Blocking and double blocking sets in finite planes

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order q2 of size q2 +2q+2 admitting ⇤This author has been supported as a postdoctoral fellow of the Research Foundation Flanders (Belgium) (FWO). †This author was supported by a Visiting Professor grant of the Special Research Fund Ghent University (BOF project number 01T00413) and a Research Grant of the Research Foundation Flanders (Belgium) (FWO) (project number 1504514N). ‡This author is a postdoctoral fellow of the Research Foundation Flanders (Belgium) (FWO). the electronic journal of combinatorics 23(2) (2016), #P2.5 1 1-,2-,3-,4-, (q + 1)and (q + 2)-secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian a ne plane of order q2 of size at most 4q2/3+5q/3, which is considerably smaller than 2q2 1, the Jamison bound for the size of a minimal blocking set in an a ne Desarguesian plane of order q2. We also consider particular André planes of order q, where q is a power of the prime p, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in 1 mod p points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.


Introduction
In finite geometry one often studies combinatorial analogues of classical substructures of Galois geometries.In case of projective planes, examples of such combinatorially defined substructures are arcs, ovals and hyperovals, (k, n)-arcs, unitals, blocking sets and multiple blocking sets.Most results regarding these are for planes coordinatized over finite fields (i.e.Desarguesian planes) and are obtained by using algebraic methods.Therefore it is an interesting question to decide whether these results remain valid for non-Desarguesian planes or not.It is not too surprising that only few results have a combinatorial proof, and in most cases one can find counterexamples showing that the strong results for Desarguesian planes cannot be extended to non-Desarguesian ones.One strategy for constructing counterexamples is to consider a substructure of a Desarguesian plane and study when this subset will be a similar inherited substructure in a suitable non-Desarguesian plane.Another strategy for constructing examples and counterexamples is to use higher-dimensional representations of non-Desarguesian planes (the André/Bruck-Bose representation).These methods can be successful and the present paper illustrates both of them but the focus is on the first one.The early results on inherited substructures were about ovals, the reader is referred to [18].More recent results in this direction can be found in [17].In the present paper we use inherited substructures for constructing blocking sets in Hall-and André planes.

Preliminaries
A blocking set B in a projective plane Π q of order q is a set of points such that every line of Π q contains at least one point of B. We also say that the set B blocks all lines.A minimal blocking set B is a blocking set such that no proper subset of B is a blocking set.An essential point of a blocking set is a point lying on at least one tangent line to B and we see that B is minimal if and only if every point of B is essential.A blocking set is called trivial if it contains a line.A t-fold blocking set is a set of points such that every line contains at least t points of B. A 1-fold blocking set is simply a blocking set and a 2-fold blocking set is mostly called a double blocking set.A Baer subplane of an arbitrary projective plane of order q 2 , is a set of q 2 + q + 1 points such that its inherited point-line structure forms a projective subplane of order q.It is well-known that any Baer subplane is a blocking set in its ambient plane.
A blocking set B in a projective plane of order q is said to be of Rédei type if there exists a line such that |B| = q + |B ∩ |.Given a set U of q points in an affine plane, the set of determined directions is the set D U of those points on the line at infinity that admit a line through them which intersects U in at least two points.It is well-known that U ∪ D U is a blocking set of Rédei type in the projective closure.For more details, we refer to [20].
If B is a blocking set in a projective plane Π and on a point P ∈ B there are t tangents to B, we may take one point on each but one, say, , of these tangents, add these points to B and remove P from B to obtain a blocking set in the affine plane Π \ of size |B| + t − 2. This well-known idea shows that Result 1.3 is equivalent with the following.
Result 1.4.Let B be a blocking set in PG(2, q).Then each essential point of B lies on at least 2q − |B| + 1 tangent lines to B.
After the third author's talk at the 37th ACCMCC, Gordon Royle asked whether the 1 modulo p result (Result 1.1) for blocking sets in Galois planes could be extended for non-Desarguesian planes.In a sense this question was the starting point for this paper.
In Section 3, we provide information about Baer subplanes of Desarguesian projective planes and their stabilisers which will be applied in Hall planes.In Section 4, we construct small blocking sets in non-Desarguesian planes and show that, as expected, the above mentioned results on blocking sets do not hold for non-Desarguesian planes in general.These results are also related to small double blocking sets in Desarguesian planes.Let us state some of our results here.
