On multiple blocking sets and blocking semiovals in finite non-Desarguesian planes

Finite Fields and Their Applications 62 (2020) 101624

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Finite Fields and Their Applications www.elsevier.com/locate/ffa

On multiple blocking sets and blocking semiovals in finite non-Desarguesian planes Angela Aguglia ∗ , Francesco Pavese Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy

a r t i c l e

i n f o

Article history: Received 9 January 2019 Received in revised form 4 October 2019 Accepted 10 November 2019 Available online xxxx Communicated by Olga Polverino MSC: primary 05B25 secondary 51E15, 51E21, 94A60 Keywords: Multiple blocking set Polarity Blocking semioval Regular Hughes plane Baer subplane

a b s t r a c t In this paper we first provide an infinite family of minimal (q − 1)-fold blocking sets of size q 3 in every affine translation plane of order q 2 . Next, we focus on the regular Hughes plane H(q 2 ) of order q 2 . It is well known that π = PG(2, q) is embedded as a Baer subplane in H(q 2 ). Let  be a line of H(q 2 ) having q + 1 points in common with π and let us denote by H(q 2 ) the affine plane obtained from H(q 2 ) by deleting  and all the points incident with . We exhibit a family of minimal (q − 1)-fold blocking sets of size q 3 in H(q 2 ) . Furthermore, we consider the polarity of H(q 2 ) which has q 3 −q + q 2 + 1 absolute points and we show that its absolute 2 points form a blocking semioval. Finally, we describe a class of Baer subplanes of H(q 2 ) distinct from π and admitting an automorphism group of order q 3 (q − 1). © 2019 Elsevier Inc. All rights reserved.

1. Introduction Let Π be a finite projective plane and let Π be the affine plane obtained from the plane Π by removing the line  and all the points incident with it. A line  of Π is a * Corresponding author. E-mail addresses: [email protected] (A. Aguglia), [email protected] (F. Pavese). https://doi.org/10.1016/j.ffa.2019.101624 1071-5797/© 2019 Elsevier Inc. All rights reserved.

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A. Aguglia, F. Pavese / Finite Fields and Their Applications 62 (2020) 101624

translation line if the group of collineations of Π consisting of elations with axis  acts transitively on the points of Π . A projective plane with a translation line is called a translation plane and the affine plane obtained by removing the translation line is called an affine translation plane. A spread S in PG(3, q) is a set of lines which partitions the points of PG(3, q). A spread S is regular if given any three lines of S, the unique regulus containing these three lines is contained in S. An alternate description of an affine translation plane of order q 2 , known as the Bruck–Bose representation, can be obtained as follows, see [7,8]. Let PG(4, q) be the projective space of dimension 4 over GF(q) and let S be a spread of a fixed hyperplane Σ∞ of PG(4, q). Consider the incidence structure A(S) whose points are the points of PG(4, q) \ Σ∞ and whose lines are the planes of PG(4, q) meeting Σ∞ in a line of S. From [7], A(S) is an affine translation plane (of dimension at most two over its kernel), and conversely, any such affine translation plane can be constructed from a spread of PG(3, q) in this way. Further, A(S) is Desarguesian if and only if S is regular. A t-fold blocking set B in a projective or affine plane is a set of points with the property that every line meets B in at least t points and some line in exactly t points. A 1-fold blocking set is simply called a blocking set and it has to satisfy the extra condition to contain no line. A point P of B is defined to be essential if B \ P is not a t-fold blocking set. When all points of B are essential then B is said to be minimal. In AG(2, q 2 ), the minimum size of a minimal (q − 1)-fold blocking set is known to be q 3 − 1; see [6]. In Section 2, by means of the Bruck–Bose representation, we provide an infinite family of minimal (q − 1)-fold blocking sets B of size q 3 in A(S) having just three intersection numbers with lines. As a byproduct, we also exhibit families of two-character multisets in the projective closure of A(S) of size q 3 + q − 1 and characters q − 1 and 2q − 1, that arise from B. Note that sets with few intersection numbers with respect to hyperplanes are connected with many theoretical and applied areas such as coding theory, strongly regular graphs, association schemes, optical multiple covering, secret sharing; furthermore certain multisets arise in various classification problems for optimal linear codes of higher dimension; see [2] and references therein. In order to introduce a class of non-Desarguesian planes given by D. Hughes [20], we recall the following notions. A right nearfield (N, +, ) is an algebraic structure for which + and  are binary operations on N , (N, +) and (N \ {0}, ) are groups, the former being abelian, and (a + b)  c = a  c + b  c, for all a, b, c ∈ N . The kernel of a nearfield N is the set of elements c ∈ N for which c  (a + b) = c  a + c  b, for all a, b ∈ N . A class of right nearfields, called regular nearfields or Dickson nearfields, were obtained by L. Dickson from a finite field by modifying the multiplication. Let q be an odd prime power, let GF(q) be the finite field of order q and let us denote with 2q the squares of GF(q). Multiplication in GF(q) will be denoted by juxtaposition. A regular nearfield N is defined by taking the elements of GF(q 2 ) with + defined as in GF(q 2 ) and  defined as follows

A. Aguglia, F. Pavese / Finite Fields and Their Applications 62 (2020) 101624

 ab=

ab

if

b ∈ 2q 2 ,

q

if

b∈ / 2q 2 .

a b

3

Note that GF(q) is the kernel of N and the center of (GF(q 2 ) \ {0}, ). H. Zassenhaus [39] proved that all but seven finite nearfields are either fields or Dickson nearfields, see also [18]. Using a finite nearfield with q 2 elements whose center is a field of order q, D. Hughes [20] introduced a class of finite projective planes, known as Hughes planes. Note that non-Desarguesian finite projective planes can be constructed from a right nearfield having dimension two over its kernel [25] and these are also called Hughes planes. Hughes planes constructed from regular nearfields are referred to as regular Hughes planes and will be denoted by H(q 2 ). In Section 3, we collect some properties of H(q 2 ). The regular Hughes plane contains π = PG(2, q) as a Baer subplane. Let  be a line of H(q 2 ) having q + 1 points in common with π and H(q 2 ) the affine plane obtained from H(q 2 ) by deleting  and all the points incident with . In Section 4, by using the properties of the regular Hughes planes H(q 2 ) and the construction of minimal (q−1)-fold blocking sets in A(S) provided in Section 2, we exhibit a family of minimal (q − 1)-fold blocking sets of size q 3 in H(q 2 ) . A polarity ⊥ of a finite projective plane Π is an involutory bijective map sending points to lines and lines to points which preserves incidence. A point P of Π is said to be absolute if P ∈ P ⊥ . In [29], T.G. Room described two distinct types of polarities 3 in the regular Hughes planes of order q 2 . These polarities have either q 2−q + q + 1 or q 3 −q 2

+ q 2 + 1 absolute points. A semioval in Π is a set of points Z with the property that for every point P ∈ Z, there exists exactly one tangent at P , i.e., a line of Π meeting Z precisely in P . A blocking semioval in Π is a set of points K which is both a blocking set and a semioval. Although blocking semiovals have their main applications in cryptography [5], they are interesting in their own right. An upper bound on the size of a blocking semioval K in a projective √ plane of order q, namely |K| ≤ q q + 1, can be found in [19] and [36]. Moreover, equality holds if and only if K is a unital. Some lower bounds on the size of a blocking semioval are provided in [12]. Another well-known example of a blocking semioval in projective planes of order greater than two, with much less points than a unital, is the vertexless triangle which has size 3q − 3. Apart from the unitals and the vertexless triangle, many other constructions of blocking semiovals are known; see [1,4,12–17,22,34,35]. However, apart from few examples [1,21,23], not many blocking semiovals are known to exist in non-Desarguesian planes. In Section 5, we consider the polarity of H(q 2 ) which has q 3 −q + q 2 + 1 absolute points and we show that its absolute points form a blocking 2 semioval. Finally, in the last section we describe a class of Baer subplanes of H(q 2 ) distinct from π and admitting an automorphism group of order q 3 (q − 1).

