discussed in the literature, see, e.g. [9].

Let us now turn our attention to non-additive I-monomials of \({{\mathbb {F}}}_{q^2}[x]\) with \(q \le 32\), non-additive I-binomials of \({{\mathbb {F}}}_{q^2}[x]\) with \(q \le 9\) and non-additive I-trinomials of \({{\mathbb {F}}}_{q^2}[x]\) with \(q \le 5\). We have verified by computer that for each \(q = p^h \le 32\) and each \(e \in E(q^2)\) not being a power of p, there exists up to equivalence a unique I-polynomial of the form \(\lambda x^e\). These I-monomials have been listed in Table 3. In this table, \(\alpha \) is a primitive element of \({{\mathbb {F}}}_{q^2}\) that is a root of the polynomial mentioned in Table 4.

We have also verified that each subset of \(E_2'(q^2)\), \(q=p^h \le 9\), and each subset of \(E_3'(q^2)\), \(q=p^h \le 5\), not entirely consisting of powers of p, is associated with either one or two I-polynomials (up to equivalence). These have been listed in Table 5. Also here, \(\alpha \) is a primitive element of \({{\mathbb {F}}}_{q^2}\) that is a root of the polynomial mentioned in Table 4.

Finally, we wish to mention that we found more subsets of the \(E_l(q^2)\)’s than the ones mentioned above. Through ad hoc searches we found, for instance, that \(\{ 1,17,33 \} \in E_3'(7^2)\) and \(\{ 1,7,13,19 \} \in E_4'(5^2)\).

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De Bruyn, B. On Ovoids of the Generalized Quadrangle \(H(3,q^2)\). Ann. Comb. 25, 495–514 (2021). https://doi.org/10.1007/s00026-021-00538-3

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Keywords

  • Ovoid
  • Locally Hermitian
  • (Hermitian) generalized
  • Indicator set
  • Polynomial

Mathematics Subject Classification

  • 51E12
  • 51E20
  • 11T06