On small minimal blocking sets in classical

generalized quadrangles

Miroslava Cimrakova a Jan De Beule b Veerle Fack a,∗aResearch Group on Combinatorial Algorithms and Algorithmic Graph Theory,

Department of Applied Mathematics and Computer Science, Ghent University,

Krijgslaan 281–S9, B–9000 Ghent, Belgium

bDepartment 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 minimalblocking sets in generalized quadrangles. Using these techniques, new results wereobtained for some classical generalized quadrangles. Moreover, some of these com-puter 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 inci-dence structure S = (P, B, I), in which P and B are disjoint (non-empty) setsof objects, called points and lines respectively, and for which I is a symmetricpoint-line incidence relation satisfying the following axioms:

(i) each point is incident with 1 + t lines (t ≥ 1) and two distinct points areincident with at most one line;

(ii) each line is incident with 1 + s points (s ≥ 1) and two distinct lines areincident with at most one point;

∗ Corresponding author.Email addresses: Miroslava.Cajkova@UGent.be (Miroslava Cimrakova),

Jan.DeBeule@UGent.be (Jan De Beule), Veerle.Fack@UGent.be (Veerle Fack).URL: http://caagt.ugent.be/∼vfack/ (Veerle Fack).

Preprint submitted to Discrete Applied Mathematics 27 June 2006

(iii) if x is a point and L is a line not incident with x, then there is a uniquepair (y, M) ∈ P × B for which x I M I y I L .

Interchanging points and lines in S yields a generalized quadrangle SD oforder (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 generalizedquadrangle of order (1, t) is called a dual grid. A generalized quadrangle withs, t > 1 is called thick.

The thick classical finite generalized quadrangles are respectively the non-singular 4-dimensional parabolic quadrics Q(4, q) of order q, the non-singular5-dimensional elliptic quadrics Q−(5, q) of order (q, q2), the non-singular 3- and4-dimensional Hermitian varieties H(3, q2) and H(4, q2) of respective orders(q2, q) and (q2, q3), and the non-singular finite generalized quadrangle W (q) oforder q consisting of the points of PG(3, q) and of the totally isotropic linesof a symplectic polarity η.

Let S = (P, B, I) be a generalized quadrangle of order (s, t). For x ∈ P denotex⊥ = {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 eachline of S is incident with exactly one point of O, necessarily, |O| = st + 1. Apartial ovoid is a set O′ of points of S such that each line of S is incident withat most one point of O′. Dually, a spread of S is a set R of lines of S such thateach point of S is incident with a unique line of R, necessarily, |R| = st+1. Apartial spread of S is a set R′ of lines of S such that each point of S is incidentwith at most one line of R′. A partial ovoid (or spread) is called maximal orcomplete 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 incidentwith at least one point of B. Necessarily |B| ≥ st + 1 for a GQ(s, t), withequality if and only if B is an ovoid. A cover of S is a set C of lines of Ssuch 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 clearthat a blocking set in S is a cover in the dual SD. A blocking set B is calledminimal if B \ p is not a blocking set for any point p ∈ B, while a cover C iscalled minimal if C \ L is not a cover for any line L ∈ C.

Let B be a blocking set of S of size st + 1 + r. We call r the excess of theblocking set. A line of S is called a multiple line if it contains at least twopoints 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 ofthe excesses of the lines of S passing through this point. Dually, for a coverthe concepts of excess, multiple point, excess of a point and weight of a line

are defined in a similar way.

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A non-singular quadric Q(2, q) is called a conic and a non-singular Hermitianvariety H(2, q2) is called a Hermitian curve.

When q is even, every non-singular parabolic quadric Q(4, q) has a nucleus, apoint 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 eachline 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 throughthis point. A pencil of a GQ(s, t) is the set of t + 1 lines on a point of thegeneralized 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 con-taining 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 generalizedquadrangles. It is easy to see that a minimal blocking set of size st + s isobtained by taking the set of points of a pencil through an arbitrary point xof S, minus x itself. Our aim is to find minimal blocking sets of S of sizesmaller 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 gen-eralized quadrangles W (q) and Q(4, q). We collect the known results on thesmallest minimal blocking sets, describe the exhaustive and non-exhaustivealgorithms which we use in the computer search, and present our new resultson small minimal blocking sets. In Section 4 we summarize the known resultson small minimal blocking sets in other classical generalized quadrangles, andadd some new results on spectra of sizes for which small minimal blocking setsof H(3, q2) 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 toW (q), for q even.

