Flag-rank-metric codes arise as a natural generalization of rank-metric codes in the context of network communication. While recent research has mainly focused on algebraic and structural properties of these codes, the combinatorial geometry underlying the flag-rank metric remains largely unexplored. In this paper, we initiate a detailed investigation of this geometry. We explicitly determine the size of spheres of small flag-rank radius in the space $\mathrm{U}(n,\mathbb{F}_q)$ of upper triangular matrices over the finite field $\mathbb{F}_q$, and consequently obtain formulas for the size of balls of radius at most $3$. Using these enumerative results, we derive a sphere-packing bound for flag-rank-metric codes and introduce the notion of perfect codes with respect to the flag-rank metric. We observe that no non-trivial perfect flag-rank-metric codes exist in $\mathrm{U}(n,\mathbb{F}_q)$ for $n\in\{2,3\}$. We then investigate the possible parameters of perfect codes in higher dimensions. For minimum distance $3$, we obtain a characterization in terms of the codimension of the code, and show that suitable maximum flag-rank distance codes with minimum distance $3$ yield non-trivial perfect codes. For minimum distances $5$ and $7$, we derive explicit quadratic and cubic conditions, respectively, that any perfect code must satisfy. Finally, using asymptotic estimates for balls of fixed radius, we prove that for fixed length $n$ and $δ\in\{3,5,7,9,11\}$, perfect linear flag-rank-metric codes with minimum distance $δ$ do not exist over $\mathbb{F}_q$ for all sufficiently large $q$.

翻译：旗秩度量码作为网络通信背景下秩度量码的自然推广而出现。尽管近期研究主要集中于这些码的代数与结构性质，但旗秩度量背后的组合几何仍未得到充分探索。本文首次对该几何结构展开详细研究。我们明确确定了有限域 $\mathbb{F}_q$ 上上三角矩阵空间 $\mathrm{U}(n,\mathbb{F}_q)$ 中小旗秩半径球面的大小，进而得到半径至多为 $3$ 的球体体积公式。利用这些计数结果，我们推导出旗秩度量码的球包界，并引入关于旗秩度量的完美码概念。我们发现，在 $n\in\{2,3\}$ 时 $\mathrm{U}(n,\mathbb{F}_q)$ 中不存在非平凡完美旗秩度量码。随后我们研究了高维空间中完美码的可能参数。对于最小距离 $3$，我们得到了基于码余维数的刻画，并证明具有最小距离 $3$ 的适定极大旗秩距离码可产生非平凡完美码。对于最小距离 $5$ 和 $7$，我们分别推导出完美码必须满足的显式二次及三次条件。最后，利用固定半径球体的渐近估计，我们证明了对于固定长度 $n$ 及 $\delta\in\{3,5,7,9,11\}$，当 $q$ 充分大时，域 $\mathbb{F}_q$ 上不存在具有最小距离 $\delta$ 的完美线性旗秩度量码。