(de Gruyter 2003
Blocking sets in PG(2, p) for small p, and partial spreads in PG(3, 7)
Aart Blokhuis, Andries E. Brouwer and Henny A. Wilbrink
Dedicated to Professor Adriano Barlotti on the occasion of his 80th birthday
Abstract.We find all minimal blocking sets of size32ðpþ1Þin PGð2;pÞfor p<41. There is one new sporadic example, for p¼13. We find all maximal partial spreads of size 45 in PGð3;7Þ.
1 Minimal nontrivial blocking sets in PG(2,p)
A blocking set in a projective plane is a set of points meeting all lines. It is called nontrivialwhen it does not contain a line. Anm-secantof a set is a line meeting the set in preciselympoints.
Blokhuis [2] shows that in a Desarguesian projective plane PGð2;pÞ of prime order p, a nontrivial blocking set has size at least 32ðpþ1Þ, and, moreover, that in case of equality each point of the blocking set lies on precisely12ðp1Þtangents (1-secants).
Nontrivial blocking sets of size32ðpþ1Þexist for allp. Indeed, an example is given by the projective triangle: the set consisting of the points ð0;1;s2Þ, ð1;s2;0Þ, ðs2;0;1ÞwithsAFp.
No nontrivial blocking set of sizeqþmin PGð2;qÞcan have ak-secant fork>m, and in particular such a set of size32ðpþ1Þin PGð2;pÞcannot have ak-secant with k>12ðpþ3Þ. The triangle has three 12ðpþ3Þ-secants. Conversely, Lova´sz and Schrijver [10] show that any nontrivial blocking set of size32ðpþ1Þwith a12ðpþ3Þ- secant must be projectively equivalent to the triangle. (They put the given secant at infinity and show that the remaining p a‰ne points can be taken to be the points ða;aðpþ1Þ=2ÞforaAFp.)
A blocking setSin PGð2;qÞis calledof Re´dei typewhen there is a lineLsuch that jSnLj ¼q. Thus, we know the blocking sets of Re´dei type meeting the Blokhuis bound in PGð2;pÞ, pprime. Let us call a nontrivial blocking set in PGð2;pÞthat meets the Blokhuis bound sporadic if it is not of Re´dei type. A single sporadic blocking set (in PGð2;7Þ) was known. Here we find a second sporadic blocking set (in PGð2;13Þ) and show that no other sporadic blocking sets exist in PGð2;pÞ, p<41.
2 The Blokhuis bound
Theorem 2.1 ([2]). Let S be a nontrivial blocking set in PGð2;pÞ, p prime. Then jSjd32ðpþ1Þ. If equality holds, then each point of S lies on precisely 12ðp1Þ tangents.
Proof. Let S¼ fðai;bi;ciÞ ji¼1;. . .;qþmg be a minimal blocking set in PGð2;qÞ, where q is a power of the prime p. The polynomial FðX;Y;ZÞ ¼ Q
iðaiXþbiYþciZÞvanishes in all pointsðx;y;zÞ, hence can be written as FðX;Y;ZÞ ¼AðX;Y;ZÞðXqXÞ þBðX;Y;ZÞðYqYÞ þCðX;Y;ZÞðZqZÞ.
Since FðX;Y;ZÞ is homogeneous, all low degree terms cancel, and we have FðX;Y;ZÞ ¼A0ðX;Y;ZÞXqþB0ðX;Y;ZÞYqþC0ðX;Y;ZÞZq, where F has de- greeqþmandA0;B0;C0have degreem. Assume thatjSj<2q, so that no cancella- tion takes place between the terms on the right hand side.
Let the lineZ¼0 containlpoints ofS, and assume thatð1;0;0ÞAS. Now divide byXand substituteX¼0,Y ¼1 to get fðZÞ ¼bðZÞ þcðZÞZqwhere f has degree qþml and factors completely, andchas degree ml andb has degree at most m1. Write fðZÞ ¼sðZÞ rðZÞ where scontains every irreducible factor of f just once, and r contains the repeated factors. Then sj ðbþcZqÞ and sj ðZqZÞ so sj ðbþcZÞ. And rjf0¼b0þc0Zq, so that f ¼rsj ðbþcZÞðb0þc0ZqÞ, and hence
f j ðbþcZÞðb0cbc0Þ.