In [9] Eisfeld, Storme, Szonyi and Sziklai proved the following results for block-ing 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| ≤ q2 + 1 +

√q, then B contains an ovoid of Q(4, q).

3

q # points Earlier results [9] Sizes found

4 85 19 19

8 585 71 69,71

16 4369 271 269,271

Table 1Sizes of minimal blocking sets found in Q(4, q), for small val-ues of q, q even. All results are obtained by heuristic search.

If q = 4, 8, 16 and |B| ≤ q2 + 1 + q+4

6, then B contains an ovoid of Q(4, q).

Theorem 2 (Eisfeld et al. [9]) Let B be a blocking set of W (q), q odd, then

|B| > q2 + 1 + (q − 1)/3.

Moreover the authors in [9] give a construction of a minimal blocking set ofW (q) of size q2 + 1 + (q − 2). For q = 3, 4, 5 they show that this is indeed thesize 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 agreedy heuristic as follows. A minimal blocking set is built step by step, start-ing 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 maintainedof points which are still allowed choices for removing. Choosing a point thatleaves the largest number of points in this allowed set, tends to build smallminimal blocking sets. Using a minimal blocking set obtained by this approach,a simple restart strategy adds some of the points and again removes pointsuntil the blocking set is minimal. Both the adding and the removing can bedone either randomly or following the above greedy heuristic.

For each q considered, Table 1 lists the sizes smaller than q2 + q for whichour program found a minimal blocking set. For comparison reasons we alsolist the sizes of the known small minimal blocking set obtained in [9]. Ourcomputer searches find a minimal blocking set of size q2 + q − 1 for each qeven considered. We also observed the existence of a minimal blocking set ofsize q2 + q − 3 for q = 8, 16 (for q = 4 this value corresponds to the size of anovoid). We now describe the structure of these blocking sets, the investigationof which is also done by computer.

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p1 p2Lq − 1 points

q/2 points

Fig. 1. Blocking set of Q(4, q) of size q2 + q − 1

2.2 Blocking set B of size q2 + q − 1 for q = 2, 4, 8, 16

There is a line L and two points p1 and p2 on this line not contained in theblocking set B. All q−1 points from L, except for the points p1 and p2, belongto B. Through the points p1 and p2 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 forma minimal blocking set of size q2 + q − 1. This is illustrated in Figure 1. Wenote that for q = 2 we get an ovoid.

In [9] a construction for a minimal blocking sets of size q2 + q − 1 in W (q) isgiven. Here we describe another construction for minimal blocking sets of sizeq2 + q − 1, which corresponds to our computer results.

Let B be the blocking set of size q2 + q described earlier, i.e., a pencil on apoint p1, where all points of the lines of the pencil belong to B, except p1. Let Lbe a line of this pencil and let p2 be a point on L different from p1. We considera conic C1, containing p2, such that its nucleus is the nucleus of Q(4, q). Thisconic intersects the pencil through the point p1 in q+1 mutually non-collinearpoints c1

1, c12, . . . c

1q+1, where c1

1 = p2. The conic C⊥

1 intersects the pencil on p2

in q +1 non-collinear points c21, c

22, . . . c

2q+1, where c2

1 = p1. Now, we remove theq+1 points c1

1, c12, . . . c

1q+1 from the blocking set and add the q points c2

2, . . . c2q+1.

The obtained set of size q2 + q−1 is indeed a blocking set, since we substitutethe points of C1 by the points of C⊥

1 . Checking the minimality is easy, sincethrough each point of the new set there is at least one line containing onlyone point of the set. Notice that the points of conics can be interchanged q−1times. This construction is illustrated in Figure 2.

2.3 Blocking set B of size q2 + q − 3 for q = 4, 8, 16

There is a line L and four points p0, p1, p2 and p3 on this line not contained inthe blocking set B. All q−3 points from L, except for the points pi, i = 0, 1, 2, 3,belong to B. Through the points pi, i = 0, 1, 2, 3 there are 4q lines (excludingthe line L) and on each line there are q/4 points of B. These q− 3+ (4q)(q/4)

5

Lp1

C1

C⊥

1

p2

Fig. 2. Construction of blocking sets of size q2 + q − 1

q/4 points

q − 3 points

L

p1 p2 p3 p4

q lines

Fig. 3. Blocking set of Q(4, q) of size q2 + q − 3

points form a minimal blocking set of size q2 + q − 3. This is illustrated inFigure 3. We note that for q = 4 we get an ovoid.

