On the existence of linear rank-metric intersecting codes

Abstract: Intersecting codes are a classical object in coding theory whose rank-metric analogue has recently been introduced. Although the definition formally parallels the Hamming-metric case, the structure and parameter constraints of rank-metric intersecting codes exhibit substantially different behavior. It was previously shown that a nondegenerate $[n,k,d]{q^m/q}$ rank-metric intersecting code must satisfy $2k-1 \le n \le 2m-3$, and the tightness of the upper bound was left open. Using the geometric interpretation of rank-metric codes via $q$-systems, we prove that the dual subspace associated with a rank-metric intersecting code must satisfy strong evasiveness properties. This connection allows us to derive new restrictions on the parameters of such codes and to show that the bound $n=2m-3$ can be attained only when $k=3$ and $m\ge 6$. More generally, we show that $n \leq 2m-\lfloor(k+4)/2\rfloor$. Moreover, we obtain a geometric characterization of these extremal codes in terms of scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}{q^m}^3$. As a consequence, the existence problem for $[2m-3,3,d]{q^m/q}$ rank-metric intersecting codes is reduced to the existence of scattered subspaces of dimension $m+3$. Using known constructions of maximum scattered subspaces, we derive existence results when $m$ is even. Finally, we prove that $[6,3,3]{q^5/q}$ rank-metric intersecting codes do not exist for any prime power $q$, thus resolving an open problem posed by Bartoli et al. in 2025.

• The paper establishes a new upper bound for rank-metric intersecting codes by leveraging dual q-systems and the properties of scattered and evasive subspaces.
• It utilizes a geometric framework based on q-systems to characterize non-2-spannable codes, linking existence conditions to strict combinatorial and finite geometry constraints.
• The paper proves existence for even m ≥ 6 and demonstrates non-existence for specific parameters (e.g., [6,3,3]_{q^5/q}), highlighting sharp limitations in the code structure.

Existence and Parameter Bounds of Linear Rank-Metric Intersecting Codes

Introduction

The paper "On the existence of linear rank-metric intersecting codes" (2604.02004) investigates the existence and structural properties of linear rank-metric codes whose nonzero codewords' rank supports pairwise intersect nontrivially—so-called rank-metric intersecting codes. Whereas the concept of intersecting codes is classical in the Hamming-metric setting, the rank-metric analogue exhibits more nuanced and restrictive parameter regimes due to intricate geometric constraints arising from finite geometry and $q$-system structure.

This essay articulates the paper's main contributions, including the derivation of new upper bounds on code parameters, geometric characterizations via evasive and scattered subspaces, and a resolution of specific open existence questions. Emphasis is placed on the geometric and combinatorial frameworks deployed, as well as the implications for existence theorems and non-existence results.

Geometric Framework and Definitions

A linear rank-metric code over $F_{q^m}$ is defined as an $F_{q^m}$-subspace of $F_{q^m}^n$, equipped with the rank distance. The rank support of a codeword is the row span (over $F_q$) of its vectorization in a chosen $F_{q^m}/F_q$ basis, and plays a pivotal role in the intersecting property: a code is rank-metric intersecting if all pairs of nonzero codewords have rank supports with nontrivial intersection.

The geometric interpretation uses $q$-systems: an $[n,k,d]_{q^m/q}$ $q$-system is an $F_q$-subspace $F_{q^m}$0 of dimension $F_{q^m}$1, with a correspondence between nondegenerate rank-metric codes and such systems via the images of generator matrices. The minimum rank distance is described as

$F_{q^m}$2

The equivalence class of a code is given by action via $F_{q^m}$3.

A critical notion is the 2-spannable property: a $F_{q^m}$4-system is 2-spannable if it equals the sum of its intersections with two $F_{q^m}$5-hyperplanes. The fundamental geometric result is that a code is rank-metric intersecting if and only if its associated $F_{q^m}$6-system is not 2-spannable.