If the factors on the right are nonzero, it follows that qþmlc2ðm1Þ þ ml1 that is,mdðqþ3Þ=2. And in case of equality the degree ofsequals the degree ofbþcZso thatð1;0;0Þlies on preciselyðq1Þ=2 tangents.
If bþcZ¼0 then f ¼c ðZqZÞ and it follows thatð1;0;0Þdoes not lie on a tangent, i.e.,Sis not minimal, contradiction.
If b0cbc0¼0 then b and cdi¤er by a p-th power. In the particular caseq¼p (andm<q) it follows that they di¤er by a constant factor, saybðZÞ ¼acðZÞ, and
fðZÞ ¼cðZÞ ðaþZÞq so thatScontains (and hence is) a line.
3 Lacunary polynomials
We see that the blocking set problem leads one to search for polynomials fðxÞ,gðxÞ, hðxÞ, where f factors completely into linear factors andgandhhave degree at most
1
2ðqþ1Þsuch that f ¼xqgþh.
(Indeed, in the proof above we found such an f given a small blocking set S, a pointPinside, and a lineLpassing through that point. Ane-fold linear factor of f corresponds to a line onPdistinct fromLmeetingSineþ1 points. The lineLmeets SinjSj degreeðfÞpoints. Below we takejSj ¼32ðqþ1Þ.)
This equation has solutions that need not correspond to blocking sets. We give a few examples.
a) (For oddq, sayq¼2rþ1.) Take fðxÞ ¼xQðxaÞ3where the product is over the nonzero squares a. Then f satisfies fðxÞ ¼xðxr1Þ3¼xqgþh with gðxÞ ¼ xr3,hðxÞ ¼3xrþ1x. This would correspond to line intersections (with frequen-
cies written as exponents) 1r224r. Forq¼7 this is the function for the blocking set ð1;0;0Þ,ð0;1;0Þ,ð0;0;1Þ,ða;b;1Þwitha;bAf1;2;4g.
b) (Forq¼4tþ1.) Take fðxÞ ¼xQ
ðxaÞQ
ðxbÞ4 where the product is over the nonzero squares a and fourth powers b. Here fðxÞ ¼xðx2t1Þðxt1Þ4 ¼ xqgþh withgðxÞ ¼x2t4xtþ5 andhðxÞ ¼ 5x2tþ1þ4xtþ1x. This would cor- respond to line intersections 12t2tþ26t.
c) (Forq¼4tþ1.) TakefðxÞ ¼xtþ1Q
ðxaÞQ
ðxbÞ2where the product is over the nonzero squares a and fourth powers b. Here fðxÞ ¼xtþ1ðx2t1Þðxt1Þ2 ¼ xqgþhwithgðxÞ ¼xt2 andhðxÞ ¼2x2tþ1xtþ1. This would correspond to line intersections 12t2t4tðtþ2Þ2. Forq¼13 this is a function for the blocking setð1;0;0Þ, ð0;1;0Þ,ð0;0;1Þ,ð1;a;0Þ,ð0;1;aÞ,ða;0;1Þ,ðb;c;1Þwitha3 ¼ 1,b3¼c3¼1.
d) (For q¼13.) Take fðxÞ ¼xQðxaÞ4Q x12a
where the product is over all a with a3¼1. Here fðxÞ ¼xðx31Þ4x318
¼xqgþh with gðxÞ ¼x3þ4 andhðxÞ ¼5x75x45x. This would correspond to line intersections 162454, and indeed this occurs.
These lacunary polynomials are just weighted subsets of the projective line, and in particular PGLð2;qÞ acts. For example, x7!1x sends xqgþh to xq~hhþgg~where k~
kðxÞ ¼xðqþ1Þ=2kðx1Þ.