2.4 Blocking set B of size q2 + q − (2k − 1) for q ≥ 2k, k > 0

In the previous cases a minimal blocking set of the smallest possible parameterwas an ovoid. If q = 8, a minimal blocking set of size q2 + q − 7 is an ovoid.However, our heuristic searches did not find a minimal blocking set of thatsize for q > 8. Now the following question naturally arises. Is it possible toconstruct a blocking set of size q2 + q − 7 for q > 8 using the previous ideas?Even more generally, are there any blocking sets of size q2 + q − (2k − 1), forq ≥ 2k, k > 0 with the following structure?

There is a line L and 2k points pi, 0 ≤ i ≤ 2k − 1, on this line not containedin the blocking set B. All q + 1 − 2k points from L, except for the points pi,belong to B. Through all points pi there are 2kq lines (excluding the line L)and on each line there are q/2k points of B. Is it possible that these (q + 1 −2k) + (2kq)(q/2k) = q2 + q − (2k − 1) points form a minimal blocking set?

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Finally we note that we are currently working on a geometrical constructionfor minimal blocking sets of size smaller than q2 + q − 1.

3 Blocking sets in Q(4, q), q odd

When q is even, Theorem 1 gives a lower bound for minimal blocking setsof Q(4, q). However, for q odd, no similar result is known. Even the questionwhether minimal blocking sets of cardinality q2 + 2 exist, has not yet beenanswered in general. Recently De Beule and Metsch [4] solved this problemfor q an odd prime and proved that, for such q, Q(4, q) has no minimal blockingset of size q2 + 2.

Here we perform a computer search for small minimal blocking sets i.e. mini-mal blocking sets of size smaller than q2 + q, of Q(4, q), q odd. Our algorithmsrely on theoretical properties of the structure of multiple lines. Before describ-ing 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 gridQ+(3, q) ⊂ Q(4, q). A regulus of Q(4, q) is a set of q + 1 pairwise disjoint linesin 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 ina 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 + 1 + 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. Thiscover dualizes to a blocking set B of S ′ = Q(4, q) (see [11]). Note that a line ofW (q) dualizes to a pencil in Q(4, q), while a line of PG(3, q) which is not a lineof 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.

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m

(a) Case I: r = 1, q odd,one pencil

m

(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 q2 +1+ 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 q2 + 1 + r, with q + r smallerthan the cardinality of the smallest non-trivial blocking sets in PG(2, q). Wenow describe in more detail the different possibilities for the structure of themultiple lines of B, for the special cases r = 1 and r = 2. This is done byapplying Theorem 3 to the cover C of W (q) and dualizing this result; here weonly 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 mare the multiple lines of B and they meet B in exactly two points. This caseis 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 mare the multiple lines of B and they meet B in exactly three points (seeFigure 4(b)).

b. There are two collinear points m, n ∈ Q(4, q) \ B, and the q + 1 lineson m resp. n meet B in exactly two points, except for the line on m andn 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 +1lines on m resp. n meet B in exactly two points (see Figure 6).

d. There is a regulus RM and its opposite regulus RN , such that every lineof RM and every line of RN meets B in exactly two points (see Figure 7).

8

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 oppositeregulus

Using the above information about the structure of multiple lines, the followingresults for q = 3, 5, 7 were obtained previously:

• De Winter [8] constructed a minimal blocking set B of size q2 +3 of Q(4, 5),which contains exactly 12 points on a hyperbolic quadric. This correspondsto Case II.d (r = 2).

• With the aid of a computer, De Beule and Storme [7] found that, if B is aminimal 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 blockingset of Q(4, q), q = 5, 7 different from an ovoid of Q(4, q), then |B| > q2 + 2.

• De Beule, Hoogewijs and Storme [6] also performed a computer search toexclude the existence of a minimal blocking set of Q(4, 7) of size q2 + 3satisfying the special property that there is one point of Q(4, 7) with q + 1lines on it being blocked by exactly three points of B. This corresponds toCase II.a.

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Here we tackle the specific questions:

• Are there other minimal blocking sets of size q2 + 3 in Q(4, 5)?• Is there any blocking set of size q2 + 3 in Q(4, 7)?

and the more general question:

• Is there any minimal blocking set of size at least q2 + 2 and smaller thanq2 + 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” (whichis not a blocking set yet) by adding the points of a set A of allowed remainingpoints 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 howto determine the allowed set and how the search can be pruned by forcingpoints. From now on, we consider only parameter values q and r for which thestructure of the multiple lines was described above.