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2. (q − 1)-fold blocking sets in affine translation planes In this section we construct an infinite family of minimal (q − 1)-fold blocking sets in every translation plane π of order q 2 (of dimension at most two over its kernel). Let PG(4, q) be the projective space of dimension 4 over GF(q) and let S be a spread of a fixed hyperplane Σ∞ of PG(4, q). Let A(S) be the translation affine plane arising from S and let Π be the projective closure of A(S). Let us denote by Σ a hyperplane of PG(4, q) sharing with Σ∞ a plane σ and let P be a point of σ such that the line  of S containing P does not lie in σ. Consider an ovoid O of Σ such that P ∈ O and σ is its tangent plane at the point P . Let us denote by C the cone of PG(4, q) with base O and vertex a point V ∈  \ {P }. In [9] Buekenhout proved that C corresponds to a unital of Π. Here, we investigate the case in which the base of the cone is a hyperbolic quadric. Precisely, let Q be a hyperbolic quadric of Σ such that σ is tangent to Q at the point P ∈ Q. Thus Q ∩ σ consists of two lines through P , say r1 and r2 . As before, let V ∈  \ {P } and let C be the cone with vertex V and base Q. Let us denote by αi the plane generated by V and ri , i = 1, 2. Note that both, α1 and α2 , are contained in C. We prove that B = C \ Σ∞ is a minimal (q − 1)-fold blocking set of A(S). Theorem 2.1. B is a minimal (q − 1)-fold blocking set of size q 3 in the affine plane A(S) of order q 2 such that each line shares either q − 1 or q or 2q − 1 points with it. There are q 2 lines meeting B in q points, q 4 − q 3 lines meeting B in q − 1 points and q 3 lines meeting B in 2q − 1 points. Proof. Since C ∩ Σ∞ = α1 ∪ α2 we have that |B| = q 3 . Now, we are going to prove that each plane passing through a line of S and that is not contained in Σ∞ meets B in either q − 1, or q or 2q − 1 points. We first consider a plane α of PG(4, q) \ Σ∞ passing through the line . Then α meets Σ in a line, say m. The intersection m ∩ Q consists of two points P and P  with P = P  . Hence α contains the line of C joining V and P  . Thus, |B ∩ α| = q. Now, let us assume that β is a plane of PG(4, q) \ Σ∞ passing through a line r of S \ . Since V is not on β we have that β meets the cone C in either a non-degenerate conic or in two incident lines. Hence |β ∩ C| = q + 1 or |β ∩ C| = 2q + 1. On the other hand the line r has exactly one point in common with αi , for i = 1, 2; therefore |r ∩ (C \ B)| = 2. It follows that the intersection of B and β consists of either q − 1 or 2q − 1 points. In order to prove that B is a minimal (q − 1)-fold blocking set of A(S) we will show that through each point Q ∈ B there passes at least one line of A(S) meeting B in q − 1 points. First of all we note that there is only one line in A(S) passing trough Q and containing q points of B, that is the line which corresponds to the above plane α. Let us denote by x the number of lines m through Q such that |m ∩ B| = q − 1 and by y the number of lines r trough Q such that |r ∩ B| = 2q − 1. Then x + y = q 2 and x(q − 2) + y(2q − 2) + q = q 3 . We get x = (q − 1)2 , y = 2q − 1 and hence x ≥ 1.

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A double counting argument on pairs (Q, m), where Q ∈ B, m is a line of A(S), |m ∩ B| = q − 1 and Q ∈ m gives q 3 (q − 1)2 = z(q − 1), where z denotes the number of lines of A(S) having q − 1 points in common with B. Hence there are q 2 lines meeting B in q points, q 4 − q 3 lines meeting B in q − 1 points and q 3 lines meeting B in 2q − 1 points. 2 Note that if S is a regular spread then Theorem 2.1 coincides with [3, Theorem 1.1 (i)]. 2.1. A class of two-character multisets A multiset in Π is a mapping M : Π −→ N from the points of Π into non-negative integers. The points of a multiset are the points P of Π with multiplicity M (P ) > 0.  For any line r of Π, the non-negative integer P ∈r M (P ) is a character of the multiset M . If the set {M (r) | r ∈ Π} consists of two non-negative integers only, then M is a two-character multiset. By considering the minimal (q − 1)-fold blocking set B of A(S) described above, we provide a family of two-character multisets in Π. Let P∞ be the point of Π which corresponds to the element  of the line spread S. Consider the multiset M arising from B by assigning multiplicity larger than 1 to the point P∞ . Precisely, the points of M with positive multiplicity are those of B ∪ P∞ where each affine point has multiplicity one, whereas P∞ has multiplicity q − 1. From the proof of Theorem 2.1 we have the following result. Theorem 2.2. M is a two-character multiset in Π of size q 3 + q − 1 with characters q − 1 and 2q − 1. 3. A description of the regular Hughes planes In this section we provide a description of the regular Hughes plane H(q 2) of order q 2 . Although many of the results presented in this section can be found in [24,31,32,38], for the convenience of the reader a proof of the assertions needed in the paper is provided. From now on, q is a power of an odd prime. Let N be a regular nearfield and let V = {(x1 , x2 , x3 ) | x1 , x2 , x3 ∈ N }. Then (V, +, ) is a three-dimensional right vector space over N (i.e., V satisfies all the axioms defining a vector space except the left distributivity law). Note that the multiplication of the vector (x1 , x2 , x3 ) ∈ V by the scalar λ ∈ N is given by (x1 , x2 , x3 )  λ = (x1  λ, x2  λ, x3  λ). Define an equivalence relation on V \ {(0, 0, 0)} by saying that two non-zero vectors P and Q are equivalent if and only if Q = P  k for some k ∈ N \ {0}, where the notation P k means (x1 k, x2 k, x3 k) if P = (x1 , x2 , x3 ) ∈ V \{(0, 0, 0)}. Thus V \{(0, 0, 0)} is partitioned into q 4 + q 2 + 1 equivalence classes, each of size q 2 − 1. Define the points of H(q 2 ) as the equivalence classes of V \ {(0, 0, 0)} introduced above. In other words,

A. Aguglia, F. Pavese / Finite Fields and Their Applications 62 (2020) 101624

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points of H(q 2 ) are the one-dimensional right vector subspaces of V with the vector (0, 0, 0) deleted. Also, we may choose a unique left-normalized (or right-normalized) representative for points of H(q 2 ). As in the classical case, we will represent points of the Hughes plane with vectors in V \ {(0, 0, 0)}. We also freely identify a point of H(q 2 ) with any of its representatives in V \ {(0, 0, 0)}. Lemma 3.1. If A ∈ GL(3, q), then the map θ : (x1 , x2 , x3 ) → (x1 , x2 , x3 )A induces a permutation on the points of H(q 2 ). Proof. Let P = (x1 , x2 , x3 ) be a point of H(q 2 ) and let A = (aij ) ∈ GL(3, q). Then 3 3 3 P θ = (x1 , x2 , x3 )A = i=1 ai1 xi , i=1 ai2 xi , i=1 ai3 xi . On the other hand, if k ∈ GF(q 2 ) \ {0} and P = (x1  k, x2  k, x3  k), then P θ = (x1  k, x2  k, x3  k)A =  =

ai1 xi  k,

i=1

3 

ai2 xi  k,

i=1

3 3 3    ai1 xi )  k, ( ai2 xi )  k, ( ai3 xi )  k ( i=1

=

 3 

 3  i=1

ai1 xi ,

i=1 3  i=1

ai2 xi ,

3 

 ai3 xi

3 

 ai3 xi  k

=

i=1

 =

i=1

 k.

