On small minimal blocking sets in classical generalized quadrangles

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1 On small minimal blocking sets in classical generalized quadrangles Miroslava Cimráková a Jan De Beule b Veerle Fack a, a Research Group on Combinatorial Algorithms and Algorithmic Graph Theory, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, B 9000 Ghent, Belgium b Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281 S22, B 9000 Ghent, Belgium Abstract We present exhaustive and non-exhaustive search algorithms for small minimal blocking sets in generalized quadrangles. Using these techniques, new results were obtained for some classical generalized quadrangles. Moreover, some of these computer results lead to a general construction of small minimal blocking sets. Key words: generalized quadrangle, blocking set, computer search 1 Introduction and preliminaries A finite generalized quadrangle of order (s, t), also denoted GQ(s, t), is an incidence structure S = (P, B, I), in which P and B are disjoint (non-empty) sets of objects, called points and lines respectively, and for which I is a symmetric point-line incidence relation satisfying the following axioms: (i) each point is incident with 1 + t lines (t 1) and two distinct points are incident with at most one line; (ii) each line is incident with 1 + s points (s 1) and two distinct lines are incident with at most one point; Corresponding author. addresses: Miroslava.Cajkova@UGent.be (Miroslava Cimráková), Jan.DeBeule@UGent.be (Jan De Beule), Veerle.Fack@UGent.be (Veerle Fack). URL: vfack/ (Veerle Fack). Preprint submitted to Discrete Applied Mathematics 27 June 2006

2 (iii) if x is a point and L is a line not incident with x, then there is a unique pair (y, M) P B for which x I M I y I L. Interchanging points and lines in S yields a generalized quadrangle S D of order (t, s), which is called the dual of S. If s = t, S is said to have order s. A generalized quadrangle of order (s, 1) is called a grid and a generalized quadrangle of order (1, t) is called a dual grid. A generalized quadrangle with s, t > 1 is called thick. The thick classical finite generalized quadrangles are respectively the nonsingular 4-dimensional parabolic quadrics Q(4, q) of order q, the non-singular 5-dimensional elliptic quadrics Q (5, q) of order (q, q 2 ), the non-singular 3- and 4-dimensional Hermitian varieties H(3, q 2 ) and H(4, q 2 ) of respective orders (q 2, q) and (q 2, q 3 ), and the non-singular finite generalized quadrangle W (q) of order q consisting of the points of P G(3, q) and of the totally isotropic lines of a symplectic polarity η. Let S = (P, B, I) be a generalized quadrangle of order (s, t). For x P denote x = {y P y x} and note that x x. If A is an arbitrary subset of P, the perp of A is denoted by A and is defined as A = {x x A}. An ovoid of a generalized quadrangle S is a set O of points of S such that each line of S is incident with exactly one point of O, necessarily, O = st + 1. A partial ovoid is a set O of points of S such that each line of S is incident with at most one point of O. Dually, a spread of S is a set R of lines of S such that each point of S is incident with a unique line of R, necessarily, R = st+1. A partial spread of S is a set R of lines of S such that each point of S is incident with at most one line of R. A partial ovoid (or spread) is called maximal or complete if it is not contained in a larger partial ovoid (or spread). A blocking set of S is a set B of points of S such that each line of S is incident with at least one point of B. Necessarily B st + 1 for a GQ(s, t), with equality if and only if B is an ovoid. A cover of S is a set C of lines of S such that each point of S is incident with at least one line of C. Necessarily C st + 1 for a GQ(s, t), with equality if and only if C is a spread. It is clear that a blocking set in S is a cover in the dual S D. A blocking set B is called minimal if B \ p is not a blocking set for any point p B, while a cover C is called minimal if C \ L is not a cover for any line L C. Let B be a blocking set of S of size st r. We call r the excess of the blocking set. A line of S is called a multiple line if it contains at least two points of B. The excess of a line is the number of points of B it contains, minus one. The weight of a point of S with respect to B is the minimum of the excesses of the lines of S passing through this point. Dually, for a cover the concepts of excess, multiple point, excess of a point and weight of a line are defined in a similar way. 2