The first step in the algorithm is to determine the excess of the lines. Since ageneralized quadrangle is transitive on the pairs of points and on the pairs oflines, 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 consideredexcess. Let midpoint be a point, where the lines of the pencil meet. In thecases where pencils appear (I, II.a, II.b and II.c), we can also directly removethe midpoint(s) of pencil(s) from the allowed set.

In each recursive step we determine the new allowed set. Let p be a pointcurrently added to B in this recursive step. If adding the point p gives riseto a line, which attains the expected excess, we can remove all still allowedpoints on this line from the allowed set. The information about the structureof multiple lines can be used to improve the pruning as follows.

Let eL be the excess of a line L and let bL be the number of points of the current“partial blocking set” on the line L. Let AL be the set of still allowed points onthe line L. Consider a step in the recursive process where the current set givesrise to a line for which eL + 1 = |AL| + bL holds. If the points of AL are notadded to the current set, then the resulting “partial blocking set” can neverbe extended to a minimal blocking set, so we can prune these possibilities andforce 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

r

L

f p

(a) Forcing apoint on a non-multiple line

p

r

L

f

(b) Forcinga point on amultiple line

p

r

L

f2

f1

(c) Forcing twopoints on a multi-ple line

Fig. 8. Illustrating the idea of forcing points

# points |B| # BII.a II.b II.c II.d

Q(4, 5) 156 28 0 0 0 1

Q(4, 7) 400 52 0 0 0 0

Table 2Classification of minimal blocking sets of size q2+3 in Q(4, 5)and Q(4, 7).

cursive step, a point r was removed from the current set of allowed points. InFigure 8(a), there is only one point f on the non-multiple line L left whichcan be added to the “partial blocking set”. Similarly, let L be a multiple linewith 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 “par-tial blocking set”, we can forcedly add these points to the set, thus pruningthe possible extensions that do not contain these points.

By exhaustive search we determined the classification of minimal blocking setsof size q2 + 3 for Q(4, 5) and Q(4, 7). The results are shown in Table 2. Forboth values of q and for each of the cases II.a, II.b, II.c and II.d, we classifiedall minimal blocking sets of size q2 + 3. We summarize these results in thefollowing lemma.

Lemma 5 There is a unique minimal blocking set of size q2 + 3 = 28 in

Q(4, 5) and there is no minimal blocking set of size q2 + 3 in Q(4, 7).

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# multiple lines r # grids

Q(4, 5) 12 2 1

Q(4, 7) 32 4 2

Q(4, 11) 96 8 4

Q(4, q) (q + 1)(q − 3) q − 3 (q − 3)/2

if q ∈ {5, 7, 11}

Table 3Structure of the blocking sets of Q(4, q), q = 5, 7, 11 of sizeq2 + q − 2.

3.3 Minimal blocking sets of size q2 + q − 2

Using the same greedy heuristic as described in Section 2 we searched forminimal blocking sets in Q(4, q) for larger values of q and r. Typically wewere interested in any minimal blocking set (different from an ovoid) with sizesmaller than q2 + q. For q = 5, 7, 9, 11, we obtain minimal blocking sets of sizeq2 + q − 2, i.e., with excess r = q − 3.

We investigated (also by computer) the structure of the obtained blockingsets, 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 multiplelines have excess one, which means that the multiple lines contain two pointsof the blocking set. For q = 5, De Winter [8] already noticed that all multiplelines form a grid. We observed that also for q = 7, 11 grids are formed bythe multiple lines, and even that more than one grid appears, as summarizedin Table 3. For these cases the number of multiple lines is 2(q + 1)(q − 3)/2and the number of grids formed by multiple lines is (q − 3)/2. Moreover, forq = 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 ofsize q2 − 1 by removing these q + 1 common points and adding two points ofthe “perp” of this set. In Q(4, 5) are there 4 possibilities of obtaining such amaximal partial ovoid by removing q + 1 points.

For q = 9 we observed that, for the maximal blocking sets found, all multiplelines have excess three. The blocking set found has 20 multiple lines and wechecked that they form a grid. Previously we found by exhaustive search [2]that no maximal partial ovoids of size q2 − 1 exist in Q(4, 9); this result wasrecently generalized by De Beule and Gacs [3] for all q = ph, p an odd prime,h > 1.