For completeness we describe the lacunary polynomials that correspond to the Re´dei type blocking set:
e) Take fðxÞ ¼xqxðqþ1Þ=2¼xðqþ1Þ=2QðxaÞ where the product is over the nonzero squaresa.
f ) Take fðxÞ ¼xq2xðqþ1Þ=2þx¼xQ
ðxaÞ2 where the product is over the nonzero squaresa.
4 Search setup
We search for lacunary polynomials as described above over the prime field Fp by exploiting the equation
f ¼xpgþh¼aðxgþhÞðgh0g0hÞ
for some constanta, where f factors into linear factors, andxgþh factors into dis- tinct linear factors, andgandhhave degree at most12ðpþ1Þ.
If we guessxgþhand the constant of proportionalityaand the constant term ofg then this relation gives a recurrence that allows us to compute all other coe‰cients of g, and thus to find f. If we take l¼1, then xgþh is a product of m¼ ðpþ3Þ=2 distinct linear factors, and there are p
m
possible choices for the set of roots of xgþh. We tried all possibilities for p<41, where PGLð2;pÞ was used to divide the computation time by roughly p3. This yields all possibilities for f, and in partic- ular the multiplicities of the roots of f, so that we know the sizes of the intersection of lines on some arbitrary point ð1;0;0Þ with S. This su‰ces to classify the pos- sible solutions. In fact, except for example d) in the previous section we only find solutions ifxgþh¼xðqþ1Þ=2x. In a seperate section we will completely classify this special case.
Looking at p¼31 took 80 minutes CPU time on an old Pentium running Linux, and p¼37 took four days.
5 Results
The results are as follows. First of all there are possibilities with a factor of multi- plicity 12ðpþ1Þ, i.e., a 12ðpþ3Þ-secant, and we have a Re´dei example, unique by Lova´sz and Schrijver.
For the primesp¼7;11;19;23;31 there is a unique non-Re´dei intersection pattern, namely 1ðp1Þ=2224ðp1Þ=2 (corresponding to the lacunary polynomial found under a) above). Counting the total number of lines on these points we see that this can be a blocking set only for p¼7. It remains to investigate the cases p¼7;13;17;29;37.
5.1 pF7.Forp¼7 there is a unique intersection pattern 132243(and no computer search is required to see that). It gives rise to a unique sporadic blocking set (of size 12) (see also [4]).
It arises as follows. The a‰ne plane AGð2;3Þ can be embedded into PGð2;qÞif and only if q¼0;1ðmod 3Þ, as one easily checks by assigning coordinates to the 9 points of AGð2;3Þ(for more details see [9] and [1]). This embedding is unique up to isomorphism. The three lines in a parallel class of AGð2;3Þare concurrent in PGð2;qÞ if and only ifq¼0ðmod 3Þ. Forq¼1ðmod 3Þthis 9-set can be found as the set of inflections of a nondegenerate cubic. Dualizing we find a dual a‰ne plane DAGð2;3Þ with 12 points, 9 4-lines (3 on each point) and 12 2-lines (2 on each point) embedded in PGð2;qÞ for q¼1ðmod 3Þ. It has ðq2þqþ1Þ 12ðqþ15Þ 912¼ ðq4Þðq7Þ0-secants, and hence is a blocking set forq¼4;7 and forq¼4 even a 2-fold blocking set.
The projective triangle in PGð2;7Þcan also be viewed as a modification of AGð2;3Þ:
it arises by taking the 9 points of AGð2;3Þand adding the 3 points of intersection of the lines of one parallel class.
There are no other possibilities: Suppose the blocking set S has ni i-secants, 1cic4. Then P
ni¼57, P
ini¼96, Pi
2
ni¼66 by standard counting. And n1¼36 since we have equality in the Blokhuis bound. Hencen2¼12,n3¼0,n4¼9.
If there are mi i-secants on a fixed point sAS, then P
mi¼8, P
ði1Þmi¼11, m1¼3 so thatm2¼2,m4¼3. This yields the DAGð2;3Þstructure.
