On Generalized Galois Cyclic Orbit Flag Codes

TL;DR

This paper studies generalized Galois cyclic orbit flag codes, focusing on generating flags with subfield subspaces, and explores their parameters, properties, and relationship with Galois flag codes.

Contribution

It introduces and analyzes generalized Galois flag codes with subfield subspaces, extending previous work on Galois flag codes and examining their parameters and properties.

Findings

Characterization of generalized Galois flag codes

Analysis of parameters and properties of these codes

Relationship established with underlying Galois flag codes

Abstract

Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper we present a new contribution to the study of such codes started in arXiv:2102.00867, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.