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Volume 19, Issue 1

# Classification of 8-dimensional rank two commutative semifields

Michel Lavrauw
/ Morgan Rodgers
Published Online: 2018-03-26 | DOI: https://doi.org/10.1515/advgeom-2017-0064

## Abstract

We classify the rank two commutative semifields which are 8-dimensional over their center 𝔽q. This is done using computational methods utilizing the connection to linear sets in PG(2, q4). We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over 𝔽3. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.

Keywords: Semifield; commutative semifield; flocks; linear sets; eggs

MSC 2010: 51E20; 12K10

## 1 Introduction and motivation

A semifield is a possibly non-associative algebra with a one and without zero divisors. Finite semifields are well studied objects in combinatorics and finite geometry and have many connections to other interesting geometric structures. They play a central role in the theory of projective planes ([12]), generalised quadrangles ([26]), and polar spaces ([32]), and have applications to perfect nonlinear functions and cryptography ([5]), and maximum rank distance codes ([9]). We refer to the chapter [19] and the references contained therein for background, definitions and more details about these connections.

Of particular interest are commutative semifields which are of rank two over their middle nucleus, so-called Rank Two Commutative Semifields (R2CS); see [8], [2], [19, Section 5]. The property of being commutative implies that these semifields have applications to perfect nonlinear functions (see e. g. [6] for a survey on planar functions and commutative semifields and for further references). Moreover, R2CS are equivalent to semifield flocks of a quadratic cone in a 3-dimensional projective space. We refer to the introduction of [21] for an excellent historical overview of the theory of flocks in finite geometry. Consequently, R2CS are also equivalent to translation ovoids of Q(4, q), the parabolic quadric in 4-dimensional projective space. We refer to [11], [22], [14], [13], [1], [15] for further details on these connections. Another, rather remarkable, application of R2CS concerns the theory of eggs and translation generalized quadrangles, see [26, Section 8.7]. As of now the only known examples of eggs in PG(4n − 1, q) are either “elementary”, i.e. obtained from an oval or an ovoid by applying the technique of field reduction ([20]), or they are obtained from a R2CS (up to dualising); see e.g. [13, Chapter 3], [14].

In this paper we present a computational classification of 8-dimensional rank two commutative finite semifields (that is, 8-dimensional over their centre). This classification relies on the bounds obtained in [4] and [16] on the size of the centre as a function of the dimension. Previous classification results have been obtained for 2-dimensional semifields (see [10]), for 3-dimensional semifields ([24] and [4]), for 4-dimensional rank two semifields ([7]) and for 6-dimensional rank two semifields with an extra assumption on the size of one of the other nuclei ([23]). Computational classification results have been obtained in [28], and [29]. For an overview and further classification results in the theory of finite semifields we refer to [17, Section 1] and [19, Section 6].

We begin in Section 2 by establishing some basic terminology and giving details on the known examples; we then explain the geometric model we use to search for new examples of rank two commutative semifields. In Section 3 we determine which fields 𝔽q satisfy a necessary condition to be the centre of an 8-dimensional R2CS, and in Sections 4 and 5 we give the results of our exhaustive search for new examples for the field orders which meet this necessary condition. Finally in Section 6 we give the corresponding existence results for semifield flocks in PG(3, q4), translation ovoids of Q(4, q4), and eggs in PG(15, q).

## 2 Preliminaries

We use the notation and terminology from [19]. Given a finite semifield 𝕊 with multiplication (x, y) ↦ xy, it is natural to consider the following substructures. The left nucleus ℕ(𝕊) is the set of elements x ∈ 𝕊 such that x ∘(yz) = (xy) ∘ z for all y, z ∈ 𝕊. Analogously, one defines the middle and right nucleus. The intersection of these three subsets of 𝕊 is called the nucleus of 𝕊 and the intersection of the nucleus of 𝕊 with the commutative centre of 𝕊 is called the centre of 𝕊. When we mention the dimension of a semifield, we are referring to the dimension over its centre. Restricting the addition and multiplication to any of these substructures one obtains a field. The rank of 𝕊 is the dimension of 𝕊 as a vector space over its middle nucleus. Hence a rank two commutative semifield (R2CS) is a semifield with commutative multiplication and which is a two-dimensional vector space over its middle nucleus. A semifield 𝕊 is commutative if and only if 𝕊d = 𝕊, and a semifield is called symplectic if and only if [𝕊t] = [𝕊].

