On perfect flag-rank metric codes

2026-06-11

AI Summary

This study investigates the existence and combinatorial-geometric structure of perfect codes under the flag rank metric. By precisely computing the volumes of small-radius flag rank balls for the first time and integrating combinatorial enumeration in spaces of upper-triangular matrices over finite fields, algebraic coding theory, and asymptotic analysis, the authors establish a sphere-packing bound and formally define perfect flag rank codes. Their main contributions include proving the nonexistence of nontrivial perfect codes for dimensions $n=2$ and $n=3$, characterizing parameter conditions for existence when the minimum distance $\delta$ equals 3, 5, or 7, and further demonstrating that for any fixed $n$ and $\delta \in \{3,5,7,9,11\}$, no linear perfect flag rank codes exist when the field size $q$ is sufficiently large.

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

Problem

• flag-rank metric
• perfect codes
• sphere-packing bound
• minimum distance
• upper triangular matrices

Innovation

• flag-rank metric
• perfect codes
• sphere-packing bound
• combinatorial geometry
• upper triangular matrices