Maximal Ferrers Diagram Codes: Constructions and Genericity Considerations

This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension of a rank-metric code with given specified Ferrers diagram shape and rank distance. While the conjecture in its generality is wide open, several cases have been established in the literature. This paper contributes further cases of Ferrers diagrams and ranks for which the conjecture holds true. In addition, probabilities for maximal Ferrers diagram codes and MRD codes are investigated. It is shown that for growing field size the limiting probability for the event that randomly chosen matrices with given shape generate a maximal Ferrers diagram code, depends highly on the Ferrers diagram. For instance, for [m× 2]-MRD codes with rank 2 this limiting probability is close to 1/e.