More generally, Ga´cs et al. showed in [6] that if a nontrivial blocking setSof size
3
2ðpþ1Þin PGð2;pÞhas ak-secant forkd12ðpþ1Þthen it is of Re´dei type, unless p¼7 and we have this dual a‰ne plane.
5.2 pF11.We already saw that for p¼11 nothing of interest happens. More gen- erally, Ga´cs [5] showed that ak-secant withk¼12ðp1Þonly occurs for sets of Re´dei type, and simple counting then shows that forp¼11 the setSmust be of Re´dei type.
5.3 pF13.For p¼13 there is a nice example again that is not of Re´dei type. Let q¼1ðmod 3Þand take in PGð2;qÞthe 9 points of an embedded AGð2;3Þtogether with the 12 points of intersection of lines that are parallel in AGð2;3Þ. This yields a
self-dual configuration. Indeed, these 21 points together with the 21 lines that contain more than two of the points have a structure that is that of PGð2;4Þin which the incidences between the 9 points of a unital (AGð2;3Þ) and the tangent at these points has been removed. There are 12 5-secants, 9 4-secants, 36 2-secants, 21ðqþ18Þ1- secants and ðq2þqþ1Þ 21ðqþ18Þ 36912¼ ðq7Þðq13Þ 0-secants, so that this is a blocking set forq¼7;13, and forq¼7 even a 2-fold blocking set.
For p¼13 we have jSj ¼21. The search shows that there are four possible inter- section patterns: a) 162246, b) 162563, c) 16234352, d) 162454. Let there beNapoints of typea, etc., andnii-secants.
IfNb >0, then there is a 6-secant, and it meets another 12 6-secants, so 13cn6 ¼ 3Nb=6 andNb>jSj, contradiction.
SoNb¼0. If alsoNa¼0 thenNcþNd ¼21,n1¼126,n2¼32Ncþ2Nd,n4¼34Nc, n5¼25Ncþ45Nd,P
ni¼132þ13þ1¼183, with unique solutionNc¼12,Nd ¼9, n2¼36, n4¼9, n5¼12. Each 4-secant meets the remaining eight, that is, the 4- secants meet pairwise (in points of type c)), and the points of type c) form a DAGð2;3Þ. A 5-secant meets the DAGð2;3Þin at most two points, so has at least three points of type d), and the points of type d) together with the 5-secants form an AGð2;3Þ. Now everything is determined, and this indeed yields a solution.
If Na>0 then at most two points do not lie on a 4-secant, so Ndc2. If Nc¼ Nd ¼0, then Na¼21 and n4 ¼6Na=4 is not integral. Contradiction. So, n5 ¼
2
5Ncþ45Nd >0. We have n4¼64Naþ34Nc, so Nc is even, and 4jn5. Each 5-secant meets at least five more, so n5d8, i.e., Ncþ2Ndd20, NcþNdd18, Nac3. If n5d12 then Ncþ2Ndd30, NcþNdd28, contradiction. So n5 ¼8. Now n4 ¼
6
4Naþ34Nc¼32ðNaþNcþNdÞ 34ðNcþ2NdÞ ¼32213420 is not integral. Con- tradiction.
So, up to isomorphism there is a unique minimal blocking set in PGð2;13Þof size 21 that is not of Re´dei type.
5.4 pF17.Forp¼17 we havejSj ¼27. There are three possible intersection pat- terns: a) 182248, b) 182664, c) 18244462.
We have NaþNbþNc¼27 and n1 ¼827¼216, and n2¼Naþ3Nbþ2Nc, n4¼2NaþNc, so n2þn4¼327¼81 and n6¼172þ17þ121681¼10.
2NbþNc¼3n6¼30, soNbd3. Now three points of type b) see twelve 6-secants, but there are only ten, so there is a 6-secant with at least two points of type b). But such a 6-secant meets at least 3þ3þ1þ1þ1þ1¼10 other 6-secants, contradiction.
