Extend the binary puncturing-inclusion construction to general finite fields

Extend the construction of optimal binary Ferrers diagram rank-metric codes obtained directly by the puncturing-inclusion construction to general finite fields, and determine for which prime powers q an optimal code exists for the corresponding infinite family of Ferrers diagrams.

Background

The paper directly constructs optimal binary Ferrers diagram rank-metric codes for the infinite family of diagrams $\mathcal F_{m,t}=[m,m,\ldots,m,m+t]$, with $m$ copies of $m$, over $\mathbb F_2$, using the puncturing-inclusion construction. The authors ask whether this construction can be generalized from the binary field to arbitrary finite fields.

The problem also asks for a classification of the prime powers $q$ for which an optimal Ferrers diagram rank-metric code exists for the corresponding infinite family. This question is motivated by the paper’s observation that optimality for related Ferrers diagrams is not guaranteed uniformly across all finite fields.