Maximal arcs in projective planes of order 16 and related designs

Abstract

The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52, 4, 1) designs arising from previously known maximal (52, 4)-arcs, as well as some newly discovered maximal (52, 4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(1.20,8,1) design associated with a maximal (1.20,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose-Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52, 4, 1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.