Theorem 1.6.Let q 2 9 be a square prime power.Then there exists an affine plane of order q 2 in which there is a blocking set of size 4q 2 /3 + 5q/3 .Note that regarding Result 1.3, there was one counterexample known: Bruen and de Resmini ( [6], 1983) constructed a blocking set of size 16 in a particular non-Desarguesian affine plane of order 9.However, Bierbrauer [3] pointed out that the construction works in all non-Desarguesian affine planes of order 9.
Theorem 1.7.Let GF(r) be a proper subfield of GF(q), and suppose that gcd(r − 2, q − 1) = 1, r 4. Then there exists a projective plane of order q in which there is a minimal blocking set of size q + 2(q − 1)/(r − 1) admitting an r + 2-secant.
Theorem 1.8.Let B be a non-trivial blocking set in PG(2, p h ), p prime, h 2, of size |B| 3  2 (p h − p h−1 ).Then there exists a blocking set of size p h + p h−1 + 1 that is disjoint from B. Consequently, if p > 5, then there exists a double blocking set in PG(2, p h ) of size 2(p h + p h−1 + 1).
Additionally, we apply the same methods to find small double blocking sets with respect to k-spaces in PG(2k, p h ).
In Section 2, we show some connections of the above results with value sets of certain polynomials.For a polynomial f ∈ GF(q)[x], let V (f ) = {f (x) : x ∈ GF(q)} denote the value set of f .As usual, we consider x −a and x q−1−a as the same functions, and thus we interpret 0 −a as zero.Determining the size of the value set of polynomials is hard in general.Cusick [7] and, based on Cusick's work, Rosendahl [16] examined the value sets of polynomials of the form s a (x) = x a (x + 1) √ q−1 , q a square, and have derived several exact results, mostly in the case when q is even.Their proofs depend on the arithmetic of GF(q) and connections to cross-correlation functions.If q is odd, then, up to our knowledge, the following are the only known corresponding results.Note that one may define these polynomials more generally with respect to any field extension GF(q) ⊂ GF(q h ) as s a (x) = x a (x + 1) q−1 .
Result 1.9 (Cusick-Müller [8]).For any prime power q and h 2, the size of the value set of s 1 (x) = x(x + 1) q−1 in GF(q h ) is Result 1.10 (Rosendal [16]).Assume that q ≡ 0 (mod 3).Then the size of the value set of Let us remark that [16, Theorem 2.8] states the above formula for q ≡ 1 (mod 3), q odd as well but, a short check using GAP [10] shows that the result actually fails in that case.Using the connections to be established in Section 2, in Section 5 we obtain the following result.
Theorem 1.11.Let q be odd.The size of the value set of s −1 (x) = x −1 (x + 1) q−1 in GF(q 2 ) is Let us remark that our methods work for q even as well, for which case the respective results have already been obtained by Cusick [7] and Rosendahl [16], see Section 5.
2 Connections of double blocking sets of PG(2, q h ), lines of André planes, and value sets of certain polynomials In the following, unless stated otherwise, q denotes an arbitrary power of a prime p, and we consider GF(q h ), h 2 as the extension of GF(q).For a field F, F * denotes its multiplicative group.We use (x : y : z) to denote a homogeneous triplet over the respective field.We call ∞ = {(1 : y : 0) : y ∈ GF(q h )} ∪ {(0 : 1 : 0)} the line at infinity of PG(2, q h ), and we let AG(2, q h ) denote the affine plane PG(2, q h ) \ ∞ .Recall that x → x q is an automorphism of GF(q h ).
We consider the following model for a particular affine André plane of order q h .The upcoming method for constructing projective planes, due to T. G. Ostrom for h = 2, is called derivation.For more information on derivation and derived planes, we refer to [12].We use D as a derivation set, that is, for all a ∈ D, we replace the lines of AG(2, q h ) with equation y = ax + b by other suitable subsets, namely those defined by the equation y = ax q + b.For our purposes, it will be convenient to introduce the following notation.f a,c (x) := ax q − ac, where a = 0.
Then an affine André plane can be constructed in the following way.Define the set P as the set of points of AG(2, q h ).Define the set L as a set of lines of two types: (i) the lines of PG(2, q h ) meeting ∞ not in D; the electronic journal of combinatorics 23(2) (2016), #P2.5 (ii) the sets B(a, c), where a ∈ D, c ∈ GF(q h ).