3 A non-singular quadric Q(2, q) is called a conic and a non-singular Hermitian variety H(2, q 2 ) is called a Hermitian curve. When q is even, every non-singular parabolic quadric Q(4, q) has a nucleus, a point on which every line has exactly one point in common with Q(4, q). A sum of lines L of PG(n, q) is a collection of lines of PG(n, q), where each line is assigned a positive integer, called its weight. Furthermore, the weight of a point with respect to L is the sum of the weights of the lines of L through this point. A pencil of a GQ(s, t) is the set of t + 1 lines on a point of the generalized quadrangle. A blocking set B of the projective plane PG(2, q) is a set of points in PG(2, q) such that each line is incident with at least one point of B. A blocking set containing a line is called trivial, otherwise it is called non-trivial. It is known [1] that the smallest non-trivial blocking sets in PG(2, q), with q an odd prime, have size 3(q + 1)/2. In this paper we concentrate on small blocking sets in the classical generalized quadrangles. It is easy to see that a minimal blocking set of size st + s is obtained by taking the set of points of a pencil through an arbitrary point x of S, minus x itself. Our aim is to find minimal blocking sets of S of size smaller than st + s, which neither are ovoids (in case S contains ovoids). The paper is organized as follows. In Sections 2 and 3 we focus on the generalized quadrangles W (q) and Q(4, q). We collect the known results on the smallest minimal blocking sets, describe the exhaustive and non-exhaustive algorithms which we use in the computer search, and present our new results on small minimal blocking sets. In Section 4 we summarize the known results on small minimal blocking sets in other classical generalized quadrangles, and add some new results on spectra of sizes for which small minimal blocking sets of H(3, q 2 ) exist, obtained by our non-exhaustive searches. 2 Blocking sets of W (q) and of Q(4, q), q even It is known [11] that Q(4, q) has ovoids for every value of q and that W (q) only has ovoids for even values of q. We recall that Q(4, q) is isomorphic to W (q), for q even. In [9] Eisfeld, Storme, Szőnyi and Sziklai proved the following results for blocking sets of W (q) and Q(4, q). Theorem 1 (Eisfeld et al. [9]) Let B be a blocking set of the quadric Q(4, q), q even. If q 32 and B q q, then B contains an ovoid of Q(4, q). 3

4 q # points Earlier results [9] Sizes found , ,271 Table 1 Sizes of minimal blocking sets found in Q(4, q), for small values of q, q even. All results are obtained by heuristic search. If q = 4, 8, 16 and B q q+4, then B contains an ovoid of Q(4, q). 6 Theorem 2 (Eisfeld et al. [9]) Let B be a blocking set of W (q), q odd, then B > q (q 1)/3. Moreover the authors in [9] give a construction of a minimal blocking set of W (q) of size q (q 2). For q = 3, 4, 5 they show that this is indeed the size of the smallest minimal blocking sets in W (q). In the rest of this section, we focus on small minimal blocking sets in Q(4, q), for even q. 2.1 Computer search The following results were obtained by a computer search, implementing a greedy heuristic as follows. A minimal blocking set is built step by step, starting from the set of all points, and removing points one by one from this set, until it is a minimal blocking set. During this process a set is maintained of points which are still allowed choices for removing. Choosing a point that leaves the largest number of points in this allowed set, tends to build small minimal blocking sets. Using a minimal blocking set obtained by this approach, a simple restart strategy adds some of the points and again removes points until the blocking set is minimal. Both the adding and the removing can be done either randomly or following the above greedy heuristic. For each q considered, Table 1 lists the sizes smaller than q 2 + q for which our program found a minimal blocking set. For comparison reasons we also list the sizes of the known small minimal blocking set obtained in [9]. Our computer searches find a minimal blocking set of size q 2 + q 1 for each q even considered. We also observed the existence of a minimal blocking set of size q 2 + q 3 for q = 8, 16 (for q = 4 this value corresponds to the size of an ovoid). We now describe the structure of these blocking sets, the investigation of which is also done by computer. 4