i=1

Hence P θ does not depend on the representative of the point P . In order to prove the result we need to show that the map θ is surjective and hence bijective on points of H(q 2 ).Indeed, if Q = (x1 , x2 , x3 ) is a pointof H(q 2 ), the preimage of 3 3 3    −1 Q under θ is the point P = = (bij ). 2 i=1 bi1 xi , i=1 bi2 xi , i=1 bi3 xi , where A Let i be a fixed element of GF(q 2 ) \ GF(q) such that iq + i = 0. Then iq+1 = −i2 and hence i2 ∈ GF(q) with i2 ∈ / 2q . Furthermore, every element ρ ∈ GF(q 2 ) can be uniquely written as ρ0 +iρ1 , where ρ0 , ρ1 ∈ GF(q). Consider the elements of V \{(0, 0, 0)} satisfying the equation H : r1 X1 + r2 X2 + r3 X3 + i  (s1 X1 + s2 X2 + s3 X3 ) = 0,

(3.1)

where ri , si ∈ GF(q), 1 ≤ i ≤ 3, not all of them zero. Note that if (x1 , x2 , x3 ) ∈ V \ {(0, 0, 0)} satisfies (3.1), then r1 (x1  k) + r2 (x2  k) + r3 (x3  k) + i  (s1 (x1  k) + s2 (x2  k) + s3 (x3  k)) = (r1 x1 )  k + (r2 x2 )  k + (r3 x3 )  k + i  ((s1 x1 )  k + (s2 x2 )  k + (s3 x3 )  k) = (r1 x1 + r2 x2 + r3 x3 + i  (s1 x1 + s2 x2 + s3 x3 ))  k = 0

A. Aguglia, F. Pavese / Finite Fields and Their Applications 62 (2020) 101624

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and hence (x1 , x2 , x3 )  k satisfies (3.1). It follows that H consists of points of H(q 2 ). Note that the elements of V satisfying (3.1) form a two-dimensional right vector subspace of V . Therefore H consists of q 2 + 1 points of H(q 2 ). If ui = ri + isi , 1 ≤ i ≤ 3, denote H by the triple [u1 , u2 , u3 ]H . Define the lines of H(q 2 ) as the pointsets [u1 , u2 , u3 ]H , where ui ∈ GF(q 2 ), (u1 , u2 , u3 ) = (0, 0, 0). We are going to prove that the line H is defined up to a non-zero scalar, that is, if k ∈ GF(q 2 ) \ {0}, then [u1 , u2 , u3 ]H = [u1 , u2 , u3 ]H  k, where [u1 , u2 , u3 ]H  k := [u1  k, u2  k, u3  k]H . Lemma 3.2. Let M, M  , L, L ∈ GF(q), then (M + iL) + i  (M  + iL ) = 0 if and only if (M + iM  ) + i  (L + iL ) = 0. Proof. Suppose first that (M + iL) + i  (M  + iL ) = 0. If L = 0, then the claim follows. Assume that L = 0, then L + iL = 0. We have (L + iL )  (M  + iL ) = L(M  + iL ) + i  (M  + iL )L = LM  + iLL − (M + iL)L = LM  − M L . Hence (M  + iL ) = (L + iL )−1 (LM  − M L ) = (LM  − M L )(L + iL )−1 . Here (L + iL )−1 denotes the multiplicative inverse of L + iL in N . It follows that LM  − M L = (M  + iL )(L+iL ) = M  L+iM  L +i(L+iL )L . Therefore −M L = iM  L +i(L+iL )L and the claim follows. The converse is similar. 2 Lemma 3.3. If k ∈ GF(q 2 ) \ {0}, then [u1 , u2 , u3 ]H = [u1 , u2 , u3 ]H  k. Proof. Let P = (x1 , x2 , x3 ) be a point of H(q 2 ) belonging to the line H , where H is given by [u1 , u2 , u3 ]H . We want to prove that P lies on [u1 , u2 , u3 ]H  k. Let ui = ri + isi , xi = yi + izi , 1 ≤ i ≤ 3. Hence 0 = r1 x1 + r2 x2 + r3 x3 + i  (s1 x1 + s2 x2 + s3 x3 ) = (M + iL) + i  (M  + iL ), where M = r1 y1 + r2 y2 + r3 y3 , M  = s1 y1 + s2 y2 + s3 y3 , L = r1 z1 + r2 z2 + r3 z3 and L = s1 z1 + s2 z2 + s3 z3 . From Lemma 3.2, we get y1 u1 + y2 u2 + y3 u3 + i  (z1 u1 + z2 u2 + z3 u3 ) = 0. It follows that y1 u1  k + y2 u2  k + y3 u3  k + i  (z1 u1  k + z2 u2  k + z3 u3  k) = 0. Therefore, if ui  k = ri + isi , 1 ≤ i ≤ 3, we have r1 x1 + r2 x2 + r3 x3 + i  (s1 x1 + s2 x2 + s3 x3 ) = 0, as required. 2 For a matrix A ∈ GL(3, q), let A−t denote the inverse transpose of A. Lemma 3.4. If A ∈ GL(3, q), then the map θ : (x1 , x2 , x3 ) → (x1 , x2 , x3 )A, θ : [u1 , u2 , u3 ]H → [u1 , u2 , u3 ]H A−t gives rise to a collineation of H(q 2 ). Proof. Let θ be a projectivity of π and let A = (aij ) be an element of GL(3, q) inducing θ. Taking into account Lemma 3.1, we need to show that the images under θ of a line of H(q 2 ) is a line of H(q 2 ) and that θ preserves incidence. Let P = (x1 , x2 , x3 ) be a point of H(q 2 ) and let H be the line of H(q 2 ) given by [u1 , u2 , u3 ]H , where ui = r i + isi , xi = yi + izi , 1 ≤ i ≤ 3. Let A−t = (bij ). Then θH is given by [u1 , u2 , u3 ]H A−t = 3 3 3 , that is the line of H(q 2 ): i=1 bi1 ui , i=1 bi2 u2 , i=1 bi3 ui H

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8 3 

bi1 ri X1 +

3 

i=1

bi2 ri X2 +

3 

i=1

 bi3 ri X3 + i 

i=1

3 

bi1 si X1 +

i=1

3 

bi2 si X2 +

i=1

3 

 bi3 si X3

i=1

= 0. Similarly to the proof of Lemma 3.1, it can be shown that θH does not depend on the representative of H and that θ is bijective on lines of H(q 2 ). If P is a point  of H , then r1 x1 + r2 x2 + r3 x3 + i (s1 x1 + s2 x2 + s3 x3 ) = 0. On the 3 3 3 θ θ other hand P = i=1 ai1 xi , i=1 ai2 xi , i=1 ai3 xi belongs to H since 3  i=1

bi1 ri

3 

aj1 xj +

j=1

3 

bi2 ri

i=1

⎛

+i⎝

=

i=1 j=1 m=1

aj2 xj +

j=1 3  i=1

3  3  3 

3 

bi1 si

3 

bi3 ri

i=1 3 

aj1 xj +

j=1

3 

aj3 xj +

j=1 3  i=1

bi2 si

3 

aj2 xj +

j=1

3  i=1

bi3 si

3 

⎞ aj3 xj ⎠ =

j=1

⎛ ⎞ 3  3 3 3  3    bim ajm ri xj + i  ⎝ bim ajm si xj ⎠ = r i xi + i  si xi = 0, i=1 j=1 m=1

since (aij ) = A, (bij ) = A−t and P ∈ H .

