Unitals in Projective Planes of Order 25

In this paper, results of a non-exhaustive computer search for unitals in the known planes of order twenty-five are reported. The 2-(126, 6, 1) designs associated with newly found unitals are studied in detail. 938 non-isomorphic unital designs are discovered and we show that three of the unital designs are embeddable in two non-isomorphic planes and 239 of them are resolvable. The findings of this study improve some well-known lower bounds on the number of such designs and provide new connections between some pairs of planes. A conjecture concerning the p-ranks of unital designs embedded in planes of order q(2) is formulated

Maximal arcs and disjoint maximat arcs in projective planes of order 16

This paper tabulates the results of a number of computer searches in projective planes of order 16. Maximal arcs of degree 4 are found in all but two of the known planes of order 16 (and their duals). Any such arc yields a resolvable 2-(52,4,1) design that admits at least 52 resolutions. Pairs of disjoint degree 4 maximal arcs are also shown to exist in certain of the planes giving rise to 104-sets of type (4, 8)