Flag codes of maximum distance and constructions using Singer groups

Abstract: In this paper we study flag codes of maximum distance. We characterize these codes in terms of, at most, two relevant constant dimension codes naturally associated to them. We do this first for general flag codes and then particularize to those arising as orbits under the action of arbitrary subgroups of the general linear group. We provide two different systematic orbital constructions of flag codes attaining both maximum distance and size. To this end, we use the action of Singer groups and take advantage of the good relation between these groups and Desarguesian spreads, as well as the fact that they act transitively on lines and hyperplanes.