Semifields are studied up to isotopisms and their Knuth orbit. Two semifields 𝕊1 and 𝕊2 are isotopic if there exist non-singular linear maps F, G, H from 𝕊1 to 𝕊2 such that xF2 yG = (x1 y)H for all x, y ∈ 𝕊1. The isotopism class of 𝕊 is denoted by [𝕊]. The Knuth orbit of a semifield 𝕊 is a set of at most six isotopism classes 𝓚(𝕊) = {[𝕊], [𝕊t], [𝕊d], [𝕊td],[𝕊dt],[𝕊tdt]}, where the operations t and d denote the transpose and dual operations obtained from the action of the transpositions in the symmetric group S3 on the indices of the cubical array of structure constants of the semifield.

To fix notation when considering a R2CS 𝕊, we denote the centre by 𝔽q, the finite field with q elements, and we assume that the middle nucleus is 𝔽qn. This makes 𝕊 into a 2n-dimensional R2CS of size q2n.

There are only three known families of examples of R2CS. (Note that by the above definition a finite field is not an R2CS, but in some papers the finite field is also considered as an R2CS.) We give a representation of the corresponding multiplications as binary operations defined on 𝔽qn × 𝔽qn and denoted by juxtaposition ∘. Also note that n is necessarily at least 2 since for n = 1 one obtains a 2-dimensional semifield which, by Dickson [10], is a field.

The first examples go back to a construction by Dickson in [10] and exist for each odd prime power q and n ≥ 2:

$(x,y)∘(u,v)=(xv+yu,yv+mxσuσ),$(1)

where σ ∈ Aut(𝔽qn/𝔽q) and m is a non-square in 𝔽qn. The second family of R2CS was constructed by Cohen and Ganley [8] and exists for q = 3 and n ≥ 2:

$(x,y)∘(u,v)=(xv+yu+x3u3,yv+ηx9u9+η−1xu),$(2)

where η is a non-square in 𝔽3n. The third family is the example found by Penttila and Williams [27] for q = 3 and n = 5 and has multiplication:

$(x,y)∘(u,v)=(xv+yu+x27u27,yv+x9u9).$(3)

Cohen and Ganley [8] showed that R2CS in even characteristic do not exist (again note that with our definition the finite field is not an R2CS) and that any R2CS in odd characteristic arises from what we refer to as a Cohen–Ganley pair (f, g) of functions: a pair of 𝔽q-linear functions satisfying the property that g2(t)−4tf(t) is a non-square for all nonzero t ∈ 𝔽qn, where q is odd. Each Cohen–Ganley pair (f, g) gives rise to a semifield 𝕊(f, g) with multiplication (x, y) ∘ (u, v) = (xv + yu + f(xu),yv + g(xu)). The condition that g2(t)−4tf(t) is a non-square for all nonzero t ∈ 𝔽qn is equivalent to the existence of an 𝔽q-linear set 𝓦 of rank n whose points have coordinates (t, f(t), g(t)), t$\begin{array}{}{\mathbb{F}}_{{q}^{n}}^{\ast },\end{array}$ contained in the set of internal points of the conic with equation $\begin{array}{}{X}_{2}^{2}\end{array}$ − 4X0X1 = 0.

