Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On the linearity of higher-dimensional blocking sets. | On the linearity of higher-dimensional blocking sets G. Van de Voorde Submitted Jun 16 2010 Accepted Nov 29 2010 Published Dec 10 2010 Mathematics Subject Classification 51E21 Abstract A small minimal k-blocking set B in PG n q q pt p prime is a set of less than 3 qk 1 2 points in PG n q such that every n k -dimensional space contains at least one point of B and such that no proper subset of B satisfies this property. The linearity conjecture states that all small minimal k-blocking sets in PG n q are linear over a subfield Fpe of Fq. Apart from a few cases this conjecture is still open. In this paper we show that to prove the linearity conjecture for k-blocking sets in PG n pt with exponent e and pe 7 it is sufficient to prove it for one value of n that is at least 2k. Furthermore we show that the linearity of small minimal blocking sets in PG 2 q implies the linearity of small minimal k-blocking sets in PG n p with exponent e with pe t e 11. Keywords blocking set linear set linearity conjecture 1 Introduction and preliminaries If V is a vectorspace then we denote the corresponding projective space by PG V . If V has dimension n over the finite field Fq with q elements q p p prime then we also write V as V n q and PG V as PG n 1 q . A k-dimensional space will be called a k-space. A k-blocking set in PG n q is a set B of points such that every n k -space of PG n q contains at least one point of B. A k-blocking set B is called small if B 3 qk 1 2 and minimal if no proper subset of B is a k-blocking set. The points of a k-space of PG n q form a k-blocking set and every k-blocking set containing a k-space is called trivial. Every small minimal k-blocking set B in PG n pt p prime has an exponent e defined to be the largest integer for which every n k -space intersects B in 1 mod pe points. The fact that every small minimal k-blocking set has an exponent e 1 follows from a result of Szonyi and Weiner and will be explained in Section 2. A minimal k-blocking set B in PG n
