Designs over the binary field from the complete monomial group

All known direct constructions of designs over finite fields arise by subgroups of the normalizer of a Singer cycle, by lifted special linear groups, or lifted general linear groups. Other types of non-trivial automorphism groups for which designs over finite fields do exist have not been mentioned so far. In this paper we construct the first non-trivial designs over finite fields admitting the lifted complete monomial group as group of automorphisms.