5 q 1 points p 1 L p 2 q/2 points Fig. 1. Blocking set of Q(4, q) of size q 2 + q Blocking set B of size q 2 + q 1 for q = 2, 4, 8, 16 There is a line L and two points p 1 and p 2 on this line not contained in the blocking set B. All q 1 points from L, except for the points p 1 and p 2, belong to B. Through the points p 1 and p 2 there are 2q lines (excluding the line L) and on each line there are q/2 points of B. These q 1 + (2q)(q/2) points form a minimal blocking set of size q 2 + q 1. This is illustrated in Figure 1. We note that for q = 2 we get an ovoid. In [9] a construction for a minimal blocking sets of size q 2 + q 1 in W (q) is given. Here we describe another construction for minimal blocking sets of size q 2 + q 1, which corresponds to our computer results. Let B be the blocking set of size q 2 + q described earlier, i.e., a pencil on a point p 1, where all points of the lines of the pencil belong to B, except p 1. Let L be a line of this pencil and let p 2 be a point on L different from p 1. We consider a conic C 1, containing p 2, such that its nucleus is the nucleus of Q(4, q). This conic intersects the pencil through the point p 1 in q +1 mutually non-collinear points c 1 1, c 1 2,... c 1 q+1, where c 1 1 = p 2. The conic C1 intersects the pencil on p 2 in q + 1 non-collinear points c 2 1, c2 2,... c2 q+1, where c2 1 = p 1. Now, we remove the q+1 points c 1 1, c1 2,... c1 q+1 from the blocking set and add the q points c2 2,... c2 q+1. The obtained set of size q 2 + q 1 is indeed a blocking set, since we substitute the points of C 1 by the points of C1. Checking the minimality is easy, since through each point of the new set there is at least one line containing only one point of the set. Notice that the points of conics can be interchanged q 1 times. This construction is illustrated in Figure Blocking set B of size q 2 + q 3 for q = 4, 8, 16 There is a line L and four points p 0, p 1, p 2 and p 3 on this line not contained in the blocking set B. All q 3 points from L, except for the points p i, i = 0, 1, 2, 3, belong to B. Through the points p i, i = 0, 1, 2, 3 there are 4q lines (excluding the line L) and on each line there are q/4 points of B. These q 3 + (4q)(q/4) 5

6 C 1 p 1 L p 2 C 1 Fig. 2. Construction of blocking sets of size q 2 + q 1 p 1 p 2 p 3 p 4 q 3 points L q/4 points q lines Fig. 3. Blocking set of Q(4, q) of size q 2 + q 3 points form a minimal blocking set of size q 2 + q 3. This is illustrated in Figure 3. We note that for q = 4 we get an ovoid. 2.4 Blocking set B of size q 2 + q (2 k 1) for q 2 k, k > 0 In the previous cases a minimal blocking set of the smallest possible parameter was an ovoid. If q = 8, a minimal blocking set of size q 2 + q 7 is an ovoid. However, our heuristic searches did not find a minimal blocking set of that size for q > 8. Now the following question naturally arises. Is it possible to construct a blocking set of size q 2 + q 7 for q > 8 using the previous ideas? Even more generally, are there any blocking sets of size q 2 + q (2 k 1), for q 2 k, k > 0 with the following structure? There is a line L and 2 k points p i, 0 i 2 k 1, on this line not contained in the blocking set B. All q k points from L, except for the points p i, belong to B. Through all points p i there are 2 k q lines (excluding the line L) and on each line there are q/2 k points of B. Is it possible that these (q k ) + (2 k q)(q/2 k ) = q 2 + q (2 k 1) points form a minimal blocking set? 6

7 Finally we note that we are currently working on a geometrical construction for minimal blocking sets of size smaller than q 2 + q 1. 3 Blocking sets in Q(4, q), q odd When q is even, Theorem 1 gives a lower bound for minimal blocking sets of Q(4, q). However, for q odd, no similar result is known. Even the question whether minimal blocking sets of cardinality q exist, has not yet been answered in general. Recently De Beule and Metsch [4] solved this problem for q an odd prime and proved that, for such q, Q(4, q) has no minimal blocking set of size q Here we perform a computer search for small minimal blocking sets i.e. minimal blocking sets of size smaller than q 2 + q, of Q(4, q), q odd. Our algorithms rely on theoretical properties of the structure of multiple lines. Before describing our algorithm and presenting our results, we summarize these properties. 3.1 Structure of multiple lines First, we define a regulus and its opposite regulus in Q(4, q). Consider a grid Q + (3, q) Q(4, q). A regulus of Q(4, q) is a set of q + 1 pairwise disjoint lines in Q + (3, q). The set of the remaining q + 1 pairwise disjoint lines in Q + (3, q) is called the opposite regulus. The following theorem describes the structure of multiple points of a cover in a generalized quadrangle. Theorem 3 (Eisfeld et al. [9]) Let C be a cover of a classical generalized quadrangle S of order (q, t) embedded in PG(n, q). Let C = qt r, with q + r smaller than the cardinality of the smallest non-trivial blocking sets in PG(2, q). Then the multiple points of C form a sum of lines of PG(n, q), where the weight of a line in this sum is equal to the weight of this line with respect to the cover, and with the sum of the weights of the lines equal to r. Consider a cover of S = W (q), satisfying the conditions of Theorem 3. This cover dualizes to a blocking set B of S = Q(4, q) (see [11]). Note that a line of W (q) dualizes to a pencil in Q(4, q), while a line of PG(3, q) which is not a line of W (q) dualizes to a regulus. Hence the sum of multiple lines of S = W (q) can be described by pencils and reguli in S = Q(4, q). In [7], De Beule and Storme prove the following lemma. 7

