Cyclic Orbit Flag Codes

TL;DR

This paper investigates cyclic orbit flag codes in network coding, determining their size, bounds on their distance, and analyzing special classes like Galois and optimum distance codes for improved performance.

Contribution

It introduces methods to compute the size and bounds of cyclic orbit flag codes and characterizes special classes with optimal distance properties.

Findings

Cardinality of cyclic orbit flag codes can be explicitly determined.

Bounds for the distance of these codes are established using the concept of the best friend subfield.

Galois and optimum distance cyclic orbit flag codes achieve extremal distance values.

Abstract

In network coding, a flag code is a set of sequences of nested subspaces of $\mathbb{F}_q^n$, being $\mathbb{F}_q$ the finite field with $q$ elements. Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of $\mathbb{F}_q^n$ are called cyclic orbit flag codes. Inspired by the ideas in arXiv:1403.1218, we determine the cardinality of a cyclic orbit flag code and provide bounds for its distance with the help of the largest subfield over which all the subspaces of a flag are vector spaces (the best friend of the flag). Special attention is paid to two specific families of cyclic orbit flag codes attaining the extreme possible values of the distance: Galois cyclic orbit flag codes and optimum distance cyclic orbit flag codes. We study in detail both classes of codes and analyze the parameters of the respective subcodes that still have a cyclic orbital structure.