Designs, Codes and Cryptography, 10, 29–39 (1997) c  1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Two Remarks on Blocking Sets and Nuclei in Planes of Prime Order ANDR ´ AS G ´ ACS Department of Computer Science, E ¨ otv¨ os Lor ´ and University, Budapest H-1088, Budapest, M´ uzeum krt. 6–8. HUNGARY P ´ ETER SZIKLAI Department of Computer Science, E ¨ otv¨ os Lor ´ and University, Budapest H-1088, Budapest, M´ uzeum krt. 6–8. HUNGARY TAM ´ AS SZ ˝ ONYI Department of Computer Science, E ¨ otv¨ os Lor ´ and University, Budapest H-1088, Budapest, M´ uzeum krt. 6–8. HUNGARY Communicated by: D. Jungnickel Received February 15, 1995; Revised October 2, 1995; Accepted October 10, 1995 Abstract. In this paper we characterize a sporadic non-R´ edei type blocking set of PG(2, 7) having minimum cardinality, and derive an upper bound for the number of nuclei of sets in PG(2, q ) having less than q + 1 points. Our methods involve polynomials over finite fields, and work mainly for planes of prime order. Keywords: blocking set, nucleus, polynomials. 1. Introduction In this paper we study two problems concerning blocking sets and nuclei in planes of prime order. The two objects are in close relation, see Blokhuis [3], Bruen [8]. The problems considered here are not directly related, but the methods used in the proofs are similar: polynomials over finite fields are used. The first problem is about blocking sets. A point-set B in a projective plane is called a blocking set if it intersects every line but contains no line. A blocking set is called minimal (irreducible) if it is minimal subject to set inclusion. For a blocking set (or more generally, for any pointset) a line is called an i -secant, if it intersects the blocking set in exactly i points. Instead of 1-secant also the term tangent will be used. So a blocking set is minimal if it has a tangent at each of its points. It is easy to prove that a line intersects a blocking set B in at most | B |− q points. If there is a line L with | B ∩ L |=| B |− q , then the blocking set is said to be of R´ edei type (or maximal type). For the size of a blocking set of PG(2, p) (the Desarguesian plane of order p), p a prime, Blokhuis [2] recently proved that | B |≥ 3( p + 1)/2. Blokhuis’ proof is algebraic; combinatorial arguments only give | B |≥ q + √ q + 1 (Bruen [7]), but they work for any plane of order q . There are blocking sets of R´ edei type satisfying | B |= 3(q + 1)/2 (see [9, Chap. 13.4], [12, Par. 36] and [4]). For planes of prime order Lov´ asz and Schrijver [10] characterized blocking sets of R´ edei