Puncturing-inclusion constructions of optimal Ferrers diagram rank-metric codes using semifield codes

Abstract: Ferrers diagram rank-metric codes are essential components of the multilevel construction of constant dimension codes for random network coding. This paper aims to construct optimal Ferrers diagram rank-metric codes, focusing on the diagram ([3,3,3,5]) with rank distance (3). We construct an optimal ([[3,3,3,5],6,3]_q) Ferrers diagram rank-metric code for every prime power (q), thereby answering an open problem posed by Etzion et al. in 2016~\cite{Etzion2016}. The construction first builds a ([3\times4,4,3]_q) MRD code and then proves that the Delsarte dual of its puncturing on the last column contains a ([3\times3,3,3]_q) MRD code, which comes from semifield codes. We further construct optimal Ferrers diagram rank-metric codes for two infinite families of Ferrers diagrams: the first relies on the ([[3,3,3,5],6,3]_q) construction and holds for every prime power (q), while the second is binary and obtained directly by the puncturing-inclusion construction.