8 m m (a) Case I: r = 1, q odd, one pencil (b) Case II.a: r = 2, q > 3 odd, one pencil Fig. 4. One pencil Lemma 4 (De Beule and Storme [7]) Suppose that C is a cover of S = W (q), of size q r, with q + r smaller than the cardinality of the smallest non-trivial blocking sets in PG(2, q), such that the multiple points of C are a sum A of lines of PG(3, q). If L is a line of A, L not a line of W (q), then L A, with the symplectic polarity corresponding to W (q). Let B be a blocking set of Q(4, q), of size q r, with q + r smaller than the cardinality of the smallest non-trivial blocking sets in PG(2, q). We now describe in more detail the different possibilities for the structure of the multiple lines of B, for the special cases r = 1 and r = 2. This is done by applying Theorem 3 to the cover C of W (q) and dualizing this result; here we only give the dual information. Case I: r = 1, for all q odd There is exactly one point m Q(4, q) \ B, such that all q + 1 lines on m are the multiple lines of B and they meet B in exactly two points. This case is illustrated for q = 3 in Figure 4(a). Case II: r = 2, for all q > 3 odd a. There is exactly one point m Q(4, q) \ B, such that all q + 1 lines on m are the multiple lines of B and they meet B in exactly three points (see Figure 4(b)). b. There are two collinear points m, n Q(4, q) \ B, and the q + 1 lines on m resp. n meet B in exactly two points, except for the line on m and n which meets B in exactly three points (see Figure 5). c. There are two non-collinear points m, n Q(4, q) \ B, such that the q + 1 lines on m resp. n meet B in exactly two points (see Figure 6). d. There is a regulus R M and its opposite regulus R N, such that every line of R M and every line of R N meets B in exactly two points (see Figure 7). 8

9 m n Fig. 5. Case II.b: r = 2, q > 3 odd, two collinear pencils m n Fig. 6. Case II.c: r = 2, q > 3 odd, two non-collinear pencils Fig. 7. Case II.d: r = 2, q > 3 odd, regulus and opposite regulus Using the above information about the structure of multiple lines, the following results for q = 3, 5, 7 were obtained previously: De Winter [8] constructed a minimal blocking set B of size q of Q(4, 5), which contains exactly 12 points on a hyperbolic quadric. This corresponds to Case II.d (r = 2). With the aid of a computer, De Beule and Storme [7] found that, if B is a minimal blocking set of Q(4, 3) different from an ovoid, then B > 11. De Beule, Hoogewijs and Storme [6] found that, if B is a minimal blocking set of Q(4, q), q = 5, 7 different from an ovoid of Q(4, q), then B > q De Beule, Hoogewijs and Storme [6] also performed a computer search to exclude the existence of a minimal blocking set of Q(4, 7) of size q satisfying the special property that there is one point of Q(4, 7) with q + 1 lines on it being blocked by exactly three points of B. This corresponds to Case II.a. 9

10 Here we tackle the specific questions: Are there other minimal blocking sets of size q in Q(4, 5)? Is there any blocking set of size q in Q(4, 7)? and the more general question: Is there any minimal blocking set of size at least q and smaller than q 2 + q in Q(4, q), q odd? 3.2 Exhaustive search algorithm and results In order to search for minimal blocking sets we use a backtracking algorithm, which tries in every recursion step to extend a partial blocking set (which is not a blocking set yet) by adding the points of a set A of allowed remaining points in a systematic way. When reaching a point where the set A is empty, a new minimal blocking set has been found. In what follows, we explain how to determine the allowed set and how the search can be pruned by forcing points. From now on, we consider only parameter values q and r for which the structure of the multiple lines was described above. The first step in the algorithm is to determine the excess of the lines. Since a generalized quadrangle is transitive on the pairs of points and on the pairs of lines, we can fix the structure, i.e., the pencils or reguli, in all considered cases. In particular, we determine the lines, which are multiple with the considered excess. Let midpoint be a point, where the lines of the pencil meet. In the cases where pencils appear (I, II.a, II.b and II.c), we can also directly remove the midpoint(s) of pencil(s) from the allowed set. In each recursive step we determine the new allowed set. Let p be a point currently added to B in this recursive step. If adding the point p gives rise to a line, which attains the expected excess, we can remove all still allowed points on this line from the allowed set. The information about the structure of multiple lines can be used to improve the pruning as follows. Let e L be the excess of a line L and let b L be the number of points of the current partial blocking set on the line L. Let A L be the set of still allowed points on the line L. Consider a step in the recursive process where the current set gives rise to a line for which e L + 1 = A L + b L holds. If the points of A L are not added to the current set, then the resulting partial blocking set can never be extended to a minimal blocking set, so we can prune these possibilities and force the points to be added to the current set. Figure 8 illustrates this idea. Suppose that after adding a point p in the re- 10