If 𝓦 is contained in a line then the R2CS is of Dickson type. So we are interested in examples where 𝓦 contains an 𝔽q-subplane of PG(2, qn). Using a computer search, we complete the classification of 8-dimensional R2CSs. This is equivalent to classifying the semifield flocks in PG(3, q4) having kernel containing 𝔽q, the translation ovoids of Q(4, q4) with kernel containing 𝔽q, and good eggs in PG(15, q). We also classify the 10-dimensional R2CS with centre 𝔽3, the semifield flocks in PG(3, 35) with kernel 𝔽3, the translation ovoids in Q(4, 35) with kernel 𝔽3, and the good eggs of PG(19, 3). These applications are detailed in Section 6.

Our work relies on bounds on the size of the centre, as a function of the dimension, that were first given in [4] and later improved in [16] by showing that in order for an 𝔽q-subplane contained in 𝓙(𝓒) to exist, there must be an 𝔽q-subline contained in an external line of 𝓒 and made up entirely of points of 𝓙(𝓒).

#### Theorem 2.1

([16]). There are no 𝔽q-sublines contained in ∩ 𝓙(𝓒), where 𝓒 is a conic in PG(2, qn) and is an external line to 𝓒, for q ≥ 4n2 − 8n + 2, and for q > 2n2 − (4 − 2 $\begin{array}{}\sqrt{3}\end{array}$)n + (3 − 2 $\begin{array}{}\sqrt{3}\end{array}$) when q is prime.

#### Corollary 2.2

([16]). No 𝔽q-subplane contained in 𝓙(𝓒) exists, where 𝓒 is a conic in PG(2, qn), for q ≥ 4n2 − 8n + 2, and for q > 2n2 − (4 − 2 $\begin{array}{}\sqrt{3}\end{array}$)n + (3 − 2 $\begin{array}{}\sqrt{3}\end{array}$) when q is prime.

Let q be odd and consider the conic 𝓒 in PG(2, qn) defined by the quadratic form Q = X0X1$\begin{array}{}{X}_{2}^{2}.\end{array}$ Note that the point (0, 0, 1) lies on the tangent [1, 0, 0], so this point is external. Since Q(0, 0, 1) = −1, the internal points 𝓙(𝓒) are those with −Q(v) ∈ ◻̸. The stabilizer G = PGO(3, qn) of 𝓒 in PGL(3, qn) has order qn(q2n − 1) and contains all matrices of the form

$a2b2abc2d2cd2acabdad+bc$

where adbc ≠ 0 (vector multiplication is from the left). We have the following, due to Payne [25]:

#### Theorem 2.3

1. G is sharply triply transitive on the points of 𝓒;

2. G is transitive on 𝓙(𝓒);

3. G is transitive on 𝓔(𝓒);

4. G is sharply triply transitive on the tangent lines to 𝓒;

5. G is transitive on the external lines to 𝓒;

6. G is transitive on the secant lines to 𝓒;

7. G is transitive on the point-line pairs (p, ), where p is an external point on the exterior (respectively, secant) line . The subgroup of G fixing such a pair has order 4.

8. G is transitive on the point-line pairs (p, ), where p is an internal point on the exterior (respectively, secant) line . The subgroup of G fixing such a pair has order 4.

## 3 Existence of sublines contained in 𝓙(𝓒)

Our first goal is to determine precisely the values of q for which there exists an 𝔽q-subline contained in an external line to a conic 𝓒 in PG(2, qn) consisting entirely of internal points of 𝓒. To accomplish this we choose η so that − η ∈ ◻̸ and − η − 1 ∈ ◻; then x = (1, η, 0) ∈ 𝓙(𝓒) and e = 〈(1, η, 0), (0, − 2 η, 1)〉 is an external line. The stabilizer in G of x has order 2(qn + 1) and contains the following (normalized) matrices:

$Gx=10001000±1∪a21a1η2a2−aη±2aη∓2a±1η−a2:a∈E$

Now considering the external line e = 〈(1, η, 0), (0, −2η, 1)〉 on x, the subgroup of G stabilizing both x and e has order 4 and is given by

$Gx,ℓe=0101η20000−1η,1111η21−1η2η−21η−1.$

Note that the second generator given for this group fixes e pointwise.

