An Assmus–Mattson theorem for rank metric codes

A t-(n, d, λ) design over Fq, or a subspace design, is a collection of d-dimensional subspaces of Fⁿ _q , called blocks, with the property that every t-dimensional subspace of Fⁿ _q is contained in the same number λ of blocks. A collection of n × m matrices over Fq is said to hold a t-design over Fq if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a t-design over Fq. We show that for F_qm-linear vector rank metric codes, the property of a code being maximum rank distance (MRD) is equivalent to its minimal weight codewords holding trivial subspace designs, and show that this characterization does not hold for Fq-linear matrix MRD codes that are not linear over F_qm. Finally, using arguments based on covering radius and external distance, we establish various existence results that apply to both the rank and the Hamming metric.