Unitals in Projective Planes of Order 25

Abstract

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.