i=1

i=1

2

Let τ be the Baer involution of PΓL(3, q 2 ) given by (x1 , x2 , x3 ) ∈ PG(2, q 2 ) → ∈ PG(2, q 2 ) and let π be the Baer subplane of PG(2, q 2 ) fixed pointwise by τ . The abovementioned map induces a collineation in both planes PG(2, q 2 ) and H(q 2 ), see Lemma 3.6. With a slight abuse of notation we will use the symbol τ to denote both the collineation of PG(2, q 2 ) and H(q 2 ). (xq1 , xq2 , xq3 )

Lemma 3.5. π is a subplane of order q of H(q 2 ). Proof. A point P = (x1 , x2 , x3 ) of the Desarguesian plane π is also a point of H(q 2 ). Let H be the line of H(q 2 ) given by [u1 , u2 , u3 ]H , where u1 , u2 , u3 ∈ GF(q), (u1 , u2 , u3 ) = (0, 0, 0). Then H ∩ π is a line of the Desarguesian plane π and every line of π can be obtained in this way. 2 Lemma 3.6. The map (x1 , x2 , x3 ) → (xq1 , xq2 , xq3 ), [u1 , u2 , u3 ]H → [uq1 , uq2 , uq3 ]H gives rise to a collineation τ of H(q 2 ) of order two fixing pointwise π. Proof. Since (α  β)q = αq  β q , we have that τ sends a point of H(q 2 ) to a point of H(q 2 ). Analogously τ maps a line of H(q 2 ) to a line of H(q 2 ). On the other hand, if P = (x1 , x2 , x3 ) belongs to the line H given by [u1 , u2 , u3 ]H , where ui = ri + isi , xi = yi + izi , 1 ≤ i ≤ 3, then r1 xq1 + r2 xq2 + r3 xq3 − i  (s1 xq1 + s2 xq2 + s3 xq3 ) =

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9

q

(r1 x1 + r2 x2 + r3 x3 + i  (s1 x1 + s2 x2 + s3 x3 )) = 0. Therefore P τ belongs to the line τH . It is straightforward to check that τ has order two and fixes pointwise π. 2 From Lemma 3.3, by an appropriate choice of the coefficients, it follows that any equation of any one of the following types is the equation of a line of H(q 2 ): (I) X1 = 0, (II) t  X1 + X2 = 0, t ∈ GF(q 2 ), (III) aX1 + t  X2 + X3 = 0, t ∈ GF(q 2 ), a ∈ GF(q), (IV) t  (X1 + bX2 ) + aX2 + X3 = 0, t ∈ GF(q 2 ) \ GF(q), a, b ∈ GF(q). Note that a line of H(q 2 ) of type (I), (II), (III), (IV) is given by [1, 0, 0]H , [t, 1, 0]H , [a, t, 1]H , [t, a + bt, 1]H , respectively. If we consider points of H(q 2 ) represented by leftnormalized vectors of V \ {(0, 0, 0)}, then points of H(q 2 ) can be identified with the points of PG(2, q 2 ). In view of this fact, it makes sense to ask how many points a line of H(q 2 ) may share with a line of PG(2, q 2 ). Here and in the sequel, we will represent lines of PG(2, q 2 ) by using non-zero triples of elements of GF(q 2 ) in square brackets. Also, in the remaining part of this section we consider points (of H(q 2 ) or of PG(2, q 2 )) represented by left-normalized vectors. Lemma 3.7. If H is a line of H(q 2 ), then there exists a unique line of PG(2, q 2 ), say , such that either  coincides with H or |H ∩ | = (q 2 + 3)/2 and |H ∩ τ | = (q 2 + 1)/2. Conversely, if  is a line of PG(2, q 2 ), then there exists a unique line of H(q 2 ), say H , such that either H coincides with  or | ∩ H | = (q 2 + 3)/2 and | ∩ τH | = (q 2 + 1)/2. Proof. If H is given by [1, 0, 0]H , then it coincides with the line  of PG(2, q 2 ) given by [1, 0, 0]. Analogously, if H is given either by [t, 1, 0]H , t ∈ GF(q 2 ), or by [a, t, 1]H , a, t ∈ GF(q), or by [t, a, 1]H , t ∈ GF(q 2 ) \ GF(q), a ∈ GF(q), then it coincides with the line  of PG(2, q 2 ) given either by [t, 1, 0], or by [a, t, 1], or by [t, a, 1], respectively. On the other hand, if H is given by [a, t, 1]H , a ∈ GF(q), t ∈ GF(q 2 ) \ GF(q), then H shares with the line  of PG(2, q 2 ) given by [a, t, 1] the following set of (q 2 + 3)/2 points: {(1, x, −a − tx) | x ∈ 2q2 } ∪ {(0, 1, −t)}. Similarly, it can be seen that  and τ share (q 2 + 1)/2 points and (1, 0, −a) ∈  ∩ τ ∩ H . Finally, if H is given by [t, a + bt, 1]H , t ∈ GF(q 2 ) \ GF(q), a, b ∈ GF(q), b = 0, then H shares (q 2 + 3)/2 points with the line  of PG(2, q 2 ) given by [t, a + bt, 1] and (q 2 + 1)/2 points with τ . Furthermore, the point (1, −b−1 , a) ∈  ∩ τ ∩ H . 2 Taking into account Lemma 3.7, if H is a line of H(q 2 ), we denote by  the unique line of PG(2, q 2 ) such that | ∩ H | ≥ (q 2 + 3)/2 and viceversa. Also, from the proof of Lemma 3.7, there are q 3 + q 2 + 1 lines of H(q 2 ) that are also lines of PG(2, q 2 ) and these are all the lines of H(q 2 ) meeting the Baer subline obtained by intersecting the line X1 = 0 with π, in at least a point.

