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On large minimal blocking sets in PG(2, q )
Author(s) -
Szőnyi Tamás,
Cossidente Antonello,
Gács András,
Mengyán Csaba,
Siciliano Alessandro,
Weiner Zsuzsa
Publication year - 2005
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20017
Subject(s) - combinatorics , mathematics , upper and lower bounds , blocking (statistics) , order (exchange) , prime (order theory) , square (algebra) , generalization , bounded function , discrete mathematics , geometry , mathematical analysis , statistics , finance , economics
The size of large minimal blocking sets is bounded by the Bruen–Thas upper bound. The bound is sharp when q is a square. Here the bound is improved if q is a non‐square. On the other hand, we present some constructions of reasonably large minimal blocking sets in planes of non‐prime order. The construction can be regarded as a generalization of Buekenhout's construction of unitals. For example, if q is a cube, then our construction gives minimal blocking sets of size q 4/3 + 1 or q 4/3 + 2. Density results for the spectrum of minimal blocking sets in Galois planes of non‐prime order is also presented. The most attractive case is when q is a square, where we show that there is a minimal blocking set for any size from the interval $[4q\,{\rm log}\, q, q\sqrt q-q+2\sqrt q]$ . © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 25–41, 2005.