Classification of some arcs of type (m,n)‎ in PG(2,19)‎

Abstract EN

This paper conclude the study of the weights of points and lines of(k,n)-arcs from 1 0 L  until 381 0 L  , since the number of points and linesof this plane is very large, so we divide this search in to three parts.

The first part contained the existence of these arcs from10L  to171 0 L  and we show that when 171 0 L  the (210,39;f)-arc of type(20,39) exist and all the other cases from 1 0 L  to 170 0 L  does not exist.In the second part we studied these arcs from 340 0 L  to 381 0 L and we saw that there are two types of these arcs when 342 0 L  and361 0 L  which are respectively (39,21;f)- arc of type (2,21) and (20,20;f)-arc of type (1,20), then proved the theorem of these special cases aregiven.Finally in the last step we prove that (k,n;f)- arc of type (n-19,n)does not exist in PG(2,19) for all values of 0 Lfrom 362 to 381.