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The non-existence of certain large minimal blocking sets. (English) Zbl 0917.51012
The paper under review is concerned with the size of minimal blocking sets. In a projective plane $$\Pi_q$$ of order $$q$$, a blocking $$k$$-set is a set of $$k$$ elements which meets every line but contains no line completely. A blocking set $$\mathcal B$$ is called minimal iff for each $$P\in \mathcal B$$, $$\mathcal B\setminus\{P\}$$ is not a blocking set. A basic problem in Finite Geometry is to find the values of $$k$$ for which a minimal blocking $$k$$-set (in a given projective plane) exists. For a nice survey on blocking sets see A. Blokhuis [Bolyai Soc. Math. Stud. 2, 133-155 (1996; Zbl 0849.51005)].
Let $$\mathcal B$$ be a minimal blocking $$k$$-set in $$\Pi_q$$. It is known that (1) $$q+\sqrt q +1\leq k\leq q\sqrt q +1$$; (2) $$k=q+\sqrt q+1$$ iff $$q$$ is a square and $$\mathcal B$$ is a Baer plane; (3) $$k=q\sqrt q +1$$ iff $$q$$ is a square and $$\mathcal B$$ is a unital [see A. A. Bruen, Bull. Am. Math. Soc. 76, 342-344 (1970; Zbl 0207.02601), SIAM J. Appl. Math. 21, 380-392 (1971; Zbl 0252.05014), A. A. Bruen and J. A. Thas, Geom. Dedicata 6, 193-203 (1977; Zbl 0367.05009)]. If, in addition, the plane $$\Pi_q$$ is Desarguesian and $$q\geq 16$$ is a square, then there do not exist minimal blocking $$k$$-sets with $$q+\sqrt q +2\leq k\leq q+2\sqrt q$$ [see S. Ball and A. Blokhuis, Finite Fields Appl. 2, 125-137 (1996; Zbl 0896.51008)]. If $$q=9$$ the foregoing holds with $$k=14$$, by a result of A. A. Bruen and R. Silverman [Eur. J. Comb. 8, 351-356 (1987; Zbl 0638.51013)]. However it is not known whether or not there exists a minimal blocking 15-set in $$\Pi_9$$.
Next step is to look at large minimal blocking $$k$$-sets with $$k$$ close to the upper bound $$q\sqrt q +1$$. A. Blokhuis and K. Metsch [Math. Soc. Lect. Note. Ser. 191, 37-52 (1993; Zbl 0797.51012)] noticed that there does not exist a blocking $$q\sqrt q$$-set in Desarguesian projective planes of square order $$q\geq 25$$.
In the paper under review the author extends this result to an arbitrary projective plane of order 9. The main ingredients of the proof are standard diophantine equations; inner and outer point equations associated to blocking sets; a previous result of the author and A. Maturo concerning blocking sets in $$\Pi_7$$ ; and a case by case elimination of 19 possible cases. Finally, the case $$q=16$$ of Blokhuis and Metsch’s result remains open and it seems difficult to extend the techniques of this paper to this case.

##### MSC:
 5.1e+21 Combinatorial structures in finite projective spaces 5.1e+22 Blocking sets, ovals, $$k$$-arcs
##### Keywords:
blocking sets; Desarguesian projective planes