The incidence I is the natural incidence.It is well known that Π A D := (P, L, I) is an affine plane of order q h (and it is easy to check as well).Its projective completion Π D is a projective André plane of order q h .Unless causing confusion, we write simply Π and Π A instead of Π D and Π A D .It is clear that the parallel classes of lines of type (ii) are the sets [a] := {B(a, c) : c ∈ GF(q h )}, where a ∈ D, as U(a, c 1 ) and U(a, c 2 ) are disjoint if c 1 = c 2 .For each a ∈ D, let (a) denote the common point of the lines of [a], and let D = {(a) : a ∈ D}.The points of ∞ \ D ⊆ PG(2, q h ) naturally correspond to the common points of the parrallel classes of lines of type (i) in Π A .Note that the point set of a line B(a, c) of type (ii) in Π is U(a, c) ∪ {(a)}.The directions in D and lines of type (ii) are also called derived directions and derived lines, respectively.
We will denote by ∞ the line at infinity in Π D .In what follows, we will often consider an object in one plane (Π or PG(2, q 2 )), and interpret it in the other plane, e.g., a line of type (ii) in Π is a blocking set of PG(2, q 2 ), an affine point of PG(2, q 2 ) is an affine point of Π, etc.Note that if h = 2, then B(a, c) is always a Baer subplane, and the plane Π is the well known Hall plane of order q 2 .In this case, the set of all lines of type (ii) in Π is the set of all Baer subplanes in PG(2, q 2 ) that contain D.
Suppose that B is a blocking set of PG(2, q h ).Then B * := B \ D blocks every line of type (i) in Π.Clearly, B * ∪ D is a blocking set of Π; however, B * alone may not block all lines of type (ii We choose a blocking set of PG(2, q h ) in the following way.Let g(x) = x q .Similarly as before, the set of directions determined by g(x) is {(1 : is a blocking set of Rédei type in PG(2, q h ), and ∞ ∩ B 0 = {(0 : 1 : 0)} / ∈ D. Recall that s −1 (x) = x −1 (x + 1) q−1 , and note that 0 ∈ V (s −1 ).Definition 2.1.For a point set T ⊂ PG(2, q h ) and any element a ∈ GF(q h ) * , let σ T i (a) = |{c ∈ GF(q h ) : |U(a, c) ∩ T | = i}|, and let the type of a (with respect to T ) be σ Proof.Let a ∈ GF(q h ) * , c ∈ GF(q h ), f = f a,c and g(x) = x q .As g(y) = 0 ⇐⇒ y = 0, D(a) ∩ U 0 and D(a) ∩ D 0 are clearly empty; thus D(a) ∩ B 0 = ∅.Note that f (x) = 0 ⇐⇒ x q = c.Thus U(a, c) ∩ D 0 is nonempty iff there exists y ∈ GF(q) * such that (y −(q−1) ) q = c.
Let t ∈ GF(q h ) * .Replacing x by t (q−1)/q x in ( ), it is easy to see that If c = 0, then ( ) has a solution if and only if a ∈ C. Suppose now c = 0. Since Let C a be the coset (−1/a q−1 )C, and let χ C (a) be one or zero depending on whether a / ∈ C or a ∈ C, respectively.As gcd(q − 2, ) is a mapping from GF(q h ) * to C a which covers each element of the image exactly k times.Thus, by Lemma 2.2 (iv), for any a ∈ D we have Recall that 0 ∈ V (s −1 ).Note that by Lemma 2.2 (3), σ U 0 0 is constant on any coset of C, hence, if the elements a 1 , . . ., a k ∈ D are representatives of the k cosets of C, then Note that each coset of C intersects D in |D|/k points.Therefore we see that Thus the size of the value set V (s −1 ) is determined by a∈D σ U 0 0 (a), which is of purely geometrical nature: it is the number of skew lines of type (ii) to U 0 in the affine André plane Π A D .In the case of h = 2, that is, Hall planes, lines of type (ii) are Baer subplanes of PG(2, q 2 ).Baer subplanes and their intersection properties are quite well understood in Desarguesian planes, which is well exploitable in the present context.