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Remark 3.8. From the proof of Lemma 3.7, it follows that H ∪ τH =  ∪ τ . We end this section with the following lemma (see [37, Lemma 2.3]), which will be very useful. Lemma 3.9. Let X be a subset of points represented by left-normalized vectors of V \ {(0, 0, 0)} and left invariant by τ . Then | ∩ X | = |H ∩ X |. Proof. If  coincides with H , then there is nothing to prove. By Remark 3.8, if  = H then  = τ . Assume that  = H , i.e., | ∩H | = (q 2 +3)/2. From the proof of Lemma 3.7, it follows that | ∩ π| = |H ∩ π| = 1 and  ∩ π = H ∩ π. Let Q ∈ X . If Q ∈ H , then either Q ∈  \ π, or Q ∈ τ \ π, or Q =  ∩ τ . In the first case Q ∈  and Qτ ∈ / ; in the second case Q ∈ /  and Qτ ∈ ; in the third case Q = Qτ ∈ . Similarly, if Q ∈ , then either Q ∈ H \ π, or Q ∈ τH \ π, or Q = H ∩ τH . In the first case Q ∈ H and Qτ ∈ / H ; τ τ in the second case Q ∈ / H and Q ∈ H ; in the third case Q = Q ∈ H . 2 4. (q − 1)-fold blocking set in H(q 2 ) Let r be the line of H(q 2 ) with equation X1 = 0 which has q+1 points in common with π = PG(2, q). Let us denote by H(q 2 )r the affine plane obtained from H(q 2 ) by deleting r and all the points incident with r. Here, we exhibit a family of minimal (q − 1)-fold blocking sets of size q 3 in H(q 2 )r . Let us consider the following set F = {(1, x, ax2 + bxq+1 + ti) | x ∈ GF(q 2 ), t ∈ GF(q)} ∪ {(0, 0, 1)}, where a, b ∈ GF(q) such that b2 − a2 ∈ / 2q . Since F is represented by left-normalized vectors, it may be considered as a pointset of H(q 2 ) or PG(2, q 2 ). Now, let Σ∞ : X1 = 0 be the hyperplane at infinity of PG(4, q) and consider the bijection φ : PG(2, q 2 ) \ r → PG(4, q) \ Σ∞ which maps the point (1, x2 + ix3 , x4 + ix5 ) to (1, x2 , x3 , x4 , x5 ). Let P = (0, 0, 0, 1, 0) and V = (0, 0, 0, 0, 1). Consider the lines  = V P  and rδ,γ = (0, 1, 0, δ, 0), (0, 0, 1, 0, γ), with δ, γ ∈ GF(q). It turns out that S = {} ∪ {rδ,γ | δ, γ ∈ GF(q)} is a regular spread of Σ∞ . Next, let Σ : X5 = 0 and consider the cone C having as vertex the point V and as base the hyperbolic quadric Q of Σ with equation (a + b)X22 + i2 (a − b)X32 − X4 X1 = 0. The plane σ : Σ∞ ∩ Σ is the tangent plane to Q at P . Let B = C \ Σ∞ . It is easy to see that a point R = (1, x, y) ∈ PG(2, q 2 ) belongs to F if and only if φ(R) lies on B. From Theorem 2.1 it follows that F \ r is a minimal (q − 1)-fold blocking set of AG(2, q 2 ). Since F τ = F, then Lemma 3.9 implies the following result.

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Theorem 4.1. F \ r is a minimal (q − 1)-fold blocking set of H(q 2 )r of size q 3 meeting each line in either q − 1 or q or 2q − 1 points. Finally we observe that a two-character multiset in H(q 2 ) arises from F as in Section 2.1. 5. Polarities and blocking semiovals in the regular Hughes planes Polarities in a regular Hughes plane were first studied by T.G. Room in [28–30], where he presented two distinct types of polarities. In [27], F. Piper proved that any polarity in a regular Hughes plane is one of two types exhibited by Room. Finally, polarities in a Hughes plane were classified by G. Nicoletti in [24]. In this section we investigate the maps of H(q 2 ): OH : (x1 , x2 , x3 ) → [x3 , x2 , x1 ]H , OH : [u1 , u2 , u3 ]H → (u3 , u2 , u1 ) and UH : (x1 , x2 , x3 ) → [xq3 , xq2 , xq1 ]H , UH : [u1 , u2 , u3 ]H → (uq3 , uq2 , uq1 ). Lemma 5.1. The maps OH and UH are polarities of H(q 2 ). Moreover, OH UH = UH OH = τ. Proof. Let P = (x1 , x2 , x3 ), R = (α1 , α2 , α3 ) be two points of H(q 2 ). First of all observe that (P OH )OH = P , for every point P of H(q 2 ). Hence OH has order two. If P belongs to ROH and xi = yi + izi and αi = βi + iγi , 1 ≤ i ≤ 3, then 0 = β3 x1 + β2 x2 + β1 x3 + i  (γ3 x1 + γ2 x2 + γ1 x3 ) = (M + iL) + i  (M  + iL ), where M = β3 y1 + β2 y2 + β1 y3 , M  = γ3 y1 + γ2 y2 + γ1 y3 , L = β3 z1 + β2 z2 + β1 z3 and L = γ3 z1 + γ2 z2 + γ1 z3 . From Lemma 3.2, we get y3 α1 + y2 α2 + y1 α3 + i  (z3 α1 + z2 α2 + z1 α3 ) = 0. Therefore R belongs to the line P OH and OH is a polarity of H(q 2 ). Note that P UH = [xq3 , xq2 , xq1 ]H coincides with P OH τ = (P OH )τ which in turns coincides with P τ OH = (P τ )OH . It follows that UH = OH τ = τ OH and UH has order two. To conclude the proof, note that P ∈ RUH = Rτ OH = (Rτ )OH if and only if Rτ ∈ P OH if and only if R ∈ (P OH )τ = P OH τ = P UH . 2 Note that the maps O : (x1 , x2 , x3 ) → [x3 , x2 , x1 ] and U : (x1 , x2 , x3 ) → [xq3 , xq2 , xq1 ] are polarities of PG(2, q 2 ) and they commute, i.e., OU = UO. In particular O is an orthogonal polarity and has as absolute points the conic C of PG(2, q 2 ) defined by 2X1 X3 + X22 = 0, whereas U is a unitary polarity and its absolute points are those of the Hermitian curve C¯ of PG(2, q 2 ) defined by X1q X3 + X1 X3q + X2q+1 = 0. Furthermore, C ∩ C¯ = C ∩ π = C¯ ∩ π and their intersection consists of the q + 1 points of the Baer conic Ω of π, see [10, Lemma 3.2]. Taking into account Lemma 3.9, since both C and C¯

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are stabilized by τ , we have that the (left-normalized) points of C and C¯ form an oval and a unital of H(q 2 ), respectively. They were found in [29] and in [33], respectively. Lemma 5.2. The union of the (left-normalized) absolute points of OH and UH coincides ¯ with the (left-normalized) points of C ∪ C. Proof. Let P be a (left-normalized) point (of H(q 2 ) or of PG(2, q 2 )). With the notation used in Lemma 3.7 and Lemma 3.9, if  and  denote the lines P O and P U of PG(2, q 2 ), respectively, observe that H and H are the lines P OH and P UH of H(q 2 ), respectively. Since OU = UO = τ then  = τ and the result follows from Lemma 5.1 and Remark 3.8. 2 ¯ be the subgroup of GL(3, q), whose elements are the matrices Let G ⎛

Aa,b,c,d

a2 ⎜ = ⎝ 2ac −2c2

ab ad + bc −2cd

⎞ −b2 /2 ⎟ −bd ⎠ , d2

¯ is isomorphic to GL(2, q). Let G be the where a, b, c, d ∈ GF(q), ad − bc = 0. Then G ¯ With a slight abuse of notation we denote group of collineations of H(q 2 ) induced by G. by G × τ the group generated by G and τ . Proposition 5.3. The group G × τ preserves both OH and UH . Proof. From Lemma 5.1, the collineation τ commutes with both OH and UH . Let P = (x1 , x2 , x3 ) be a point of H(q 2 ) and let θ denote the collineation of H(q 2 ) induced by  OH  O θ Aa,b,c,d . Straightforward calculations show that P θ = P H = [−b2 x1 /2 − bdx2 +  UH  θ = P UH = d2 x3 , abx1 + (ad + bc)x2 − 2cdx3 , a2 x1 + 2acx2 − 2c2 x3 ]H and P θ [−b2 xq1 /2, −bdxq2 + d2 xq3 , abxq1 + (ad + bc)xq2 − 2cdxq3 , a2 xq1 + 2acxq2 − 2c2 xq3 ]H . 2 Let E or S denote the set of (left-normalized) points of C¯ \ π lying on some line H of H(q 2 ), where |H ∩ π| = q + 1 and |H ∩ Ω| = 0 or |H ∩ Ω| = 2. We have that ¯ on points of C¯ in the |E| = |S| = (q 3 − q)/2. The action of the group induced by G Desarguesian case has been considered in [10, Lemma 3.1]. In the next proposition we study the action of G on points of C¯ in H(q 2 ) and some differences will arise. Proposition 5.4. The group G × τ has two orbits of equal size on points of C¯ \ Ω, that are E and S. Moreover, G × τ is transitive on points of C \ π. Proof. We will prove that S is always a G-orbit, whereas E is a G-orbit if q ≡ −1 (mod 4) and an orbit under the action of G × τ if q ≡ 1 (mod 4). First of all observe that G acts on points and lines of π (viewed as a subplane of ¯ acts on points and H(q 2 )) in the same way as the subgroup of PGL(3, q) induced by G

