Blocking sets in PG ( 2 , p ) for small p , and partial spreads in PG ( 3 , 7 )

We find all minimal blocking sets of size 32 ðpþ 1Þ in PGð2; pÞ for p < 41. There is one new sporadic example, for p 1⁄4 13. 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 nontrivial when it does not contain a line. An m-secant of a set is a line meeting the set in precisely m points. Blokhuis [2] shows that in a Desarguesian projective plane PGð2; pÞ of prime order p, a nontrivial blocking set has size at least 3 2 ðpþ 1Þ, and, moreover, that in case of equality each point of the blocking set lies on precisely 1 2 ðp 1Þ tangents (1-secants). Nontrivial blocking sets of size 3 2 ðpþ 1Þ exist for all p. Indeed, an example is given by the projective triangle: the set consisting of the points ð0; 1; sÞ, ð1; s; 0Þ, ð s; 0; 1Þ with s A Fp. No nontrivial blocking set of size qþm in PGð2; qÞ can have a k-secant for k > m, and in particular such a set of size 3 2 ðpþ 1Þ in PGð2; pÞ cannot have a k-secant with k > 1 2 ðpþ 3Þ. The triangle has three 1 2 ðpþ 3Þ-secants. Conversely, Lovász and Schrijver [10] show that any nontrivial blocking set of size 3 2 ðpþ 1Þ with a 1 2 ðpþ 3Þsecant 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=2Þ for a A Fp.) A blocking set S in PGð2; qÞ is called of Rédei type when there is a line L such that jSnLj 1⁄4 q. Thus, we know the blocking sets of Rédei type meeting the Blokhuis bound in PGð2; pÞ, p prime. Let us call a nontrivial blocking set in PGð2; pÞ that meets the Blokhuis bound sporadic if it is not of Rédei 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 jSjd 3 2 ðpþ 1Þ. If equality holds, then each point of S lies on precisely 1 2 ðp 1Þ tangents. Proof. Let S 1⁄4 fðai; bi; ciÞ j i 1⁄4 1; . . . ; qþmg be a minimal blocking set in PGð2; qÞ, where q is a power of the prime p. The polynomial F ðX ;Y ;ZÞ 1⁄4 Q iðaiX þ biY þ ciZÞ vanishes in all points ðx; y; zÞ, hence can be written as F ðX ;Y ;ZÞ 1⁄4 AðX ;Y ;ZÞðX q XÞ þ BðX ;Y ;ZÞðY q Y Þ þ CðX ;Y ;ZÞðZ ZÞ. Since FðX ;Y ;ZÞ is homogeneous, all low degree terms cancel, and we have F ðX ;Y ;ZÞ 1⁄4 A0ðX ;Y ;ZÞX q þ B0ðX ;Y ;ZÞY q þ C0ðX ;Y ;ZÞZ, where F has degree qþm and A0;B0;C0 have degree m. Assume that jSj < 2q, so that no cancellation takes place between the terms on the right hand side. Let the line Z 1⁄4 0 contain l points of S, and assume that ð1; 0; 0Þ A S. Now divide by X and substitute X 1⁄4 0, Y 1⁄4 1 to get f ðZÞ 1⁄4 bðZÞ þ cðZÞZq where f has degree qþm l and factors completely, and c has degree m l and b has degree at most m 1. Write f ðZÞ 1⁄4 sðZÞ rðZÞ where s contains every irreducible factor of f just once, and r contains the repeated factors. Then s j ðbþ cZÞ and s j ðZ ZÞ so s j ðbþ cZÞ. And r j f 0 1⁄4 b 0 þ c 0Zq, so that f 1⁄4 rs j ðbþ cZÞðb 0 þ c 0ZqÞ, and hence f j ðbþ cZÞðb 0c bc 0Þ. If the factors on the right are nonzero, it follows that qþm lc 2ðm 1Þþ m l 1 that is, md ðqþ 3Þ=2. And in case of equality the degree of s equals the degree of bþ cZ so that ð1; 0; 0Þ lies on precisely ðq 1Þ=2 tangents. If bþ cZ 1⁄4 0 then f 1⁄4 c ðZq ZÞ and it follows that ð1; 0; 0Þ does not lie on a tangent, i.e., S is not minimal, contradiction. If b 0c bc 0 1⁄4 0 then b and c di¤er by a p-th power. In the particular case q 1⁄4 p (and m < q) it follows that they di¤er by a constant factor, say bðZÞ 1⁄4 a cðZÞ, and f ðZÞ 1⁄4 cðZÞ ðaþ ZÞ so that S contains (and hence is) a line. 3 Lacunary polynomials We see that the blocking set problem leads one to search for polynomials f ðxÞ, gðxÞ, hðxÞ, where f factors completely into linear factors and g and h have degree at most 1 2 ðqþ 1Þ such that f 1⁄4 xgþ h. (Indeed, in the proof above we found such an f given a small blocking set S, a point P inside, and a line L passing through that point. An e-fold linear factor of f corresponds to a line on P distinct from L meeting S in eþ 1 points. The line L meets S in jSj degreeð f Þ points. Below we take jSj 1⁄4 3 2 ðqþ 1Þ.) This equation has solutions that need not correspond to blocking sets. We give a few examples. a) (For odd q, say q 1⁄4 2rþ 1.) Take f ðxÞ 1⁄4 x Q ðx aÞ where the product is over the nonzero squares a. Then f satisfies f ðxÞ 1⁄4 xðxr 1Þ 1⁄4 xgþ h with gðxÞ 1⁄4 x 3, hðxÞ 1⁄4 3xrþ1 x. This would correspond to line intersections (with frequenAart Blokhuis, Andries E. Brouwer and Henny A. Wilbrink S246