3 On Baer subplanes in PG(2, q 2 ) It is well-known that every subplane of a Desarguesian projective plane is itself a Desarguesian plane and hence, is coordinatised by the elements of a subfield of GF(q).In particular, we get that a Baer subplane of PG(2, q 2 ) corresponds to a set of q 2 + q + 1 points whose homogeneous coordinates, with respect to a well-chosen frame of PG(2, q 2 ), are in the subfield GF(q) of GF(q 2 ); a frame of a projective plane is a set of four points, no three of which are collinear.Likewise, a frame of PG(1, q 2 ) is a set of 3 distinct points and a Baer subline of PG(1, q 2 ) is defined as a set of q + 1 points such that the coordinates with respect to a frame of PG(1, q 2 ), are in GF(q).For a point P ∈ PG(2, q 2 ), let P denote the set of lines through P ; that is, the pencil with carrier P .For two distinct points P and R, let P, R denote the line of PG(2, q 2 ) connecting P and R. Recall that given a Baer subplane B of PG(2, q 2 ), every line of PG(2, q 2 ) intersects B in one or q + 1 points.The lines of the latter type are called long secants (of B).Let P B denote the set of long secants of B through P .It is well-known that if P ∈ B, then | P B | = q + 1, and if P / ∈ B, then Definition 3.1.Let P be a point of PG(2, q 2 ) and let l be any line such that P ∈ l.Choose a Baer subline m contained in l.Then the Baer subpencil determined by P and m is the set of q + 1 lines on P meeting l in a point of m.
Note that if B is a Baer subplane and P ∈ B, then P B is also a Baer subpencil of P , as for any long secant line l of B not containing P , P B is determined by P and the Baer subline l ∩B.Using the fact that the points of a frame in PG(2, q 2 ) or PG(1, q 2 ) determine a unique Baer subplane or subline respectively, we obtain the following well-known facts.Lemma 3.2.Given three collinear points in PG(2, q 2 ), there exists a unique Baer subline containing them.Dually, given three concurrent lines in PG(2, q 2 ), there exists a unique Baer subpencil containing them.Four points, no three of which are collinear, in PG(2, q 2 ), are contained in a unique Baer subplane.
Let us remark that we could use not only D but any Baer subline of ∞ to construct the Hall plane.However, using D does not cause loss of generality, as the collineation group of PG(2, q 2 ) is transitive on the Baer sublines of a given line.We prove a couple of such transitivity results in connection with Baer subplanes and Baer sublines which might be known, yet seem to be hard to find a direct reference for.We use the wellknown fact that the group PGL(3, q 2 ) of projective linear transformations of PG(2, q 2 ) is sharply transitive on the frames.For basic information on the collineations of PG(2, q 2 ) the electronic journal of combinatorics 23(2) (2016), #P2. 5 and PG(1, q 2 ) we refer to [11].If a group G acts on a set Ω and A 1 , . . ., A n are subsets of Ω, we denote by Stab G (A 1 , . . ., A n ) the subgroup of G that stabilises setwise each of the A i 's, 1 i n.If A i = {P } is a single point, we write simply P instead of {P }.Lemma 3.3.Let B be an arbitrary Baer subplane of PG(2, q 2 ).For a point Q ∈ B, L Q denotes the set of tangent lines to B through Q.For a line tangent to B, X denotes the set of all Baer sublines of , X = {R ∈ X : B ∩ R = ∅}.For any line , Y denotes the set of all Baer subplanes that intersect in precisely one point, and for any fixed Baer subline R of , Y ,R = {B ∈ Y l : B ∩R = ∅}.Let us interpret PGL(3, q 2 ) and PGL(2, q 2 ) as the groups of all projective linear transformations of PG(2, q 2 ) and (after is fixed), respectively.Then the following hold. 1 3. Let be a tangent to B with tangency point Q.Then the actions of Stab PGL(3,q 2 ) (B, ) and Stab PGL(2,q 2 ) (Q) on are the same.4. Let be a tangent to B. Then Stab PGL(3,q 2 ) (B, ) is transitive on X . 5. Let R be a Baer subline of an arbitrary line .Then Proof.Let Q ∈ B, and let be a line tangent to B on Q.Let ι be the mapping of Stab PGL(3,q 2 ) (B, ) into Stab PGL(2,q 2 ) (Q) given by the natural action of Stab PGL(3,q 2 ) (B, ) on .First we show that the kernel of this mapping is trivial.Suppose to the contrary that there exist a collineation ϕ ∈ Stab PGL(3,q 2 ) (B, ) that fixes pointwise, and that there exists a point A ∈ B such that ϕ(A) = A. Let e be a long secant of are two different long secants to B through R, a contradiction.Thus ι is an injection.As PGL(2, q 2 ) is sharply transitive on the triplets of , |Stab PGL(2,q 2 ) (Q)| = q 2 (q 2 − 1), whence |Stab PGL(3,q 2 ) (B, )| q 2 (q 2 − 1).Let A, B and C be three points of B such that Q, A, B and C are in general position.As Stab PGL(3,q 2 ) (B, Q) is sharply transitive on such triplets, the image of A, B and C can be chosen in (q 2 + q)q 2 (q 2 − 2q + 1) = q 3 (q + 1)(q − 1) 2 ways, so this is the order of q. Hence |Stab PGL(3,q 2 ) (B, )| q 2 (q 2 − 1).Consequently, equality holds, and thus Stab PGL(3,q 2 ) (B, Q) is transitive on L Q , and ι is a bijection, so Stab PGL(3,q 2 ) (B, ) acts on as Stab PGL(2,q 2 ) (Q) does.Thus (1) and (3) follow.(2) follows from (1) as PG(2, q 2 ) is self-dual.
