We use the notation and terminology from [19]. Given a finite semifield 𝕊 with multiplication (*x*, *y*) ↦ *x* ∘ *y*, it is natural to consider the following substructures. The *left nucleus* ℕ(𝕊) is the set of elements *x* ∈ 𝕊 such that *x* ∘(*y* ∘ *z*) = (*x* ∘ *y*) ∘ *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 𝕊^{d} = 𝕊, and a semifield is called *symplectic* if and only if [𝕊^{t}] = [𝕊].

Semifields are studied up to isotopisms and their Knuth orbit. Two semifields 𝕊_{1} and 𝕊_{2} are *isotopic* if there exist non-singular linear maps *F*, *G*, *H* from 𝕊_{1} to 𝕊_{2} such that *x*^{F} ∘_{2} *y*^{G} = (*x* ∘_{1} *y*)^{H} for all *x*, *y* ∈ 𝕊_{1}. 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], [𝕊^{t}], [𝕊^{d}], [𝕊^{td}],[𝕊^{dt}],[𝕊^{tdt}]}, where the operations *t* and *d* denote the *transpose* and *dual* operations obtained from the action of the transpositions in the symmetric group *S*_{3} 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 𝔽_{q}, the finite field with *q* elements, and we assume that the middle nucleus is 𝔽_{qn}. This makes 𝕊 into a 2*n*-dimensional R2CS of size *q*^{2n}.

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 𝔽_{qn} × 𝔽_{qn} 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:

$$\begin{array}{}(x,y)\circ (u,v)=(xv+yu,yv+m{x}^{\sigma}{u}^{\sigma}),\end{array}$$(1)

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

$$\begin{array}{}(x,y)\circ (u,v)=(xv+yu+{x}^{3}{u}^{3},yv+\eta {x}^{9}{u}^{9}+{\eta}^{-1}xu),\end{array}$$(2)

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

$$\begin{array}{}(x,y)\circ (u,v)=(xv+yu+{x}^{27}{u}^{27},yv+{x}^{9}{u}^{9}).\end{array}$$(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 𝔽_{q}-linear functions satisfying the property that *g*^{2}(*t*)−4*tf*(*t*) is a non-square for all nonzero *t* ∈ 𝔽_{qn}, 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 *g*^{2}(*t*)−4*tf*(*t*) is a non-square for all nonzero *t* ∈ 𝔽_{qn} is equivalent to the existence of an 𝔽_{q}-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 $\begin{array}{}{X}_{2}^{2}\end{array}$ − 4*X*_{0}*X*_{1} = 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 𝔽_{q}-subplane of PG(2, *q*^{n}). Using a computer search, we complete the classification of 8-dimensional R2CSs. This is equivalent to classifying the semifield flocks in PG(3, *q*^{4}) having kernel containing 𝔽_{q}, the translation ovoids of *Q*(4, *q*^{4}) with kernel containing 𝔽_{q}, and good eggs in PG(15, *q*). We also classify the 10-dimensional R2CS with centre 𝔽_{3}, the semifield flocks in PG(3, 3^{5}) with kernel 𝔽_{3}, the translation ovoids in *Q*(4, 3^{5}) with kernel 𝔽_{3}, 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 𝔽_{q}-subplane contained in 𝓙(𝓒) to exist, there must be an 𝔽_{q}-subline contained in an external line of 𝓒 and made up entirely of points of 𝓙(𝓒).

#### Theorem 2.1

([16]). *There are no* 𝔽_{q}-*sublines contained in* *ℓ* ∩ 𝓙(𝓒), *where* 𝓒 *is a conic in* PG(2, *q*^{n}) *and* *ℓ* *is an external line to* 𝓒, *for* *q* ≥ 4*n*^{2} − 8*n* + 2, *and for* *q* > 2*n*^{2} − (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* 𝔽_{q}-*subplane contained in* 𝓙(𝓒) *exists*, *where* 𝓒 *is a conic in* PG(2, *q*^{n}), *for* *q* ≥ 4*n*^{2} − 8*n* + 2, *and for* *q* > 2*n*^{2} − (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, *q*^{n}) defined by the quadratic form Q = *X*_{0}*X*_{1} − $\begin{array}{}{X}_{2}^{2}.\end{array}$ 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, *q*^{n}) of 𝓒 in PGL(3, *q*^{n}) has order *q*^{n}(*q*^{2n} − 1) and contains all matrices of the form

$$\begin{array}{}\left[\begin{array}{ccc}{a}^{2}& {b}^{2}& ab\\ {c}^{2}& {d}^{2}& cd\\ 2ac& abd& ad+bc\end{array}\right]\end{array}$$

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

#### Theorem 2.3

*G* *is sharply triply transitive on the points of* 𝓒;

*G* *is transitive on* 𝓙(𝓒);

*G* *is transitive on* 𝓔(𝓒);

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

*G* *is transitive on the external lines to* 𝓒;

*G* *is transitive on the secant lines to* 𝓒;

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

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

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