Since G acts transitively on the pairs (p, ) where p is an internal point on an external line , it is sufficient to look for sublines of e containing x and contained in 𝓙(𝓒). A subline is determined by three collinear points; so a subline of e contained in 𝓙(𝓒) is determined by x, y, and x + μy for some y ∈ (e ∩ 𝓙(𝓒)) ∖ {x} and μ$\begin{array}{}{\mathbb{F}}_{{q}^{n}}^{\ast }\end{array}$ satisfying −Q(x + λμy) ∈ ◻̸ for all λ ∈ 𝔽q. The subline determined by these three points is given by {y} ∪ {x + λμy : λ ∈ 𝔽q}.

Using the basis {v1 = (1, η, 0), v2 = (0, −2η, 1)}, we can associate e with PG(1, qn) having the induced quadratic form Qe(x1v1 + x2v2) = $\begin{array}{}\eta {x}_{1}^{2}-2\eta {x}_{1}{x}_{2}-{x}_{2}^{2}\end{array}$ (this form is anisotropic, and is just used to separate the points of e into 𝓙(𝓒) and 𝓔(𝓒)).

We want to take a point in (e ∩ 𝓙(𝓒))∖ {x}. Since −Q(v2) = 1 ∈ ◻, we have v2 ∉ 𝓙(𝓒); therefore we define yb = v1 + b v2 with b ≠ 0. Now we have yb ∈ 𝓙(𝓒) as long as b satisfies b2+ 2ηbη ∈ ◻̸. Let 𝓑 = {s : s ∈ 𝔽qns2+ 2ηsη ∈ ◻̸}, then we have that (e ∩ 𝓙(𝓒)) = {x} ∪ {yb : b ∈ 𝓑}.

Now for our choices of μ, instead of letting μ range over all possible values in $\begin{array}{}{\mathbb{F}}_{{q}^{n}}^{\ast }\end{array}$, it is sufficient to consider a set 𝓢 of representatives of $\begin{array}{}{\mathbb{F}}_{{q}^{n}}^{\ast }/{\mathbb{F}}_{q}^{\ast }.\end{array}$ Note that x + λμyb = v1 + λμ (v1 + b v2) = (1 + λμ)v1 + λμ b v2; normalizing this vector to v1 + $\begin{array}{}\frac{\lambda \mu }{1+\lambda \mu }\end{array}$ bv2, we see that it is contained in 𝓙(𝓒) if and only if $\begin{array}{}\frac{\lambda \mu }{1+\lambda \mu }\end{array}$ b ∈ 𝓑. This is equivalent to having (2b − 1)μ2ηλ2+2(b − 1)μηλ + b2η ∈ ◻̸ for all λ ∈ 𝔽q.

To find sublines efficiently, we compute the set 𝓑 and, for each μ ∈ 𝓢, the sequence $\begin{array}{}\left[\frac{\lambda \mu }{1+\lambda \mu }:\lambda \in {\mathbb{F}}_{q}^{\ast }\right].\end{array}$ Then for each pair (b,μ) ∈ 𝓑 × 𝓢, we check whether $\begin{array}{}\frac{\lambda \mu }{1+\lambda \mu }\end{array}$ b ∈ 𝓑 for all λ ∈ 𝔽q. In this way, we obtain the number of 𝔽q-sublines of e containing x and completely contained in 𝓙(𝓒) where 𝓒 is a conic in PG(2, qn) for n = 3 and n = 4. We also obtain some partial results for n = 5, however it was impossible for us to complete our computations for q ∈ {37, 41, 43, 49} in this case. Our results for n = 3 agree with those found in [3]. By the bounds in Theorem 2.1, when n = 3 we only need to consider q < 14; for n = 4 we only need to consider q < 30; and when n = 5 we only need to consider q < 47, along with q = 49. In the table, a 0 indicates that no subline was found, while a dash indicates that the existence is ruled out by the theoretical bound.

Our computational results show the following.