Parameter Bounds, Evasiveness, and Scattered Subspaces

Prior work established bounds $F_{q^m}$7 on the parameters of nondegenerate $F_{q^m}$8 rank-metric intersecting codes (with $F_{q^m}$9), with the lower bound tight but the upper bound's attainability previously unresolved.

The paper's principal theoretical advancement is to leverage the dual $F_{q^m}$0-system's geometric and combinatorial properties, particularly those associated with evasive and scattered subspaces, to refine these bounds. The dual $F_{q^m}$1-system $F_{q^m}$2 must be a $F_{q^m}$3-evasive $F_{q^m}$4-subspace. This property translates to strict constraints on the dimension and structure of $F_{q^m}$5 given the parameters $F_{q^m}$6, strengthening the upper bound on $F_{q^m}$7 relative to $F_{q^m}$8 and $F_{q^m}$9.

The central new bound established is

$F_{q^m}^n$0

Importantly, the maximal admissible length $F_{q^m}^n$1 is only attainable when $F_{q^m}^n$2 and $F_{q^m}^n$3; in this case, $F_{q^m}^n$4 must be a scattered $F_{q^m}^n$5-subspace of $F_{q^m}^n$6 of dimension $F_{q^m}^n$7. This characterization sets the minimum distance of such codes to $F_{q^m}^n$8.

The existence of such scattered subspaces is known for even $F_{q^m}^n$9 by constructions in [bartoli2018maximum, csajbok2017maximum], confirming that the upper bound is attained for all even $F_q$0. For odd $F_q$1, existence remains unresolved, contingent on the existence of sufficiently large scattered subspaces.

Explicit Existence and Non-Existence Results

By correlating extremal codes with scattered subspaces, the paper reduces the existence problem for $F_q$2 rank-metric intersecting codes to the known theory of scattered subspaces.

• Existence for even $F_q$3: The presence of maximum scattered subspaces (dimension $F_q$4) in $F_q$5 implies the existence of scattered subspaces of dimension $F_q$6, and hence $F_q$7 rank-metric intersecting codes.
• Non-existence for $F_q$8: The paper resolves the previously open case for $F_q$9, $F_{q^m}/F_q$0, $F_{q^m}/F_q$1 by leveraging detailed combinatorial arguments on the associated linear sets and projective planes. Applying weight distribution properties of $F_{q^m}/F_q$2-linear sets and ruling out the requisite scattered subspaces, it is shown that such a code cannot exist for any $F_{q^m}/F_q$3.

Theoretical and Practical Implications

These findings have several structural and combinatorial implications:

• Universal Upper Bound Except for $F_{q^m}/F_q$4: The tightness of $F_{q^m}/F_q$5 is highly restricted, indicating that intersecting codes are far rarer in the rank-metric domain compared to the Hamming-metric case.
• Reduction to Geometric Problems: The identification of extremal codes with scattered subspaces means that further progress on existence for odd $F_{q^m}/F_q$6 or other parameter ranges depends directly on advances in finite geometry, particularly in the construction and classification of scattered and evasive subspaces.
• Negative Existence Results: The non-existence theorem for $F_{q^m}/F_q$7 robustly resolves a structural gap and demonstrates that in some parameter ranges, intersecting codes are precluded by deep combinatorial contradictions.
• Potential Extensions: The methods generalize to the context of $F_{q^m}/F_q$8-scattered subspaces, suggesting further connections between the theory of $F_{q^m}/F_q$9-analogs of combinatorial designs and extremal coding theory. Improvements in scattered subspace constructions (especially for odd $q$0) may yield new code existence results.

Conclusion

The paper advances the understanding of intersecting codes in the rank-metric setting by establishing new parameter bounds, providing precise geometric characterizations, and settling critical existence questions. It demonstrates that the extremal parameter regime is sharply limited to a specific configuration where the dual $q$1-system is a scattered subspace and proves both positive existence results (for even $q$2) and a comprehensive non-existence result in the critical open case $q$3. Future progress hinges on deeper developments in the geometry of $q$4-systems, evasive, and scattered subspaces, with open questions remaining for odd $q$5 and parameter values adjacent to the established maxima.