An infinite family of non-extendable MRD codes

TL;DR

This paper introduces an infinite family of non-extendable MRD codes that are optimal yet cannot be extended further, revealing new geometric connections and self-duality properties.

Contribution

It constructs the first infinite family of non-extendable MRD codes that are not of maximum length and proves their self-duality up to equivalence.

Findings

Introduces the first infinite family of non-extendable MRD codes.

Establishes a geometric interpretation via scattered subspaces.

Proves these codes are self-dual up to equivalence.

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.