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arXiv · 2606.12934

On perfect flag-rank metric codes

Gianira N. Alfarano·Usman Mushrraf·Ferdinando Zullo
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Abstract

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 $\delta\in\{3,5,7,9,11\}$, perfect linear flag-rank-metric codes with minimum distance $\delta$ do not exist over $\mathbb{F}_q$ for all sufficiently large $q$.

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BibTeXRIS

Gianira N. Alfarano, Usman Mushrraf, Ferdinando Zullo. 2026-06-11. On perfect flag-rank metric codes. https://arxiv.org/abs/2606.12934

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