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lines of π (considered as a subplane of PG(2, q 2 )). Hence G is transitive on the q(q + 1)/2 and q(q − 1)/2 lines of π that are secant and external to Ω. Let H be a line of H(q 2 ) such that H ∩ π is a line of π. Suppose first that H is secant to Ω. Then we may assume that H is given by [0, 1, 0]H . The stabilizer of the line H in G is the group D1 of order 2(q − 1), generated by the collineations associated with the matrices A0,−2,1,0 and A1,0,0,d , d ∈ GF(q) \ {0}. An easy calculation shows that the stabilizer in D1 of a (left-normalized) point of H ∩ (C¯ \ Ω) is generated by the involution induced by A1,0,0,−1 . It follows that D1 is transitive on points of H ∩ (C¯ \ Ω) and G is transitive on S. If H is external to Ω, then we may assume that H is given by [1/2s, 0, 1]H , where s is a non-square element of GF(q). In this case the stabilizer of H in G is the group D2 of order 2(q + 1), generated by the collineations associated with the matrices A0,1,s,0 , A1,0,0,−1 and Aa,b,sb,a , where a, b ∈ GF(q), a2 − sb2 = ±1. Let P = (1, α, −1/2s) be a ¯ Note that α ∈ GF(q 2 ) \ GF(q), otherwise P ∈ π and αq+1 = 1/s since point of  ∩ C. ¯ In particular α ∈ P ∈ C. / 2q2 . Denote by ϕ the collineation of H(q 2 ) induced by A0,1,s,0 . Then it is easily seen that P ϕ = P . Let θ be the collineation of H(q 2 ) induced by Aa,b,sb,a and let ξ = a2 + sb2 + 2sabα. We have that the point P θ coincides with P if and only if α = (2ab + (a2 + sb2 )α)  ξ −1 , that is α  ξ = 2ab + (a2 + sb2 )α. Here ξ −1 denotes the multiplicative inverse of ξ in N . If ξ ∈ 2q2 , and θ is non-trivial, then the unique possibility is a = 0. However in this case θ coincides with ϕ. If ξ ∈ / 2q2 , then necessarily a2 +sb2 = 0 and this forces q ≡ −1 (mod 4).     Since a2 −sb2 = ±1, we get a = ± 1/2, b = ± −1/2s or a = ± −1/2, b = ± 1/2s / 2q , respectively. It follows that, if q ≡ 1 (mod 4), then the according as 2 ∈ 2q or 2 ∈ stabilizer of a (left-normalized) point of H ∩ C¯ in D2 is generated by the involution ϕ. Therefore in this case D2 is transitive on points of H ∩ C¯ and G is transitive on E. On the other hand, if q ≡ −1 (mod 4), then the stabilizer of a (left-normalized) point of H ∩ C¯ in D2 is a group of order four generated by the collineation φ induced by A1,−1/s,s−1/s,1 or A−1/2,1/2s,s1/2s,−1/2 , according as 2 ∈ 2q or 2 ∈ / 2q . Indeed 2 φ has order four and φ coincides with ϕ. Hence if q ≡ −1 (mod 4) the group G has two orbits of size (q 3 − q)/4 on E. Now, note that every element in G × τ can be uniquely written as the product of an element of G and an element of τ . Next, consider the stabilizer of H in G × τ , that is D2 × τ . Then, by repeating the previous argument, similar calculations to those provided above will give that the stabilizer of a point P of H in D2 × τ has order four. Therefore D2 × τ is transitive on points of H ∩ C¯ and G × τ is transitive on E. Let α ∈ GF(q 2 ) \ GF(q) such that α2 = 1/s. Then P = (1, α, −1/2s) ∈ H ∩ (C \ Ω). To conclude the proof note that the collineation of H(q 2 ) induced by A1,0,0,−1 does not fix P . Therefore D2 is transitive on points of H ∩ (C \ Ω) and G is transitive on C \ Ω. 2

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Theorem 5.5. If q ≡ 1 (mod 4), then E ∪ Ω is the set of absolute points of OH and S ∪ C is the set of absolute points of UH . If q ≡ −1 (mod 4), then E ∪ C is the set of absolute points of OH and S ∪ Ω is the set of absolute points of UH . Proof. It is straightforward to check that a point of Ω is absolute with respect to both OH and UH . Moreover, from Lemma 5.1, if a point P of H(q 2 ) belongs to both P OH and P UH , then P ∈ π. Therefore, from Lemma 5.2, Proposition 5.3 and Proposition 5.4, it is enough to show that a point of E is absolute with respect to OH , a point of S is absolute with respect to UH and a point of C \ π is absolute with respect to OH or to UH , according as q ≡ −1 (mod 4) or q ≡ 1 (mod 4), respectively. Let P be the point of S given by (1, 0, α). Then α ∈ GF(q 2 ) \ GF(q) and α + αq = 0. It is easy to chek that P ∈ P UH . On the other hand, let P be the point given by (1, α, −1/2s), where α = β + iγ ∈ GF(q 2 ) \ GF(q) and s ∈ / 2q . The point P belongs to E or to C \ π, according q+1 2 as α = 1/s or α = 1/s, respectively. Hence in the former case α ∈ / 2q2 and in the latter case α ∈ / 2q2 if q ≡ 1 (mod 4), or α ∈ 2q2 if q ≡ −1 (mod 4). Since P OH is the line −X1 /2s+βX2 +X3 +i(γX2 ) = 0 and P UH is the line −X1 /2s+βX2 +X3 −i(γX2 ) = 0, we obtain that P ∈ P OH if P ∈ E. Also, if P ∈ C \ π, then P ∈ P OH or P ∈ P UH , according as q ≡ −1 (mod 4) or q ≡ 1 (mod 4). 2 Note that if q ≡ 1 (mod 4), then OH has q 3 −q 2

q 3 −q 2

+ q + 1 absolute points and UH has

+ q + 1 absolute points; if q ≡ −1 (mod 4), then OH has 2

q 3 −q 2

+ q 2 + 1 absolute

points and UH has q 2−q + q + 1 absolute points. In what follows we rephrase a result found by J.M. Dover, K.E. Mellinger and K.L. Wantz in [15]. 3

Theorem 5.6. [15, Theorem 3.4] The set obtained by glueing together the points of C and the points of C¯ that are internal with respect to C is a blocking semioval of PG(2, q 2 ). Theorem 5.7. In PG(2, q 2 ), the set S ∪ C or E ∪ C is a blocking semioval according as q ≡ 1 (mod 4) or q ≡ −1 (mod 4), respectively. Proof. In view of Theorem 5.6, it is enough to show that the (left-normalized) points of S or of E are internal with respect to C according as q ≡ 1 (mod 4) or q ≡ −1 (mod 4), respectively. Let P = (1, 0, α) ∈ S, where α ∈ GF(q 2 ), α + αq = 0. We observe that α ∈ 2q2 or α ∈ / 2q2 , according as q ≡ −1 (mod 4) or q ≡ 1 (mod 4). Then P O is external to C if and only if α ∈ / 2q2 . Hence we have that if q ≡ 1 (mod 4), then P is internal to C. Assume that P = (1, α, −1/2s), for some s ∈ GF(q), s ∈ / 2q and α = β + iγ ∈ GF(q 2 ) \ GF(q) and αq+1 = 1/s. Then P O is external to C if and only if α2 − 1/s ∈ / 2q2 . Note that α2 − 1/s = α2 − αq+1 = 2iβγ ∈ / 2q2 if q ≡ −1 (mod 4), as required. 2 We are ready to prove our main result.

