Flag codes of maximum distance and constructions using Singer groups

TL;DR

This paper investigates maximum distance flag codes, characterizes them via associated constant dimension codes, and introduces systematic constructions using Singer groups and their relation to spreads and transitive actions.

Contribution

It provides a characterization of maximum distance flag codes and introduces new systematic constructions using Singer groups and their properties.

Findings

Characterization of maximum distance flag codes in terms of constant dimension codes

Two systematic orbital constructions of flag codes with maximum distance and size

Utilization of Singer groups and their relation to spreads for code construction

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.