In other words, [P ] is the set of lines of type (ii) through P , and [D] is the set of all lines of type (ii) in the Hall plane.The next result is well-known (basically this verifies the correctness of the construction of the Hall plane) yet we include a short proof.
Proof.Let P, R ∈ AG(2, q 2 ), and let Q 1 , Q 2 ∈ D \ P, R arbitrary.By Lemma 3.2 there is a unique Baer subplane Now let P ∈ AG(2, q 2 ) be arbitrary.Then the Baer subplanes in [P ] partition the q 2 − 1 points of each line P, Q \ {P, Q}, Q ∈ D into subsets of size q − 1.We may conclude that |[P ]| = q + 1. Definition 3.6.Suppose that P, R ∈ AG(2, q 2 ).If P, R ∩ ∞ ∈ D, then let [P, R] denote the unique Baer subplane containing {P, R} ∪ D (cf.Proposition 3.5).We will say that [P, R] exists if P, R ∩ ∞ ∈ D; otherwise we will say that [P, R] does not exist.Definition 3.7.Given a Baer subplane B of PG(2, q 2 ) such that B ∩ ∞ = {Q}, Q ∈ D, let S = S B be the set of lines of PG(2, q 2 ) meeting ∞ in a point of D and meeting B in q + 1 points.
An oval of a projective plane of order q is a set of q + 1 points, no three of which are collinear.It is easy to see that every point of an oval O lies on a unique tangent line to O; if q is odd, the tangent lines to an oval form a dual oval; and if q is even, all tangent lines to the oval O are concurrent.Moreover, there are (q+1)q 2 secant lines to O and (q−1)q 2 external lines to O. Lemma 3.10.Let B be a Baer subplane of PG(2, q 2 ) such that B∩ ∞ = {Q}, Q / ∈ D. Let P ∈ D be a point of the Hall plane Π D , and let t i (P ) denote the number of i-secant lines (of type (ii)) through P to B * := B\{Q}.If q ≡ 2 (mod 3), then ∀ P ∈ D : t 0 (P ) = (q 2 −q)/3.If q ≡ 2 (mod 3), then for (q + 1)/3 points P ∈ D : t 0 (P ) = (q 2 − q − 2)/3, and for 2(q + 1)/3 points P ∈ D : t 0 (P ) = (q 2 − q + 1)/3.
By Lemma 3.9, each of the q + 1 points in O determines q 2-secants to B through some points of D .Moreover, all these 2-secants are different and every 2-secant line of type (ii) is obtained in this way, so we get P ∈D t 2 (P ) = (q + 1)q.Thus, since |D | = q + 1, and |t 2 (P )| |O| = q + 1, either there exists a point P ∈ D such that t 2 (P ) = q + 1 ≡ q (mod 3) and hence q ≡ 2 (mod 3), or for all points P ∈ D : t 2 (P ) = q.
Suppose now q ≡ 2 (mod 3) (that is, k = 3).By Lemma 3.3 (5), we may assume without loss of generality that B = B 0 .If P = (a) then t i (P ) = σ B 0 i (a), hence by Lemma 2.2 (3) we have that D can be partitioned into three sets of size (q + 1)/3, and on each of these sets, t 2 (P ) takes on the same value.As 3 | t 2 (P ) and there must be some P ∈ D with t 2 (P ) < q+1, it is easy to see that t 2 (P ) = q−2 on one of these sets, and t 2 (P ) = q+1 on the two remaining sets.

