Multiple blocking sets in \(\text{PG}(n,q)\), \(n \geq 3\)
Let \(\text{PG}(n, q)\) be the projective space of dimension \(n\) over the finite field \(\mathbb F_q\) of order \(q\). An \(s\)-fold blocking set \(B\) of \(\text{PG}(n, q)\) is a set of points of \(\text{PG}(n, q)\) such that each hyperplane contains at least \(s\) points of \(B\). A \(1\)-fold blocking set is also called a blocking set, a \(2(3)\)-fold blocking set is called a double (triple) blocking set. An \(s\)-fold blocking set is minimal if no proper subset of \(B\) is still an \(s\)-fold blocking set. In the literature we can find a number of papers devoted to the study of \(s\)-fold blocking sets in \(\text{PG}(2, q)\) [see for instance \textit{S. Ball}, Eur. J. Comb. 17, No. 5, 427--435 (1996; Zbl 0858.51003); J. Lond. Math. Soc., II. Ser. 54, No.~3, 581--593 (1996; Zbl 0904.51002); \textit{S. Ball, A. Blokhuis}, Finite Fields Appl. 2, No.~2, 125--137 (1996; Zbl 0896.51008); \textit{A. Bruen}, SIAM J. Appl. Math. 21, 380--392 (1971; Zbl 0252.05014); \textit{A. Bruen}, Conv. Roma 1983, Symp. Math. 28, 15--29 (1986; Zbl 0598.51007); \textit{A. Gács, T. Szönyi}, Double blocking sets and Baer subplanes (unpublished manuscript, 1995)]. In the paper under review, minimal \(s\)-fold blocking sets \(B\) in \(\text{PG}(n, q)\), \(q = p^h\), \(q\geq 661\), \(n\geq 3\), of size \(| B| < s q + c_pq^{2/3} - (s - 1)(s- 2)/2\), (\(s < \min(c_p q^{1/6}, q^{1/4/2}))\), are discussed. It is proved that these \(s\)-fold blocking sets contain the disjoint union of \(s\) lines and/or Baer subplanes. To obtain their results, the authors first study \(s\)-fold blocking sets in \(\text{PG}(2, q)\) consisting of points with given positive multiplicities, then via projection arguments they prove that small \(s\)-fold blocking sets in \(\text{PG}(3, q)\) contain the disjoint union of a collection of \(s\) lines and/or Baer subplane. Finally, they extend this latter result to projective spaces of arbitrary dimension. The paper ends with a section on the connection of \(s\)-fold blocking sets and codes meeting the Griesmer bound, and in particular they show that their results improve those of \textit{N. Hamada} and \textit{T. Helleseth} [Math. Jap. 38, No.~5, 925--939 (1993; Zbl 0786.05016)].
scientific article; zbMATH DE number 2091554