Optimal Ferrers diagram rank-metric codes from MRD codes

Subspace codes, motivated by their extensive application in random network coding, have become one of central topics in algebraic coding theory during the last 10 years. Due to the significant application in subspace codes, Ferrers diagram rank-metric (FDRM) codes also have drawn a lot of attention. In this paper, we introduce two new constructions based on subcodes of MRD codes. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. Apply the first construction to solve the optimality of [ℱ,4]_q -FDRM codes, where ℱ=[2,2,4,4,… ,2l,2l] , which was raised in Etzion et al. (IEEE Trans Inf Theory 62:1616–1630, 2016). By the restricted Gabidulin codes and improving the way to select the subcodes, the second construction is presented, which unifies and generalizes all known constructions based on subcodes of Gabidulin codes. By the second construction, we can give new families of optimal FDRM codes, whose numbers of codewords are unequal to q^v_0 . This paper also shows new families of FDRM codes whose optimality cannot be obtained by the constructions based on subcodes of 𝔽_q^m -linear MRD codes.