Some blocking and double blocking sets
First, starting from the Hall plane, we give the construction of interesting small blocking sets in non-Desarguesian planes.
Proof.Let D be a Baer subline of ∞ in PG(2, q 2 ) and let, as before, D denote the set of derived directions.Let B be a Baer subplane of PG(2, q 2 ) such that B ∩ ∞ is a single point Q ∈ D. Then it is clear that T := B ∪ D is a blocking set of H of size q 2 + 2q + 2. We claim that T is a minimal blocking set.By Lemma 3.10, the points of D are essential for T .The points of B (including Q) are also essential, as there are at least the electronic journal of combinatorics 23(2) (2016), #P2.5 2q − 1 > 0 tangents on each point P of B. This proves our claim.Using Lemma 3.9, we see that a line of type (ii) meets T in either 2, 3 or 4 points.A line of type (i) meets T in 1 or q + 1 points and ∞ meets T in q + 2 points.
Remark 4.2.Using similar ideas, we could also show that in the projective Hall plane of order q 2 , q 3, there also exists a minimal blocking set of size q 2 + 2q + 1 or q 2 + 2q.
Theorem 4.3.Let q be a prime power.There exists a non-Desarguesian affine plane of order q 2 in which there is a blocking set of size at most 4q 2 /3 + 5q/3.
Proof.Consider the blocking set T of size q 2 + 2q + 2 in the Hall plane H, constructed in Theorem 4.1.By Lemma 3.10, we may choose a point P ∈ D that has at most (q 2 − q − 2)/3 or exactly (q 2 − q)/3 tangents to T through it, according to whether q ≡ 2 (mod 3) or not, respectively.Let be one of these tangents.By putting one point on all skew lines of T \ {P } but , we obtain an affine blocking set in H \ of size at most 4q 2 /3 + 5q/3.
Note that the ratio of the size of the above constructed affine blocking set and the order of the plane tends to 4/3, which is notably smaller than the ratio 2 in case of Desarguesian affine planes.
Remark 4.4.We may also use multiple derivation to achieve similar results.That is, let D 1 , . . ., D n be pairwise disjoint Baer sublines of ∞ in PG(2, q 2 ), q 3, 1 n < q − 2, replace the lines intersecing and consider the resulting projective plane Π.Take a Baer subplane is a blocking set in Π for which all points of T ∩ ∞ are essential; moreover, as on each point of B \ {Q} there are at least q 2 + 1 − (n + 1)(q + 1) > 0 tangents to T , T is minimal.Thus in the resulting projective plane we obtain a minimal blocking set of size q 2 +(n+1)(q +1); and, as in Theorem 4.3, we find an affine plane of order q 2 with a blocking set of size at most 4q 2 /3 + (3n + 2)q/3 + n − 1.
Next we discuss blocking sets of the André planes Π D and double blocking sets of PG(2, q h ).In the proof of the next theorem, we will use the following result.
Result 4.5 (Bacsó-Héger-Szőnyi [1]).Let F be a finite field of characteristic p, and let H F * be a multiplicative subgroup of m elements.Suppose that g Theorem 4.6.Suppose that k = gcd(q − 2, q h − 1) = 1, h 2. Then for each a ∈ D, there exists c ∈ GF(q h ) * such that B(a, c) and B 0 are disjoint.
Proof.It is clear that B := B 0 ∪ D is a blocking set in Π of size q h + 2(q h − 1)/(q − 1).By Theorem 4.6, all points of D are essential.On the other hand, by Result 1.4, in PG(2, q h ) there are at least q h − 2(q h − 1)/(q − 1) + 1 tangents to B 0 in each point of B 0 .As at least q h − 3(q h − 1)/(q − 1) + 1 > 0 of these tangents are also lines of Π, the points of B 0 are also all essential.The line ∞ is a (q h − 1)/(q − 1) + 1-secant of Π.
Just as in case of starting with a Hall plane, one may use the same ideas to construct small blocking sets in non-Desarguesian affine planes coming from the André plane Π D .If for a point P ∈ D there are t 1 tangents to B 0 ∪ D , we may obtain an affine plane of order q h admitting a blocking set of size q h +2(q h −1)/(q −1)+t 1 −2 ≈ (1+2/(q −1)+t 1 /q h )q h .Using the GAP package FinInG [10,2], we have computed the number of tangents on points of D for several values of q and h.The results are shown in Table 1.
