The Lower Bounds of Eight and Fourth Blocking Sets and Existence of Minimal Blocking Sets
Journal of education and science
This paper contains two main results relating to the size of eight and fourth blocking set in PG(2,16). First gives new example for (129,9)complete arc. The second result we prove that there exists (k,13)complete arc in PG(2,16), k≤197. We classify the minimal blocking sets of size eight in PG(2,4).We show that Rédei -type minimal blocking sets of size eight exist in PG(2, 4). Also we classify the minimal blocking sets of size ten in PG(2, 5), We obtain an example of a minimal blocking set of
... l blocking set of size ten with at most 4-secants.We show that Rédeitype minimal blocking sets of size ten exists in PG(2, 5). مالحظة : األطروحة من مستل البحث The Lower Bounds of Eight and Fourth ... 100 Introduction: A (k ,n)-arc K in PG(2,q) is a set of k points such that there is some n but no n+1 of them are collinear. A (k ,n)-arc K is complete if there is no (k+1,n)-arc containing it. The maximum value of k which a (k ,n)-arc K exist in PG(2,q) will be denoted by m(n) 2,q  . A t-fold blocking set B in a projective plane, is a set of points such that each line contains at least t points of B and some line contains exactly t points of B . For t=1,a1-fold blocking set is called a blocking set. A trivial blocking set B is a blocking set containing a line of PG(2,q). A t-blocking set is called minimal (irreducible)when no proper subset of it is a t-blocking set  . For t=2,3,4,...then tblocking set is called respectively double blocking set, triple blocking set , fourth blocking set...etc. (k ,n)-arcs and t-blocking sets are in fact just complements of each other in a projective plane , with n + t = q + 1. Richardson was the first one to look at larger planes  . He showed that the minimal size of a blocking set in PG(2 ,3) is 6,and noted that Baer subplanes are examples of blocking sets of size q+ q +1 in projective planes of square order. After that things were quiet for 13 years until Di paola introduced the idea of a projective triangle, which gives an example of a blocking set of size 3(q+1)/2 in Desargusian planes of odd order. That projective triangles exist in these planes was shown by Bruen , who also obtained the general lower bound q+ q +1 for the size of a blocking set in arbitrary projective plane of odd order q. Further results obtained by Bruen [3 ], giving the upper bound q q +1 for a minimal blocking set in any projective plane of order q, and make the connection with Re'dei s work on lacunary polynomials[ 10 ]. The fundamental results are for the structure of blocking sets however was only realized much later and in this course the emphasis will be to explain in some detail the recent developments and the connection between Re'dei s work on lacunary polynomials and small blocking sets and multiple blocking sets in Desargusian projective planes.