An infinite family of non-extendable MRD codes

2026-03-29

AI Summary

This study investigates whether maximum rank distance (MRD) codes can be extended while preserving optimality, with a focus on constructing MRD codes that are non-extendable yet fall short of the maximum possible length. By linking non-extendability to scattered subspaces with respect to hyperplanes in finite geometry and leveraging tools from linear algebra over finite fields and rank-metric coding theory, the authors construct, for the first time, an infinite family of non-extendable $[4,2,3]_{q^5/q}$ MRD codes. Moreover, they prove that these codes are self-dual up to equivalence. This work not only presents the first known infinite family of non-extendable MRD codes but also advances the understanding of their existence conditions and structural properties.

Abstract

In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one while preserving its optimality. This work investigates $\mathbb{F}_{q^m}$-linear MRD codes that are non-extendable but do not attain the maximum possible length. Geometrically, these correspond to scattered subspaces with respect to hyperplanes that are maximal with respect to inclusion but not of maximum dimension. By exploiting this geometric connection, we introduce the first infinite family of non-extendable $[4,2,3]_{q^5/q}$ MRD codes. Furthermore, we prove that these codes are self-dual up to equivalence.

Problem

• MRD codes
• non-extendable
• rank-metric codes
• maximum length
• scattered subspaces

Innovation

• non-extendable MRD codes
• scattered subspaces
• self-dual codes
• rank-metric codes
• geometric construction