On the 2-ranks of a class of unitals

Let UθUθ be a unital defined in a shift plane of odd order q2q2, which are constructed recently in [40]. In particular, when the shift plane is desarguesian, UθUθ is a special Buekenhout–Metz unital formed by a union of ovals. We investigate the dimensions of the binary codes derived from UθUθ. By using Kloosterman sums, we obtain a new lower bound on the aforementioned dimensions which improves Leung and Xiang's result [32] and [33]. In particular, for q=3mq=3m, this new lower bound equals 23(q3+q2−2q)−1 for even m   and 23(q3+q2+q)−1 for odd m.