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Theorem 5.8. If q ≡ 1 (mod 4), the absolute points of UH form a blocking semioval of H(q 2 ). If q ≡ −1 (mod 4), the absolute points of OH form a blocking semioval of H(q 2 ). 3 In both cases the blocking semioval has size q 2−q + q 2 + 1. Proof. Apply Theorem 5.5, Theorem 5.7 and Lemma 3.9.

2

6. Baer subplanes of H(q 2 ) Subplanes of the regular Hughes plane of order nine have been classified by Denniston in 1968 [11]. In particular he proved that H(9) contains three non-equivalent subplanes of order 2 and four non-equivalent subplanes of order 3. The results provided by Denniston regarding Baer subplanes of H(9) match with those given by Penttila and Royle in [26]. Note that a pointset Z of size q 2 + q + 1 of a projective plane Π of order q 2 such that every line of Π meets Z in either 1 or q + 1 points is a Baer subplane (i.e., a subplane of order q) of Π. In this section we describe a class of Baer subplanes of H(q 2 ) distinct from π = PG(2, q) and admitting an automorphism group of order q 3 (q − 1). Let r denote the Baer subline of π consisting of the q + 1 points {(1, c, 0) | c ∈ GF(q)} ∪{(0, 1, 0)}. Let x be a fixed element of GF(q 2 ) \GF(q), and consider the following pointset of H(q 2 ): B = {(x + λ, μ, 1) | λ, μ ∈ GF(q)} ∪ r. Theorem 6.1. B is a Baer subplane of H(q 2 ). Proof. Let H be a line of H(q 2 ). If H is the line X1 = 0, then H ∩ B = H ∩ r = {(0, 1, 0)}. Let H : t  X1 + X2 = 0, for some t ∈ GF(q 2 ). In the case in which t = 0 then H ∩ B = {(x + λ, 0, 1) | λ ∈ GF(q)} ∪ {(1, 0, 0)}. If t ∈ GF(q) \ {0} then H ∩ B = ¯ μ ¯, 1)}, where H ∩ r ={(1, −t, 0)}. Supposet ∈ GF(q 2 ) \GF(q). Then H ∩ B = {(x + λ, 2 2 2 2 2 2 i t1 −t0 t0 −i t1 t0 x1 −t1 x0 t0 x1 +t1 x0 ¯ ¯ (λ, μ ¯) = , x1 t1 ¯) = − , x 1 t1 / 2q2 or (λ, μ , according as t ∈ t1 t1 or t ∈ 2q2 , respectively. Here and in the sequel x = x0 + ix1 and t = t0 + it1 , with x0 , x1 , t0 , t1 ∈ GF(q). Let H be the line aX1 + t  X2 + X3 = 0, for some a ∈ GF(q) and t ∈ GF(q 2 ). If a = t = 0, then H ∩ B = r. If a = 0 and t ∈ GF(q) \ {0}, then H ∩ B = {(x + λ, −1/t, 1) | λ ∈ GF(q)} ∪ {(1, 0, 0)}. If a = 0 and t ∈ GF(q 2 ) \ GF(q), then H ∩ B = H ∩ r = {(1, 0, 0)}. If a = 0 and t ∈ GF(q), then H ∩ B = H ∩ r = {(t, −a, 0)}  and if a = 0 and t ∈ 2 0 t1 −t1 1 ¯ ¯ GF(q ) \ GF(q), then H ∩ B = {(x + λ, μ ¯, 1)}, where (λ, μ ¯) = ax1 t0 −ax , − ax . at1 t1 Finally let H be the line t  (X1 + bX2 ) + aX2 + X3 = 0, where a, b ∈ GF(q) ¯ μ and t ∈ GF(q 2 ) \ GF(q). If a = 0 and t ∈ 2q2 , then H ∩ B = {(x + λ, ¯, 1)}, where 2 2 2 2 2 2 bx1 (i t1 −t0 )−a(x0 t1 +t0 x1 )+bt1 x1 (t0 −i t1 )−t1 ¯ , and μ ¯ = . In the case in which a = 0 λ = at1

at1

¯ μ ¯ = and t ∈ / 2q2 , then H ∩ B = {(x + λ, ¯, 1)}, where λ

bx1 (t20 −i2 t21 )−a(x0 t1 −t0 x1 )+bt1 , at1

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x (i2 t2 −t2 )−t

and μ ¯ = 1 1at1 0 1 . Now assume a = 0, t ∈ 2q2 . If (t20 − i2 t21 )x1 − t1 H ∩ B = {(x − t0 x1 +t1tx1 0 +bt1 μ , μ, 1) | μ ∈ GF(q)} ∪ {(−b, 1, 0)}, otherwise H ∩ r = {(−b, 1, 0)}. Next suppose a = 0, t ∈ / 2q2 . If (t20 − i2 t21 )x1 + t1 t0 x1 −t1 x0 −bt1 μ H ∩ B = {(x + , μ, 1) | μ ∈ GF(q)} ∪ {(−b, 1, 0)}, otherwise t1 H ∩ r = {(−b, 1, 0)}. 2

= 0 then H ∩ B = = 0 then H ∩ B =

If we consider points of H(q 2 ) represented by left-normalized vectors of V \ {(0, 0, 0)}, then B = B1 ∪ B2 ∪ r, where   1 μ 2 , | λ, μ ∈ GF(q), x + λ ∈ / 2 , q xq + λ xq + λ    1 μ B2 = , | λ, μ ∈ GF(q), x + λ ∈ 2q2 . 1, x+λ x+λ 

B1 =

1,

The next result shows that B1 ∪ B2 ∪ r considered as a set of points of PG(2, q 2 ) is not isomorphic to PG(2, q). Proposition 6.2. B1 ∪ B2 ∪ r is not PΓL(3, q 2 )-equivalent to PG(2, q). Proof. In order to prove the result we will exhibit a line  of PG(2, q 2 ) meeting B1 ∪B2 ∪r ˜ in exactly two points. μ2 be fixed elements of GF(q), with μ1 = μ2 . Let   Let λ, μ1 ,  μ1 μ2 1 1 P1 = 1, xq +λ˜ , xq +λ˜ ∈ B1 , P2 = 1, x+ ∈ B2 and let  be the line of PG(2, q 2 ) ˜ , x+λ ˜ λ having equation   ˜ − μ1 (x − xq ) X3 = 0. (μ1 − μ2 )X1 + (x − xq )X2 + (μ2 − μ1 )(xq + λ) Then it is easily seen that  is the line joining P1 and P2 and that | ∩ r| = 0. Let  1 P = 1, xqμ+λ , xq +λ be a point of B1 , for some λ , μ ∈ GF(q), and assume that P  ˜ 2 −μ1 ), belongs to . A straightforward calculation shows that (x−xq )(μ −μ1 ) = (λ −λ)(μ ˜ 2 − μ1 ) ∈ GF(q) and x − xq ∈ GF(q 2 ) \ GF(q). It follows that λ = λ ˜ where (λ − λ)(μ   μ 1  and μ = μ1 , that is P = P1 . Similarly, if P = 1, x+λ , x+λ is a point of B2 belonging ˜ 2 − μ1 ). In this case λ = λ ˜ and to , then we have that (x − xq )(μ − μ2 ) = (λ − λ)(μ μ = μ2 . Therefore P = P2 and | ∩ (B1 ∪ B2 ∪ r)| = 2. 2