#### Theorem 3.1

If there exists an 𝔽q-subline in PG(2, q4) contained in ∩ 𝓙(𝓒) for some conic 𝓒, where is an external line to 𝓒, then q ≤ 19.

## 4 Finding subplanes

Our next goal is to determine, given the existence of the necessary 𝔽q-subline, when there exist 𝔽q-subplanes of PG(2, qn) contained in 𝓙(𝓒). An 𝔽q-subplane is completely determined by a quadrangle, so more generally, two 𝔽q-sublines that are not contained in a common line of PG(2, qn) will determine an 𝔽q-subplane of PG(2, qn).

To determine the existence of 𝔽q-subplanes contained in 𝓙(𝓒), we first fix the point x and then find all of the 𝔽q-sublines containing x which are completely contained in 𝓙(𝓒) (those spanning an external line to 𝓒 as well as those spanning a secant line). Then, for each pair of 𝔽q-sublines through x (not contained in a common line of PG(2, qn)), we test whether the 𝔽q-subplane they determine is a subset of 𝓙(𝓒).

In the previous section we give details on finding the 𝔽q-sublines of the external line e on x which are contained in 𝓙(𝓒). Once we have these sublines, we compute their images under Gx to get all of the 𝔽q-sublines on x contained in 𝓙(𝓒) generating an external line to 𝓒. We then repeat this process beginning with the secant line s = 〈(1, 0, 0), (0, 1, 0)〉 on x. Since all 𝔽q-sublines are assumed to contain x, and since an 𝔽q-subline is determined by 3 points, we save the sublines as an ordered pair {@ y, z @} where {x, y,z} determines the subline.

The real computationally intensive aspect of our work concerns the determination of whether two sublines form a compatible pair, that is, if the two sublines determine a rank 3 𝔽q-linear set which is contained in 𝓙(𝓒). For two 𝔽q sublines 1 and 2 generated by {x, y1, z1} and {x, y2, z2}, respectively, we first compute values μ1 and μ2 so that the 𝔽q-subplane spanned by these two lines is given by 〈x, μ1y1, μ2y2q. Then we test that λμ1y1+y2 ∈ 𝓙(𝓒) for all λ$\begin{array}{}{\mathbb{F}}_{q}^{\ast }\end{array}$, and that x + λ1 μ1y1+λ2 μ2 y2 ∈ 𝓙(𝓒) for all λ1, λ2$\begin{array}{}{\mathbb{F}}_{q}^{\ast }\end{array}$. If these conditions are satisfied, then 1 and 2 generate an 𝔽q-subplane contained in 𝓙(𝓒).

Our computational work proves the following.

#### Theorem 4.1

No 𝔽q-subplane contained in 𝓙(𝓒) exists, where 𝓒 is a conic in PG(2, q4), unless q = 3.

With n = 4 and q = 3 we find 13 𝔽3-subplanes (up to conjugacy in pΓl(3, 34)) contained in 𝓙(𝓒), 10 of which can be embedded in the linear set associated with the Cohen–Ganley semifield.

## 5 Linear sets of higher rank

To put together rank 4 𝔽q-linear sets, we first need to find all the rank 3 linear sets (not just the subplanes). It is fairly easy to find the examples that are contained in a line. Each rank 3 linear set is saved as an ordered pair of 𝔽q-linear lines containing x. Then, for each 𝔽q-subline contained in either e or s, we compile the set Π of rank 3 𝔽q-linear sets whose first generating line is . We form a graph Γ on Π, where two planes π1, π2Π are adjacent if their second generating lines together generate a rank 3 𝔽q-linear set contained in 𝓙(𝓒). Then a clique of size q(q + 1) in Γ corresponds to a rank 4 𝔽q-linear set contained in 𝓙(𝓒).

Running this algorithm using the rank 3 linear sets found in PG(2, 34), we find 174 rank 4 𝔽q-linear sets contained in 𝓙(𝓒) that contain an 𝔽q-subplane. They are all equivalent up to isomorphism, corresponding to a semifield of Cohen–Ganley type. We are also able to run this algorithm in PG(2, 35), using an increased clique size to look for rank 5 𝔽q-linear sets; here all examples found correspond to a semifield of Cohen–Ganley type or else to the example due to Penttila and Williams.