So, no non-Re´dei sets occur for p¼17.
5.5 pF29.Forp¼29 we havejSj ¼45. There are three possible intersection pat- terns: a) 11422414, b) 1142967, c) 114274792.
If type c) occurs then there are 9-secants, and each 9-secant meets another nine, so 10cn9¼2Nc=9 andNcd45 so that all points are of type c). But thenn4¼7Nc=4 is not integral. Contradiction.
SoNc¼0. There are 14Na=4 4-secants, soNa is even. There are 7Nb=6 6-secants, soNbis even. ButNaþNb ¼45. Contradiction.
So, no non-Re´dei sets occur for p¼29.
5.6 pF37 and larger p. For p¼37 we have jSj ¼57. There are three possible intersection patterns: a) 11822418, b) 11821169, c) 1182949112, and as before no non- Re´dei set can exist.
Let us prove more generally that no sporadic blocking set exists in PGð2;pÞ, p¼4tþ1d37, when only the three patterns a) 12t2242t, b) 12t2tþ26t and c) 12t2t4tðtþ2Þ2do occur. We havejSj ¼6tþ3.
If type c) occurs then there are ðtþ2Þ-secants, and each meets tþ2 more, so tþ3cntþ2¼2Nc=ðtþ2Þc2jSj=ðtþ2Þ<12, contradiction. So Nc¼0. Now NaþNb¼ jSj and n1þn2þn4þn6¼p2þpþ1 determines all values: Na¼12, Nb¼6t9, n1¼12t2þ6t, n2¼3t2þ32tþ3, n4¼6t, n6 ¼t232t. Now a 4-line meets 4ð2t1Þother 4-lines, contradictingn4¼6t.
So, for a new sporadic blocking set we need a new factorizing lacunary polynomial.
6 The special casexgBhFx(pB1)/2Cx In this section we consider the modular di¤erential equation
xpgþh¼aðxgþhÞðg0hh0gÞ;
wherexgþhfactors into distinct linear factors, andg;hAFp½xare both of degree at
most ðpþ1Þ=2, not both zero, and a is a nonzero constant. Write s:¼xgþh and
t:¼ ðxpxÞ=s. Thenh¼sxgands0tþst0¼ 1. Rewrite the original equation as ðxpxÞg¼sðag0sags0þag21Þ:
Division bysgives
tg¼ag0sags0þag21¼ag0sags0þag2þst0þs0t:
This may be rewritten as
sðag0þt0Þ ¼ ðagtÞðgs0Þ:
We now consider the special case s¼xnþ1x, where n:¼ ðp1Þ=2. Then t¼xnþ1, and our equation simplifies to
ðxn1Þ xag01 2xn
¼ ðxnþ1agÞ gþ11 2xn
:
Ifuis a square inFp(so thatun1¼0) thengðuÞA12;2a
. Comparing degrees we see thatghas degree at mostn. Moduloxn this equation reduces further to
xg0¼ g1 a
ðgþ1Þ modxn:
Note thatgð0Þ 1a
ðgð0Þ þ1Þ ¼0.
Consider more generally the equation xg0¼ ðgbÞðgcÞmodxn, say with gð0Þ ¼b. Then we getg¼cþ ðbcÞ=ð1CxbcÞmodxnfor some constantC.
(Indeed, the equation xg0¼ ðgbÞðgcÞsu‰ces to determine all coe‰cients of gin terms of earlier coe‰cients, except the coe‰cient ofxiwherei¼bc.)
In the above, 1=ð1CxdÞwas to be interpreted in F½½x. We get a solution in polynomial form by replacing it byð1CmxdmÞ=ð1CxdÞ, for somem such that dmdn.
Thus, in our case,
g¼cþd1Cmxdm 1Cxd þexn;
for certain constants c;d;e, where either d ¼0 and the middle term is absent, or C00, 0<d <p,md2,dðm1Þ<ncdm.