11 r r r L f p L f p L f 2 f 1 p (a) Forcing a point on a nonmultiple line (b) Forcing a point on a multiple line (c) Forcing two points on a multiple line Fig. 8. Illustrating the idea of forcing points # points B # B II.a II.b II.c II.d Q(4, 5) Q(4, 7) Table 2 Classification of minimal blocking sets of size q 2 +3 in Q(4, 5) and Q(4, 7). cursive step, a point r was removed from the current set of allowed points. In Figure 8(a), there is only one point f on the non-multiple line L left which can be added to the partial blocking set. Similarly, let L be a multiple line with excess 1, as shown in Figures 8(b) and 8(c). Since there is only one point (resp. two points) left on the multiple line L which can be added to the partial blocking set, we can forcedly add these points to the set, thus pruning the possible extensions that do not contain these points. By exhaustive search we determined the classification of minimal blocking sets of size q for Q(4, 5) and Q(4, 7). The results are shown in Table 2. For both values of q and for each of the cases II.a, II.b, II.c and II.d, we classified all minimal blocking sets of size q We summarize these results in the following lemma. Lemma 5 There is a unique minimal blocking set of size q = 28 in Q(4, 5) and there is no minimal blocking set of size q in Q(4, 7). 11

12 # multiple lines r # grids Q(4, 5) Q(4, 7) Q(4, 11) Q(4, q) (q + 1)(q 3) q 3 (q 3)/2 if q {5, 7, 11} Table 3 Structure of the blocking sets of Q(4, q), q = 5, 7, 11 of size q 2 + q Minimal blocking sets of size q 2 + q 2 Using the same greedy heuristic as described in Section 2 we searched for minimal blocking sets in Q(4, q) for larger values of q and r. Typically we were interested in any minimal blocking set (different from an ovoid) with size smaller than q 2 + q. For q = 5, 7, 9, 11, we obtain minimal blocking sets of size q 2 + q 2, i.e., with excess r = q 3. We investigated (also by computer) the structure of the obtained blocking sets, focusing on the multiple lines and the way in which they are structured. The case q = 9 turns out to be different and will be treated separately. For q = 5, 7, 11 we observed that, for the blocking sets found, all multiple lines have excess one, which means that the multiple lines contain two points of the blocking set. For q = 5, De Winter [8] already noticed that all multiple lines form a grid. We observed that also for q = 7, 11 grids are formed by the multiple lines, and even that more than one grid appears, as summarized in Table 3. For these cases the number of multiple lines is 2(q + 1)(q 3)/2 and the number of grids formed by multiple lines is (q 3)/2. Moreover, for q = 7, 11 there are q + 1 points of B common to all grids. Finally we observed that it is possible to obtain a maximal partial ovoid of size q 2 1 by removing these q + 1 common points and adding two points of the perp of this set. In Q(4, 5) are there 4 possibilities of obtaining such a maximal partial ovoid by removing q + 1 points. For q = 9 we observed that, for the maximal blocking sets found, all multiple lines have excess three. The blocking set found has 20 multiple lines and we checked that they form a grid. Previously we found by exhaustive search [2] that no maximal partial ovoids of size q 2 1 exist in Q(4, 9); this result was recently generalized by De Beule and Gács [3] for all q = p h, p an odd prime, h > 1. 12