As a consequence of Theorem 4.6, we see that there exist two disjoint blocking sets of size q h + (q h − 1)/(q − 1) in PG(2, q h ) (namely, B 0 and B(a, c) with properly chosen parameters); thus their union is a small double blocking set.Such a double blocking set was already obtained in [1] in which the following theorem is shown.
In the same paper, the authors use this bound on τ 2 , the size of the smalles double blocking set, to determine the so-called upper chromatic number of the projective plane of order q, which is easily seen to be at least q 2 + q + 2 − τ 2 .
Next we describe other small double blocking sets obtained as the union of two disjoint blocking sets.As a corollary, we will construct a minimal blocking set in the André plane Π D of order q h , strictly containing a blocking set of the corresponding Desarguesian Table 1: B 0 ∪ D is a blocking set in the André plane Π D of order q h , t 1 is the minimum of the number of tangents to B 0 ∪ D through P , where P ranges through D , B A is the affine blocking set constructed from B 0 ∪ D in the mentioned way.Note that t 1 is particularly small if q is even and, for fixed h, |B A |/q h is decreasing in q if q is prime.Let us remark that for q = 32, h = 3, |B A | is smaller than 4 3 q h + 5 projective plane, but which is smaller than the one obtained in Corollary 4.7 if q is a prime.Moreover, if q = p h , where p and h are prime, we find a smaller upper bound for τ 2 (PG(2, p h )) than the one of Result 4.8, which also improves in that case the results on the upper chromatic number.For this construction, we will need the notion of linear blocking sets.By field reduction, the points of PG(2, q h ) are in one-to-one correspondence with the elements of a Desarguesian (h − 1)-spread S in PG(3h − 1, q) (as both kinds of objects can be interpreted as subspaces of a 3h dimensional vectorspace over GF(q)).Let B(µ) denote the set of spread elements of S meeting a subspace µ of PG(3h − 1, q).We often identify the element of S with the corresponding point in PG(2, q h ).The spread element corresponding to a point P of PG(2, q h ) will be denoted as P .If π is an h-space of PG(3h − 1, q), then it is clear that B(π) is a blocking set in PG(2, q h ), and such a blocking set is an GF(q)-linear blocking set.Note that a linear blocking set is necessarily minimal and its size is at most (q h+1 − 1)/(q − 1).For more information on field reduction and linear sets, we refer to [14].
We say that B(π) is of vertex-type if it is non-trivial and there exists one spread element of S, say V , which meets the h-space π in an (h−2)-space.It is easy to see that a blocking set of vertex-type consists of q +1 (q h−1 +1)-secants through the point V and thus has size q h + q h−1 + 1.It follows from [15] that B(π) is projectively equivalent to the set of points {(Tr(x) : 1 : x)|x ∈ GF(q h )} ∪ {(Tr(x) : 0 : x) | x ∈ GF(q h ) * }, where Tr denotes the trace function from GF(q h ) to GF(q), i.e.Tr : GF(q h ) → GF(q), x → x + x q + x q2 + . . .+ x q h−1 .Theorem 4.9.Let p > 5 and let B be a non-trivial blocking set in PG(2, p h ), p prime, of size |B| 3 2 (p h − p h−1 ) (e.g., a non-trivial linear blocking set with p > 5), then there exists a blocking set of vertex-type B(π) such that B ∩ B(π) = ∅.
Proof.Let Q be a point of PG(2, p h ), not contained in B. Using that every line through Q meets B in 1 mod p points by Result 1.1, this implies that there are at least p h + 1 − (|B| − p h − 1)/p tangent lines through Q to B. It is clear that the intersection points of the tangent lines with B cannot be collinear, since otherwise either B would be a trivial blocking set, or the size of B would be at least 2p h + 1 − (|B| − p h − 1)/p >3 2 (p h − p h−1 ).Now consider 3 non-collinear points P 1 , P 2 , P 3 such that QP i is a tangent line to B.
Choose a point T of Q (i.e. the spread element corresponding to Q), and consider the 2h-space τ through P1 , P2 , and the point T , then τ meets Q only in the point T .Denote the point set of the spread elements of B by B. Let µ be an (h − 2)-space in Q, not through T .We may consider τ to be the quotient space PG(3h − 1, p)/µ; every point R of PG(3h − 1, p), not in µ corresponds to the projection of R from µ onto the space τ .