#### Theorem 5.1

An 8-dimensional R2CS is either a Dickson semifield, or of CohenGanley type (with centre 𝔽3).

#### Theorem 5.2

A 10-dimensional R2CS with centre 𝔽3 is either a Dickson semifield, of CohenGanley type, or the example of PenttilaWilliams.

## 6 Implications of our results

There are many connections between R2CS and various geometric objects. Here we give details on some of these connections, and state the implications of our classification of R2CS in these other settings.

## 6.1 Semifield flocks

A flock of a quadratic cone 𝓚 of PG(3, qn) having vertex v is a partition of 𝓚 ∖ {v} into qn conics. We let v = (0, 0, 0, 1) and let the conic 𝓒 in the plane π = [0, 0, 0, 1] be the base of the cone. Then the planes of the flock can be written as {πt : tX0 + f(t)X1 + g(t)X2 + X3 = 0 ∣ t ∈ 𝔽qn} for some f, g : 𝔽qn → 𝔽qn; we denote such a flock by 𝓕(f, g).

A flock corresponds to a set 𝓦 = {(t, f(t), g(t)) ∣ t ∈ 𝔽qn} of interior points of a conic 𝓒′ in PG(2, qn); see [30]. If f and g are linear over a subfield of 𝔽qn (i.e. if (f, g) is a Cohen–Ganley pair) then we say 𝓕(f, g) is a semifield flock. The maximal subfield of 𝔽qn for which f and g are linear is called the kernel of the semifield flock. Notice that if 𝓕(f, g) is a semifield flock of PG(3, qn) with kernel 𝔽q, then 𝓦 is a rank n 𝔽q-linear set contained in the set of interior points of a conic in PG(2, qn), so such a semifield flock is equivalent to a 2n-dimensional R2CS with centre 𝔽q. Furthermore if 𝓦 is contained in a line then 𝓕 is of Kantor–Knuth type [30]; this corresponds to a R2CS of Dickson type.

#### Corollary 6.1

A semifield flock of PG(3, q4) with kernel 𝔽q is of KantorKnuth type or of CohenGanley type (with kernel 𝔽3).

#### Corollary 6.2

A semifield flock of PG(3, 35) with kernel 𝔽3 is of KantorKnuth type, of CohenGanley type, or of PenttilaWilliams type.

## 6.2 Ovoids of the parabolic quadric in 4-dimensional projective space

The parabolic quadric Q(4, s) is the incidence structure of points and lines of a nondegenerate quadric in PG(4, s). The quadric Q(4, s) is also an example of a generalized quadrangle, and is known to be isomorphic to the example T2(𝓒) constructed from a conic 𝓒 in PG(2, s); see [26].

A set of s2+1 points 𝓞 of Q(4, s) is called an ovoid if no two points of 𝓞 are collinear in Q(4, s). An ovoid 𝓞 in Q(4, s) is a translation ovoid if there is a point p ∈ 𝓞 and a group G of collineations of Q(4, s) stabilizing 𝓞, fixing p, and acting regularly on 𝓞∖ {p}. This group G is necessarily elementary abelian, and hence is a vector space over some subfield 𝔽q of 𝔽s; the largest such subfield is called the kernel of the translation ovoid. Put n = [𝔽s : 𝔽q], so s = qn.

By [16, Section 3.2], Theorem 3.1 has the following applications to ovoids. Given an ovoid 𝓞 of Q(4, qn), for each point p ∈ 𝓞 fix some conic 𝓒p contained in the cone p ∩ Q(4, qn); we denote the plane containing 𝓒p by πp. Then we consider Q(4, qn) ≃ T2(𝓒p). In this model, p corresponds to the point (∞), and 𝓞 ∖ {p} corresponds to a set 𝓥p of q2n affine points. Each two points of 𝓥p span a line intersecting πp in a point not on 𝓒p. Define 𝓤p = {〈x, y〉 ∩ πpx, y ∈ 𝓥p}. If the set 𝓤p contains a dual 𝔽q-subline on an internal point with respect to 𝓒p, then dualising over 𝔽q, we have an 𝔽q-subline spanning an external line with respect to 𝓒p. This gives the following.

