In the network coding framework, given a prime power $q$ and the vector space $\mathbb{F}_q^n$, a constant type flag code is a set of nested sequences of $\mathbb{F}_q$-subspaces (flags) with the same increasing sequence of dimensions (the type of the flag). If a flag code arises as the orbit under the action of a cyclic subgroup of the general linear group over a flag, we say that it is a cyclic orbit flag code. Among the parameters of such a family of codes, we have its best friend, that is the largest field over which all the subspaces in the generating flag are vector spaces. This object permits to compute the cardinality of the code and estimate its minimum distance. However, as it occurs with other absolute parameters of a flag code, the information given by the best friend is not complete in many cases due to the fact that it can be obtained in different ways. In this work, we present a new invariant, the best friend vector, that captures the specific way the best friend can be unfolded. Furthermore, throughout the paper we analyze the strong underlying interaction between this invariant and other parameters such as the cardinality, the flag distance, or the type vector, and how it conditions them. Finally, we investigate the realizability of a prescribed best friend vector in a vector space.

翻译：在网络编码框架下，给定一个素数幂$q$和向量空间$\mathbb{F}_q^n$，常类型旗码是一组$\mathbb{F}_q$-子空间（旗）的嵌套序列，这些序列具有相同的递增维数序列（旗的类型）。若一个旗码作为一般线性群的循环子群作用于某个旗的轨道出现，则称其为循环轨道旗码。在这类码族的参数中，我们拥有其最佳友元，即生成旗中所有子空间均视为向量空间的最大域。该对象可用于计算码的基数并估计其最小距离。然而，与旗码的其他绝对参数类似，由于最佳友元可通过不同方式获取，其提供的信息在许多情况下并不完整。本文提出一个新不变量——最佳友元向量，用以刻画最佳友元的具体展开方式。进一步地，我们分析了这一不变量与基数、旗距离或类型向量等其他参数之间的强底层相互作用，以及它如何制约这些参数。最后，我们探讨了在向量空间中实现给定最佳友元向量的可能性。