An infinite family of non-extendable MRD codes
Solves an open problem in rank-metric coding theory by providing explicit examples of non-extendable MRD codes that are not of maximum length.
The authors introduce the first infinite family of non-extendable MRD codes of length 4 and dimension 2 over a specific field, proving they are self-dual up to equivalence.
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.