Since g takes at most two values on nonzero squares, the same holds for
1Cmxdm
1Cxd (when d00). Thus, there are constants A;B such that xn1 divides ð1CmxdmAð1CxdÞÞð1CmxdmBð1CxdÞÞ. This remains true if we re- placexdmbyxdmn, so eithernc2d,m¼2,d¼n=2, or the right hand side vanishes andA¼0,dm¼n,Cm¼1.
In the former case we have (with new constants) g¼cþdxn=2þexn with c¼ 1 or c¼1=a. Substitution and comparison of coe‰cients gives ða;c;d;eÞ ¼ ð2;1;0;1=2Þorða;c;d;eÞ ¼ ð2;1;0;0Þorða;c;d;eÞ ¼ ð4=3;1;G1=2;0Þor ða;c;d;eÞ ¼ ð2;1=2;0;0Þ or ða;c;d;eÞ ¼ ð4=3;3=4;0;1=4Þ or ða;c;d;eÞ ¼ ð4=5;5=4;G1;1=4Þ, and these correspond to the examples f ), e), c), f ), a), b), respectively.
In the latter case we haveg¼cþd 1xn
1Cxdþexn, wheren¼dm,Cm¼1 and with- out loss of generalitymd3. The two values taken bygon the set of nonzero squares di¤er by 2aþ12¼Gn¼H1
2, so that a¼ 2 and cþe¼ 1=2. Comparing leading coe‰cients we findeAf0;1=4g. Comparing constants we findcþd Af1;1=2g.
The four possible values of dturn out to be 0;n=2;n;3n=2, and we already handled those.
Altogether the conclusion is that if xpgþh¼aðxgþhÞðg0hh0gÞ andxgþh¼ xnþ1x, withg;h both of degree at mostnþ1, then we have one of the examples from Section 3.
7 Partial spreads in PG(3, 7)
Aspreadin a point-line geometry is a partition of the point set into lines. Apartial spreadis a collection of pairwise disjoint lines. Given a partial spread in a point-line geometry, we shall call a point not covered by one of its lines ahole.
Hirschfeld [8] (Section 17.6) shows that PGð3;qÞhas a maximal partial spread of sizeq2qþ2 forq>3 (and a maximal partial spread of size 7 forq¼3). No larger maximal partial spreads (that are not spreads) are known, except for q¼7, where Heden [7] constructed a maximal partial spread of size 45.
The relation with blocking sets in PGð2;qÞis as follows:Given a maximal partial spread of size q2þ1dinPGð3;qÞ,whered>0,we find a nontrivial blocking set of size qþdinPGð2;qÞ.
(Indeed, we find such a blocking set by taking the set of holes in a plane that does not contain a line of the partial spread.)
Since nontrivial blocking sets in PGð2;7Þ have size at least 12, it follows that a partial spread in PGð3;7Þthat is not a full spread has at most 45 lines, that is, has at least 40 holes.
We did a complete search for partial spreads with 40 holes and find that there are precisely 879 nonisomorphic such partial spreads. The table below gives group order, number of isomorphism classes and total number of partial spreads.
order # total
1 174 4 510 080 2 383 4 963 680
3 7 60 480
4 175 1 134 000
6 35 151 200
8 39 126 360
10 9 23 328
12 40 86 400
20 1 1 296
24 11 11 880
60 1 432
120 4 864
total 879 11 070 000
Soicher [11] had already determined the partial spreads with 40 holes and an auto- morphism group of order 5.
The geometry of the setHof 40 holes (complement of the union of a maximal partial spread S of size 45) is uniquely determined, as was already remarked by Heden.
Indeed, each plane must meet H in either 5 or 12 points (depending on whether it contains a line ofSor not), and the holes form a blocking set in each planepwith 12 holes. (Otherwise there would be a lineLinpdisjoint fromH, and looking at the 8 planes onL they must all have precisely 5 points of H, contradiction.) Thus, the planes with 12 holes are either of the triangle or of the DAGð2;3Þtype.
Now all planes with 12 holes must be of the same type. Indeed, let anm-linebe a line with m holes. A plane of triangle type does not have 4-lines, while a plane of DAGð2;3Þtype does not have 5-lines. In particular, a 4-line cannot meet a 5-line.
