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 Gluesing-Luerssen et al. (Adv Math Commun 9(2):177–197, 2015), 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.

在网络编码中，标志码是\({{\mathbb {F}}}_q^n\) 的一组嵌套子空间序列，其中\({{\mathbb {F}}}_q\)是有限的具有q 个元素的字段。定义为作用于\({{\mathbb {F}}}_q^n\)标志的一般线性群的循环子群的轨道的标志码称为循环轨道标志码。受到 Gluesing-Luerssen 等人的想法的启发。(Adv Math Commun 9(2):177–197, 2015)，我们确定循环轨道标志码的基数，并借助最大子域提供其距离的界限，在该子域上，标志的所有子空间都是向量空间（最好的朋友国旗）。特别注意获得距离的极端可能值的两个特定循环轨道标志码系列：伽罗瓦循环轨道标志码和最佳距离循环轨道标志码。我们详细研究了这两类代码，并分析了仍然具有循环轨道结构的各个子代码的参数。