Abstract
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 size ten with at most 4-secants.We show that Rédei –type minimal blocking sets of size ten exists in PG(2, 5).
Highlights
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
The maximum value of k which a (k,n)-arc K exist in PG(2,q) will be denoted by m(n)2,q[6]
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 [1]
Summary
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. 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 Redei s work on lacunary polynomials[ 10 ]. Let B be a fourth blocking set in PG(2, q) , q is square, such that through each of its points there are q+1 lines, each containing at least q+4 points of B and forming a dual Baer subline. 2( q+4)+4(q-1)= 4q+2 q+4 points and the desired bound is obtained .So assume that two long lines meet in B . Consider long lines through a point on a 4-secant to Q These meet l in another Baer subline not containing Q. Fourth blocking set must have at least 76 points, so since n=q+1-t, ( k,13)-arcs have k 197
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