We claim that there exists a line in PG(3h − 1, p)/µ, skew from B/µ. Suppose to the contrary that B/µ is a blocking set with respect to lines in PG(3h − 1, p)/µ ∼ = PG(2h, p).The set B/µ contains at most p h −1 p−1 Suppose that line T, R i contains a point X = R i of B/µ.Then µ, X would contain a point Y of B, such that B(Y ) is a point of B, different from P i .Now B(Y ) lies on the tangent line Q, P i since µ, X lies in Q, Pi .It follows from the fact that X = R i that B(Y ) is different from P i .Hence, T, R i only contains the point R i of B/µ which implies that the points of P1 , P2 are essential points and we recall all essential points are contained in a hyperplane, which is then necessarily P1 , P2 .But since P 1 , P 2 , P 3 are not collinear, the point R 3 , which is clearly essential, is not contained in P1 , P2 , a contradiction.This proves our claim.
So we find a line in τ , skew from B/µ.The space π = , µ is an h-space such that B(π) ∩ B = ∅.Since π meets Q in the (h − 2)-space µ, B(π) is a blocking set of vertex-type meeting the required conditions.Corollary 4.10.If B is a blocking set in PG(2, p h ), p > 5 prime, of size at most 3  2 (p h − p h−1 ), then there exists a double blocking set in PG(2, p h ) of size |B| + p h + p h−1 + 1.In particular, if p > 5, then there exist double blocking sets in PG(2, p h ) of size 2p h +2p h−1 +2 and τ 2 (PG(2, p h ) 2 p h + p h−1 + 1 .
Corollary 4.11.Consider the André plane Π D of order p h derived from PG(2, p h ), p prime, h 2. Then there exists a non-trivial minimal blocking set in Π D of size at least p h + p h−1 + 2 and at most p h + p h−1 + p h −1 p−1 + 1. Proof.Consider the blocking set B 1 = B(1, 0) of PG(2, p h ) as defined earlier.From Theorem 4.9, we obtain that there is a blocking set B 2 of size p h + p h−1 + 1 skew from B 1 .Clearly, T := B 2 ∪ D forms a blocking set of size p h + p h−1 + p h −1 p−1 + 1 in Π.Since B 1 (considered as a line of Π) and T are skew, at least one point of D is essential to the blocking set T .Moreover, it is clear that every point of B 2 is essential to T and the statement follows.
As an addition, using the same method, we construct small double blocking sets with respect to k-spaces in PG(2k, p h ).Note that a GF(p)-linear blocking set with respect to k-spaces in PG(2k, p h ), p prime, is a set B(π), where π is an hk-dimensional subspace of PG(h(2k + 1) − 1, p) and that such a linear blocking set is necessarily minimal.• p h −1 p−1 < 3 2 (p hk+h−1 + 1).This implies that, if S blocks all hk-spaces, it is a small blocking set with respect to hk-spaces in PG(h(2k + 1) − 1, p), and hence, S is an (hk + h − 1)-dimensional subspace of PG(h(2k + 1) − 1, p).This implies that B(π) is the set of all spread elements contained in an (h(k + 1) − 1)-dimensional subspace spanned by spread elements, hence, B(π) corresponds to a k-space of PG(2k, p h ).This implies that if π is an hk-dimensional subspace of PG(h(2k + 1) − 1, p) such that B(π) does not correspond to a k-space of PG(2k, p h ), i.e. if B(π) defines a non-trivial blocking set with respect to k-spaces, then there exists an hk-space π with B(π) ∩ B(π ) = ∅.

x
contains at most one point of D 0 .If P ∈ O − ∩ D 0 , then every line of type (ii) through P is tangent to B 0 , hence skew to U 0 .Similarly, if P ∈ O ∩ D 0 or P ∈ O + ∩ D 0 , then there is exactly one or zero line of type (ii) skew to U 0 through P , respectively.Let L O denote the set of lines of type (ii) that are skew to U 0 and intersect D 0 in a point of O. Then |L O | = |O ∩ D 0 | and δ(a) = |O − ∩ D 0 | + |[a] ∩ L O |, whence a∈D δ(a) = |D| • |O − ∩ D 0 | + |O ∩ D 0 |.