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 $\mathbb{F}_q<sup>n$, where $q$ is a prime power and $\mathbb{F}_q$, the finite field of size $q$. In this paper we study the construction on $\mathbb{F}_q<sup>{2k}$ 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 $\mathbb{F}_q<sup>{2k}$ 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 $\mathbb{F}_q<sup>{2k}$ that strongly depends on the characteristic of the field.