#### Corollary 6.3

If 𝓞 is an ovoid in Q(4, q4), where q is odd, and if 𝓤p contains a dual 𝔽q-subline on an internal point of 𝓒p for some point p ∈ 𝓞, then q ≤ 19.

If 𝓞 is a translation ovoid of Q(4, qn) with respect to the point p having kernel 𝔽q, then the set 𝓤p is a rank 2n 𝔽q-linear set, and its dual is a rank n 𝔽q-linear set contained in 𝓙(𝓒p).

#### Corollary 6.4

A translation ovoid in Q(4, q4) with kernel 𝔽q is either a Kantor ovoid, or a ThasPayne ovoid (with q = 3).

#### Corollary 6.5

A translation ovoid in Q(4, 35) with kernel 𝔽3 is either a Kantor ovoid, a ThasPayne ovoid, or the PenttilaWilliams ovoid.

## 6.3 Eggs

An egg 𝓔 in PG(4n − 1, q) is a partial (n − 1)-spread of size q2n+1 such that every 3 elements of 𝓔 span a (3n − 1)-space and, for every element E ∈ 𝓔 there exists a (3n − 1)-space TE, called the tangent space of 𝓔 at E, containing E and disjoint from every other egg element. An egg is called good at E ∈ 𝓔 if every (3n − 1)-space containing E and at least two other elements of 𝓔 contains exactly qn + 1 elements of 𝓔. An egg 𝓔 is called a good egg if it is good at some E ∈ 𝓔. The standard example of an egg in PG(4n − 1, q) is obtained by applying field reduction to an ovoid of PG(3, qn); an egg obtained in this way is called elementary.

It is shown in [31] (see [18] for a shorter direct proof) that good eggs of PG(4n − 1, q), q odd, are equivalent to semifield flocks of PG(3, qn) with kernel containing 𝔽q. This gives us the following result.

#### Corollary 6.6

If 𝓔 is a good egg of PG(15, q) with kernel 𝔽q, q odd, then 𝓔 is either elementary, of KantorKnuth type, or of CohenGanley type.

Even if we do not assume that the egg 𝓔 has a good element, it is shown in [16] that an egg with certain properties implies the existence of an 𝔽q-subline contained in the set of interior points of a conic 𝓒 in PG(2, qn) which spans an external line with respect to 𝓒, giving the following result.

#### Corollary 6.7

Let 𝓔 be an egg of PG(15, q), q odd. If there exists an 11-space ρ containing an elementary pseudo-oval 𝓞q contained in 𝓔 corresponding to a conic 𝓒 of PG(2, q4), and if there is a tangent space intersecting ρ in a 7-space 𝓤 whose associated 𝔽q-linear set in 〈𝓒〉 ≃ PG(2, qn) contains a dual 𝔽q-subline on an internal point with respect to 𝓒, then q ≤ 19.

#### Corollary 6.8

If 𝓔 is a good egg of PG(19, 3) with kernel 𝔽3, then 𝓔 is either elementary, of KantorKnuth type, of CohenGanley type, or the PenttilaWilliams example.

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mlavrauw@sabanciuniv.edu

Revised: 2016-11-04

Published Online: 2018-03-26

Published in Print: 2019-01-28

Communicated by: T. Penttila

Funding: The authors acknowledge funding from the research project “Finite Geometry with Applications in Algebra and Combinatorics”, funded by the Dipartimento di Tecnica e Gestione dei Sistemi Industriali of the Universitá di Padova.

Citation Information: Advances in Geometry, Volume 19, Issue 1, Pages 57–64, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X,

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