An Orbital Construction of Optimum Distance Flag Codes

Abstract

Flag codes are multishot network codes consisting of sequences of nested subspaces (flags) of a vector space $\\mathbb{F}_q^n$, where $q$ is a prime power and $\\mathbb{F}_q$, the finite field of size $q$. In this paper we study the construction on $\\mathbb{F}_q^{2k}$ of full flag codes having maximum distance (optimum distance full flag codes) that can be endowed with an orbital structure provided by the action of a subgroup of the general linear group. More precisely, starting from a subspace code of dimension $k$ and maximum distance with a given orbital description, we provide sufficient conditions to get an optimum distance full flag code on $\\mathbb{F}_q^{2k}$ having an orbital structure directly induced by the previous one. In particular, we exhibit a specific orbital construction with the best possible size from an orbital construction of a planar spread on $\\mathbb{F}_q^{2k}$ that strongly depends on the characteristic of the field.