An Orbital Construction of Optimum Distance Flag Codes

TL;DR

This paper presents a method to construct optimal distance flag codes with an orbital structure on finite vector spaces, leveraging subgroup actions of the general linear group to achieve maximum distance properties.

Contribution

It introduces a new orbital construction technique for full flag codes with maximum distance, based on subspace codes and group actions, especially from planar spreads.

Findings

Constructed full flag codes with maximum distance using orbital methods.

Provided conditions for orbital structures to yield optimal flag codes.

Demonstrated dependence of construction on the characteristic of the finite field.

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.