A General Construction for Blocking Sets in Finite Affine Geometries
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A General Construction for Blocking Sets in Finite Affine Geometries Ivan Landjev and Assia Rousseva To the memory of Professor Heinrich Wefelscheid
Abstract. A t-fold affine blocking set is a set of points in AG(n, q) intersecting each hyperplane in at least t points. In this paper we present a general construction of affine blocking sets in AG(n, q). The construction uses an arc in an r-dimensional subspace of PG(n, q) and a blocking set in the affine part ∼ = AG(n − r − 1, q) of its complementary subspace to produce a t-fold affine blocking set in AG(n, q). The infinite class of t-fold affine blocking sets with t = q − n + 2 meeting Bruen’s bound is obtained as a special case of this construction. It gives also several optimal affine blocking sets whose cardinality meets the lower bound provided by Ball’s improvement of Bruen’s bound. These are the first examples for blocking sets meeting this new bound. The construction produces also many examples of affine blocking sets lying close to the lower bounds by Bruen, Ball-Blokhuis, and Ball. Mathematics Subject Classification. 51E21, 51E22, 51E15, 51E23, 94B65. Keywords. t-Fold blocking set, t-fold intersecting set, Bruen’s bound, finite affine geometry, finite projective geometry, linear code.
1. Introduction Consider the affine geometry AG(n, q), where n is a positive integer and q = ph – a prime power. The pointset B in AG(n, q) is called a t-fold blocking set with respect to the hyperplanes if every hyperplane in AG(n, q) meets B in at least t points. A t-fold blocking set in AG(n, q) having cardinality N will be referred to as an affine (N, t)-blocking set. Let us note that some papers use the notion of t-fold intersection set instead of t-fold blocking set (cf. [1,5,12]). 0123456789().: V,-vol
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I. Landjev and A. Rousseva
Results Math
In this paper we deal with the problem of determining the minimal size of a t-fold blocking set in AG(n, q). Here the positive integer t and the affine geometry AG(n, q) are fixed. Affine t-fold blocking sets of minimal size are called optimal. The adjective minimal is reserved for blocking sets that are minimal with respect to inclusion. The first results on affine blocking sets were obtained independently by Jamison [7] and Brouwer and Schrijver [4]. They proved that for an 1-fold blocking set B in AG(n, q) one has the lower bound |B| ≥ n(q − 1) + 1.
(1)
This bound is sharp for all dimensions n. One example of such a set is provided by n concurrent independent lines. A generalization of this bound was given by Bruen in [5]. He proved that if B is a t-fold blocking set with respect to the hyperplanes in AG(n, q) then |B| ≥ (n + t − 1)(q − 1) + 1.
(2)
In what follows we call this bound Bruen’s bound. The bound is non-trivial for values of t satisfying 1 ≤ t ≤ (n − 1)(q − 1). For t > (n − 1)(q − 1) Bruen’s bound becomes worse than the trivial bound |B| ≥ tq, obtained by counting the points of the blocking set on q parallel hyperplanes. It was pointed out by Zanella [12] that if (n − 1)(q − 1) + 1 2 blocking sets me
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