Given $\mathbb{F}_q$ the finite field with $q$ elements and an integer $n\geq 2$, a flag is a sequence of nested subspaces of $\mathbb{F}_q^n$ and a flag code is a nonempty set of flags. In this context, the distance between flags is the sum of the corresponding subspace distances. Hence, a given flag distance value might be obtained by many different combinations. To capture such a variability, in the paper at hand, we introduce the notion of distance vector as an algebraic object intrinsically associated to a flag code that encloses much more information than the distance parameter itself. Our study of the flag distance by using this new tool allows us to provide a fine description of the structure of flag codes as well as to derive bounds for their maximum possible size once the minimum distance and dimensions are fixed.

翻译：$\ mathbb{F ⁇ {F ⁇ q$$ $q $, 带有 $q美元的元素和整数 $\ geq 2$ 的限定字段, 国旗是嵌套子空间的序列 $mathbb{F ⁇ q ⁇ n$, 国旗代码是非空的旗帜组。 在这方面, 国旗之间的距离是相应的子空间距离的总和。 因此, 许多不同的组合都可能获得一定的旗帜距离值 。 为了捕捉这种变异性, 在手头的纸张中, 我们引入了距离矢量的概念, 认为距离矢量是一个与国旗代码内在关联的代数对象, 该代数包含的信息比距离参数本身要多得多。 我们通过使用这一新工具对国旗距离进行的研究, 使我们能够对国旗代码的结构做出精细的描述, 并在最小距离和尺寸固定之后, 为其最大可能大小的界限产生界限 。