Improved Bounds on ？

An ( )-arc is a set of n points of a projective plane such that some r, but no of them, are collinear. The maximum size of an ( )-arc in PG(2, q) is denoted by (2, q). In this paper, a new (286, 16)-arc in PG(2,19), a new (341, 15)-arc, and a (388, 17)-arc in PG(2,25) are constructed, as well as a (394, 16)-arc, a (501, 20)-arc, and a (532, 21)-arc in PG(2,27). Tables with lower and upper bounds on (2, 25) and (2, 27) are presented as well. The results are obtained by nonexhaustive local computer search. 1. Introduction Let denote the Galois field of elements, and let be the vector space of row vectors of length three with entries in . Let be the corresponding projective plane. The points of are the 1-dimensional subspaces of . Subspaces of dimension two are called lines. The number of points in is and so is the number of lines. There are points on every line and lines through every point. Definition 1. An -arc is a set of points of a projective plane such that some , but no of them, are collinear. The maximum size of a -arc in is denoted by . Definition 2. An -blocking set in is a set of points such that every line of intersects in at least points, and there is a line intersecting in exactly points. Note that an -arc is the complement of a -blocking set in a projective plane and conversely. Definition 3. Let be a set of points in any plane. An -secant is a line meeting in exactly points. A 0-secant is also called skew line. Define as the number of -secants to a set . In terms of , the definitions of an -arc and an -blocking set become the following:？an -arc is a set of points of a projective plane for which for , and when ; ？an -blocking set is a set of points of a projective plane for which for , and when . In 1947, Bose [1] proved that From the result of Barlotti [2], it follows that for odd and there exists an For background on , see Hirschfeld [3]. A survey of -arcs with the best known results was presented in [4]. After this publication, many improvements were obtained in [5–7]. Summarizing these improvements, Ball and Hirschfeld [8] presented a new table with bounds on for . As we can see from these tables, the exact values of are known only for ？？ (see Table 1). Table 1: Values of . Some new improvements were made in recent years. A (79,6) arc in PG(2,17) and a (126,8) arc in PG(2,19) are given in [9]. A -arc, and a -arc, a -arc in PG(2,17) and a -arc and a -arc in PG(2,19) have been presented in [10]. In 2010, Gulliver constructed an optimal (78,8) arc in PG(2,11) (see [11]). A table for , is maintained by Ball [11]. To obtain good -arcs, we