Moreover, B1 ∪ B2 ∪ r considered as a set of points of H(q 2 ) is distinct from π. Proposition 6.3. There is no collineation of H(q 2 ) mapping π to B. Proof. Let G be the collineation group of H(q 2 ). From [31], every element of G stabilizes π. Assume by contradiction that there is a collineation α of G mapping π to B. Then α−1 Gα = G should fix B. On the other hand, τ is an element of G and Bτ = B, a contradiction. 2

A. Aguglia, F. Pavese / Finite Fields and Their Applications 62 (2020) 101624

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¯ be the subgroup of GL(3, q), whose elements are the matrices Let K ⎛

1 0 ⎜ ⎝a b c d

⎞ 0 ⎟ 0⎠, 1

where a, b, c, d ∈ GF(q), b = 0. Let K be the group of collineations of H(q 2 ) induced by ¯ Then K is a group of order q 3 (q − 1). In particular, it is not difficult to check that K. K leaves invariant B. Proposition 6.4. The Baer subplane B is left invariant by the group K. Finally, observe that distinct elements of GF(q 2 ) \ GF(q) gives equivalent Baer subplanes in H(q 2 ). Indeed if x, y ∈ GF(q 2 ) \ GF(q), where x = x0 + ix1 and y = y0 + iy1 , then the collineation of H(q 2 ) induced by 

y1 /x1 0 0

0 0 1 0 0 1



maps the point (x + λ, μ, 1) to the point (y + λ , μ, 1), where λ = (x0 y1 − x1 y0 + λy1 )/x1 . Hence it sends {(x + λ, μ, 1) | λ, μ ∈ GF(q)} ∪ r to {(y + t, μ, 1) | t, μ ∈ GF(q)} ∪ r. References [1] A. Aguglia, A. Cossidente, F. Pavese, Blocking structures in finite projective planes, J. Comb. Des. 26 (7) (2018) 356–366. [2] A. Aguglia, L. Giuzzi, Intersection sets, three-character multisets and associated codes, Des. Codes Cryptogr. 83 (2) (2017) 269–282. [3] A. Aguglia, G. Korchmáros, Multiple blocking set and multiset in Desarguesian planes, Des. Codes Cryptogr. 56 (2010) 177–181. [4] D. Bartoli, F. Pavese, Blocking semiovals in PG(2, q 2 ), q odd, admitting PGL(2, q) as an automorphism group, Finite Fields Appl. 54 (2018) 315–334. [5] L.M. Batten, Determining sets, J. Comb. 22 (2000) 167–176. [6] A. Blokhuis, On multiple nuclei and a conjecture of Lunelli and Sce, Bull. Belg. Math. Soc. Simon Stevin 1 (1994) 349–353. [7] R.H. Bruck, R.C. Bose, The construction of translation planes from projective spaces, J. Algebra 1 (1964) 85–102. [8] R.H. Bruck, R.C. Bose, Linear representations of projective planes in projective spaces, J. Algebra 4 (1966) 117–172. [9] F. Buekenhout, Exisistence of unitals in finite translation planes of order q 2 with kernel of order q, Geom. Dedic. 5 (1976) 189–194. [10] A. Cossidente, G.L. Ebert, Permutable polarities and a class of ovoids of the Hermitian surface, Eur. J. Comb. 25 (7) (2004) 1059–1066. [11] R.H.F. Denniston, Subplanes of the Hughes plane of order 9, Proc. Camb. Philos. Soc. 64 (1968) 589–598. [12] J.M. Dover, A lower bound on blocking semiovals, Eur. J. Comb. 21 (2000) 571–577. [13] J.M. Dover, Some new results on blocking semiovals, Australas. J. Comb. 52 (2012) 269–280. [14] J.M. Dover, K.E. Mellinger, Semiovals from unions of conics, Innov. Incid. Geom. 12 (2011) 61–83. [15] J.M. Dover, K.E. Mellinger, K.L. Wantz, Blocking semiovals containing conics, Adv. Geom. 13 (1) (2013) 29–40.

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[16] J.M. Dover, B.B. Ranson, Blocking semiovals in PG(2, 7) and beyond, Eur. J. Comb. 24 (2003) 183–193. [17] N. Durante, A. Siciliano, Some blocking semiovals of homology type in planes of square order, Des. Codes Cryptogr. 72 (2014) 185–193. [18] T. Grundhöfer, C. Hering, Finite nearfields classified, again, J. Group Theory 20 (4) (2017) 829–839. [19] X. Hubaut, Limitation du nombre de points d’un (k, n)-arc régulier d’un plan projectif fini, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 48 (1970) 490–493. [20] D.R. Hughes, A class of non-Desarguesian projective planes, Can. J. Math. 9 (1957) 378–388. [21] C. Jacobs, A new blocking semioval, Bull. Inst. Comb. Appl. 42 (2004) 19–24. [22] G. Kiss, A survey on semiovals, Contrib. Discrete Math. 3 (2008) 81–95. [23] N. Nakagawa, C. Suetake, On blocking semiovals with an 8-secant in projective planes of order 9, Hokkaido Math. J. 35 (2006) 437–456. [24] G. Nicoletti, Le polarità dei piani di Hughes finiti, Boll. Unione Mat. Ital. (4) 9 (1974) 281–296. [25] T.G. Ostrom, Vector spaces and construction of finite projective planes, Arch. Math. 19 (1968) 1–25. [26] T. Penttila, G.F. Royle, Sets of type (m, n) in the affine and projective planes of order nine, Des. Codes Cryptogr. 6 (3) (1995) 229–245. [27] F. Piper, Polarities in the Hughes plane, Bull. Lond. Math. Soc. 2 (1970) 209–213. [28] T.G. Room, The combinatorial structure of the Hughes plane, Proc. Camb. Philos. Soc. 68 (1970) 291–301. [29] T.G. Room, Polarities and ovals in the Hughes plane, J. Aust. Math. Soc. 13 (1972) 196–204. [30] T.G. Room, The combinatorial structure of the Hughes plane. II, Proc. Camb. Philos. Soc. 72 (1972) 135–139. [31] L.A. Rosati, I gruppi di collineazioni dei piani di Hughes, Boll. Unione Mat. Ital. (3) 13 (1958) 505–513. [32] L.A. Rosati, Unicità e autodualità dei piani di Hughes, Rend. Semin. Mat. Univ. Padova 30 (1960) 316–327. [33] L.A. Rosati, Disegni unitari nei piani di Hughes, Geom. Dedic. 27 (3) (1988) 295–299. [34] C. Suetake, Some blocking semiovals which admit a homology group, Eur. J. Comb. 21 (2000) 967–972. [35] C. Suetake, Two families of blocking semiovals, Eur. J. Comb. 21 (2000) 973–980. [36] J.A. Thas, On semi ovals and semi ovoids, Geom. Dedic. 3 (1974) 229–231. [37] K.L. Wantz, A new class of unitals in the Hughes plane, Geom. Dedic. 70 (2) (1998) 125–138. [38] G. Zappa, Sui gruppi di collineazioni dei piani di Hughes, Boll. Unione Mat. Ital. (3) 12 (1957) 507–516. [39] H. Zassenhaus, Über endliche Fastkörper, Abh. Math. Semin. Univ. Hamb. 11 (1936) 187–220.