Large (k,3)-arcs in PG(2,19) and the related linear codes

Abstract

A (n,r)-arc in a projective plane PG(2,q) is a set of n points such that some r, but no r+1 of them, are collinear. A (n,r)-arc is called complete if it is not contain in a (n+1,r)-arc. A linear -code over a finite field is a k-dimensional subspace of with minimum hamming distance d and length n. A code with parameters with Griesmer bound , is called Griesmer code. The major aim of this research is to find large size for the complete (k,3)-arcs in the projective plane of order nineteen PG(2,19) using the method of secants distributions, and the disjoint union of arcs, as well as, adding and removing points to (from) particular conic respectively. Also, we find the Griesmer codes that correspond to each large complete (k,3)-arcs, k=29,30,31. We introduced 20 inequivalent (29,3)-arcs up to secant distribution, 10 of them are complete. Also, we introduced 8 inequivalent (30,3)-arcs up to secant distribution, 2 of them are complete. Moreover, we construct 3 inequivalent complete (31,3)-arcs up to secant distribution, and then find the corresponding linear codes to some of the (29,3)-arcs, (30,3)-arcs and (31,3)-arcs. In particular, we established 3 types of Griesmer codes, and we find the weight enumerator that correspond to each one of them.