Each hole in a plane of DAGð2;3Þtype is on some 4-line, so no such hole can be on a 5-line. On a 4-line there are 8 planes, four of DAGð2;3Þtype, and we find at least 36 holes on a 4-line, no room for a 5-line.
Not all planes can be of triangle type. Indeed, suppose this is the case. Each 3-line is on three planes with 12 holes and in each of these planes each of the three holes of
the 3-line lies on a unique 5-line. It follows that each hole is on precisely three 5-lines (so that there are 24 5-lines in all). On the other hand, the projective transformations that fix the set of non-holes on a 5-line have two orbits on the 5 holes, so that the two
‘corners’ on that line in a triangle do not depend on the choice of triangle, so that these corners would be on six 5-lines, contradiction.
Thus, all planes are dual a‰ne planes. We have a geometry with points and 4-lines, where two intersecting 4-lines determine a plane, and each plane is dual a‰ne of order 3. By Cuypers [3] this is the geometry of points and hyperbolic lines and dual a‰ne planes of the Spð4;3Þgeometry. This is again a self-dual configuration that lives in PGð3;qÞfor all prime powersq¼1ðmod 3Þ. (For example, in PGð3;4Þit lives as the nonisotropic points of a Uð4;2Þ geometry.) Explicit coordinates: take the 4 points ð1;0;0;0Þand the 36 pointsð0;1;a;bÞwhere a3¼b3¼1 and the coordinates may be permuted cyclically.
References
[1] A. Bichara, G. Korchma´ros,n2-sets in a projective plane which determine exactlyn2þn lines.J. Geom.15(1980), 175–181. MR 82j:51016 Zbl 0459.51007
[2] A. Blokhuis, On the size of a blocking set in PGð2;pÞ.Combinatorica14(1994), 111–114.
MR 96b:51010 Zbl 0803.05011
[3] H. Cuypers, Symplectic geometries, transvection groups, and modules.J. Combin. Theory Ser. A65(1994), 39–59. MR 94m:51003 Zbl 0824.51003
[4] J. W. Di Paola, On minimum blocking coalitions in small projective plane games.SIAM J. Appl. Math.17(1969), 378–392. MR 40 #1140 Zbl 0191.49601
[5] A. Ga´cs, A remark on blocking sets of almost Re´dei type.J. Geom.60 (1997), 65–73.
MR 99e:51009 Zbl 0897.51004
[6] A. Ga´cs, P. Sziklai, T. Szo˝nyi, Two remarks on blocking sets and nuclei in planes of prime order.Des. Codes Cryptogr.10(1997), 29–39. MR 97j:51015 Zbl 0874.51002
[7] O. Heden, A maximal partial spread of size 45 in PGð3;7Þ. Des. Codes Cryptogr. 22 (2001), 331–334. MR 2002a:51007 Zbl 0982.51005
[8] J. W. P. Hirschfeld,Finite projective spaces of three dimensions. Oxford Univ. Press 1985.
MR 87j:51013 Zbl 0574.51001
[9] L. M. Kelly, S. Nwankpa, A‰ne embeddings of Sylvester-Gallai designs. J. Combin.
Theory Ser. A14(1973), 422–438. MR 47 #3207 Zbl 0282.05018
[10] L. Lova´sz, A. Schrijver, Remarks on a theorem of Re´dei.Studia Sci. Math. Hungar.16 (1983), 449–454. MR 85e:51017 Zbl 0535.51009
[11] L. Soicher, Computation of partial spreads.
http://www.maths.qmw.ac.uk/~leonard/partialspreads
Received 7 January, 2003; revised 5 April, 2003
A. Blokhuis, A. E. Brouwer, H. A. Wilbrink, Dept. of Math., Techn. Univ. Eindhoven, P.O.
Box 513, 5600MB Eindhoven, Netherlands Email: {aartb, aeb, wsdwhw}@win.tue.nl
