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.

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 $\{ 1,7,13,19 \} \in E_4'(5^2)$.