Communications in Mathematics and Applications Vol. 10, No. 3, pp. 361–368, 2019 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com DOI: 10.26713/cma.v10i3.1275 Research Article Weighted (k,n)-arcs of Type (n – q,n) and Maximum Size of (h,m)-arcs in PG(2,q ) Mustafa T. Yaseen 1 , Ali Hasan Ali 2, * and Ibrahim A. Shanan 3 1 Department of Business Administration, Shatt Al-Arab University College, Basrah, Iraq 2 Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah, Iraq 3 Business Management Techniques Department, Management Technical College, Southern Technical University, Basrah, Iraq *Corresponding author: aliaha1@yahoo.com Abstract. In this paper, we introduce a generalized weighted (k, n)-arc of two types in the projective plane of order q, where q is an odd prime number. The sided result of this work is finding the largest size of a complete (h, m)-arcs in PG(2, q), where h represents a point of weight zero of a weighted (k, n)-arc. Also, we prove that a ( q( q-1) 2 + 1, q+1 2 ) -arc is a maximal arc in PG(2, q). Keywords. (k, n)-arcs; Weighted (k, n)-arc; PG(2, q); PG(2, prime); Projective plane; Galois plane; Algebraic geometry MSC. 51E21; 14N99; 51A05 Received: July 15, 2019 Accepted: September 13, 2019 Copyright © 2019 Mustafa T. Yaseen, Ali Hasan Ali and Ibrahim A. Shanan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The concept of weighted ( k, n)-arcs was originally established by Tallini-Scafati [10] in 1971. In order nine Galois plane, Wilson [11] in 1986 mentioned that there is a (88, 14, f )-arc of class (11, 14). In addition, a (10, 7, f )-arc of type (4, 7) in PG(2, 3) was proved by Wilson. In 1989, Hameed [4] studied the existence and non-existence of weighted ( k, n)-arcs in PG(2, 9) as well as he proved that there exist a (81, 12, f )-arc of type (9, 12) and a (85, 13, f )-arc of type (10, 13). Hill and Love [6] in 2003 discussed the (22, 4)-arcs in PG(2, 7). They discussed the optimal linear codes and arcs in projective geometries. In 2012, Hamilton [5] constructed a new maximal arcs