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Finally we also want to mention our effort, which was not successful yet, totry and find a blocking set of size q2 + q − 2 for larger q, especially for q = 13.If we suppose that this blocking set would be of the same structure, it shouldcontain 5 grids with 14 points in the intersection. So far we have found noblocking set of the given size by our (not exhaustive!) computer search. Weexpect that no blocking set of Q(4, 13) of size q2 + q − 2 exists.

4 Blocking sets in H(d, q2), q = 3, 4 and in Q−(5, q)

As we described in the introduction, one can construct a minimal blocking setof size st + s of any generalized quadrangle of order (s, t) by taking the set ofpoints of a pencil through an arbitrary point x, minus the point x itself. Thefollowing theorems show that these construction yields the (unique) smallestminimal blocking set of size st + s for the generalized quadrangles Q−(5, q)and H(4, q2).

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| ≥q3 + q. If |B| = q3 + q, then B = p⊥ \ {p} for some point p ∈ Q−(5, q).

Recently, a similar theorem was proved for H(4, q2).

Theorem 7 (De Beule and Metsch [5]) Suppose that B is a minimal set

of points of H(4, q2) with the property that B meets every line of H(4, q2).Then |B| ≥ q5 + q2. If |B| = q5 + q2, then B = p⊥ \ {p} for some point

p ∈ H(4, q2).

To our knowledge there is nothing known about small minimal blocking setsin H(3, q2). In Table 4 we present some new results found by our heuristicsearches. We list the sizes smaller than q3 + q2 for which our program foundminimal blocking sets. For comparison reasons we also list the size q3 + 1 ofan ovoid and the size q3 + q2 of the blocking set described in the introduction.

For q = 2 we classified all minimal blocking sets of small size. There are twominimal blocking sets of size 10, twelve minimal blocking sets of size 11 andthirty minimal blocking sets of size 12.

Finally, we can construct a spectrum of minimal blocking sets for each valuefrom q3 + r, r = 1, . . . , q2.

Consider a Hermitian curve H contained in H(3, q2). This curve is containedin a plane, denoted by π. Consider any point p ∈ H. Consider the q2 + 1lines of π on p. Exactly q2 of them intersect H in a Hermitian variety H on

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q # points q3 + 1 q3 + q2 Sizes found

2 45 9 12 10,11

3 280 28 36 29..35

4 1105 65 80 72..79

Table 4Sizes for minimal blocking sets of H(3, q2) smaller than q3 +q2, for small values of q, obtained by heuristic search. Thenotation a..b means that for all values in the interval [a, b] aminimal blocking set of that size has been found.

a line (this is a H(1, q2)). It is clear that H⊥ is again a Hermitian variety ona line. Consider r different lines Li on p and their corresponding Hermitianvarieties Hi. Consider the r Hermitian varieties H⊥

i , they are mutually skewand lie all in p⊥. It is clear that the set H \ (∪r

i=1Hi) ∪ri=1 H⊥

i is a minimalblocking set of size q3 + r of H(3, q2). Since we have exactly q2 suitable lineson p, we can construct a spectrum of minimal blocking sets of size q3 + r,r = 1, . . . , q2.

References

[1] A. Blokhuis, On the size of blocking sets in PG(2, p). Combinatorica 14, 273-276, 1994.

[2] M. Cimrakova, S. De Winter, V. Fack, and L. Storme, On the smallest maximalpartial 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. Gacs, Complete (q2 − 1)-arcs of Q(4, q), q = ph, p oddprime, h > 1, do not exist. Finite Fields Appl. (2005), submitted.

[4] J. De Beule and K. Metsch, Minimal blocking sets of size q2 + 2 of Q(4, q), q

an odd prime, do not exist. Finite Fields Appl., 11, 305-315, 2005.

[5] J. De Beule and K. Metsch, The smallest point sets that meet all generators ofH(2n, q). Discrete Math., 294, 75-81, 2005.

[6] J. De Beule, A. Hoogewijs and L. Storme, On the size of minimal blocking setsof Q(4, q), for q = 5, 7. ACM SIGSAM Bulletin, 38, 67-84, 2004.

[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, 83-107, 2005.

[8] S. De Winter, Private communication, 2005.

[9] J. Eisfeld, L. Storme, T. Szonyi and P. Sziklai, Covers and blocking sets ofclassical generalized quadrangles. Discrete Math., 238, 35-51, 2001.

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