On the existence of linear rank-metric intersecting codes

2026-04-02

AI Summary

This study investigates the existence and parameter bounds of linear rank-metric intersecting codes, with a focus on the attainability of the classical bound $n = 2m - 3$. By employing the geometric framework of $q$-systems and scattered $\mathbb{F}_q$-subspaces, the authors relate intersecting codes to strong avoidance properties of their dual subspaces, thereby deriving a tighter upper bound $n \leq 2m - \lfloor (k+4)/2 \rfloor$. In particular, they completely characterize the extremal case $k=3$: they prove that an $[2m-3,3,d]_{q^m/q}$ code can exist only if $m \geq 6$, establish its existence for even $m$ via constructions from maximum scattered subspaces, and rule out the existence of a $[6,3,3]_{q^5/q}$ code for any $q$.

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.

Problem

• rank-metric codes
• intersecting codes
• existence problem
• parameter bounds
• scattered subspaces

Innovation

• rank-metric intersecting codes
• q-systems
• scattered subspaces
• evasive subspaces
• parameter bounds