13 Finally we also want to mention our effort, which was not successful yet, to try and find a blocking set of size q 2 + q 2 for larger q, especially for q = 13. If we suppose that this blocking set would be of the same structure, it should contain 5 grids with 14 points in the intersection. So far we have found no blocking set of the given size by our (not exhaustive!) computer search. We expect that no blocking set of Q(4, 13) of size q 2 + q 2 exists. 4 Blocking sets in H(d, q 2 ), q = 3, 4 and in Q (5, q) As we described in the introduction, one can construct a minimal blocking set of size st + s of any generalized quadrangle of order (s, t) by taking the set of points of a pencil through an arbitrary point x, minus the point x itself. The following theorems show that these construction yields the (unique) smallest minimal blocking set of size st + s for the generalized quadrangles Q (5, q) and H(4, q 2 ). Theorem 6 (Metsch [10]) Suppose that B is a minimal set of points of Q (5, q) with the property that B meets every line of Q (5, q). Then B q 3 + q. If B = q 3 + q, then B = p \ {p} for some point p Q (5, q). Recently, a similar theorem was proved for H(4, q 2 ). Theorem 7 (De Beule and Metsch [5]) Suppose that B is a minimal set of points of H(4, q 2 ) with the property that B meets every line of H(4, q 2 ). Then B q 5 + q 2. If B = q 5 + q 2, then B = p \ {p} for some point p H(4, q 2 ). To our knowledge there is nothing known about small minimal blocking sets in H(3, q 2 ). In Table 4 we present some new results found by our heuristic searches. We list the sizes smaller than q 3 + q 2 for which our program found minimal blocking sets. For comparison reasons we also list the size q of an ovoid and the size q 3 + q 2 of the blocking set described in the introduction. For q = 2 we classified all minimal blocking sets of small size. There are two minimal blocking sets of size 10, twelve minimal blocking sets of size 11 and thirty minimal blocking sets of size 12. Finally, we can construct a spectrum of minimal blocking sets for each value from q 3 + r, r = 1,..., q 2. Consider a Hermitian curve H contained in H(3, q 2 ). This curve is contained in a plane, denoted by π. Consider any point p H. Consider the q lines of π on p. Exactly q 2 of them intersect H in a Hermitian variety H on 13

14 q # points q q 3 + q 2 Sizes found , Table 4 Sizes for minimal blocking sets of H(3, q 2 ) smaller than q 3 + q 2, for small values of q, obtained by heuristic search. The notation a..b means that for all values in the interval [a, b] a minimal blocking set of that size has been found. a line (this is a H(1, q 2 )). It is clear that H is again a Hermitian variety on a line. Consider r different lines L i on p and their corresponding Hermitian varieties H i. Consider the r Hermitian varieties Hi, they are mutually skew and lie all in p. It is clear that the set H \ ( r i=1 H i) r i=1 H i is a minimal blocking set of size q 3 + r of H(3, q 2 ). Since we have exactly q 2 suitable lines on p, we can construct a spectrum of minimal blocking sets of size q 3 + r, r = 1,..., q 2. References [1] A. Blokhuis, On the size of blocking sets in PG(2, p). Combinatorica 14, , [2] M. Cimráková, S. De Winter, V. Fack, and L. Storme, On the smallest maximal partial ovoids and spreads of the generalized quadrangles W (q) and Q(4, q). European J. Combin. (2006), to appear. [3] J. De Beule and A. Gács, Complete (q 2 1)-arcs of Q(4, q), q = p h, p odd prime, h > 1, do not exist. Finite Fields Appl. (2005), submitted. [4] J. De Beule and K. Metsch, Minimal blocking sets of size q of Q(4, q), q an odd prime, do not exist. Finite Fields Appl., 11, , [5] J. De Beule and K. Metsch, The smallest point sets that meet all generators of H(2n, q). Discrete Math., 294, 75-81, [6] J. De Beule, A. Hoogewijs and L. Storme, On the size of minimal blocking sets of Q(4, q), for q = 5, 7. ACM SIGSAM Bulletin, 38, 67-84, [7] J. De Beule and L. Storme, On the smallest minimal blocking sets of Q(2n, q), for q an odd prime. Discrete Math., 294, , [8] S. De Winter, Private communication, [9] J. Eisfeld, L. Storme, T. Szőnyi and P. Sziklai, Covers and blocking sets of classical generalized quadrangles. Discrete Math., 238, 35-51,

15 [10] K. Metsch. A Bose-Burton theorem for elliptic polar spaces. Des. Codes Cryptogr., 17(1-3): , [11] S.E. Payne and J.A. Thas, Finite Generalized Quadrangles. Pitman Res. Notes Math. Ser Longman,