An Assmus-Mattson Theorem for Rank Metric Codes

A $t$-$(n,d,\lambda )$ design over ${\mathbb{F}}_q$, or a subspace design, is a collection of $d$-dimensional subspaces of ${\mathbb{F}}_q^n$, called blocks, with the property that every $t$-dimensional subspace of ${\mathbb{F}}_q^n$ is contained in the same number $\lambda$ of blocks. A collection of matrices in over ${\mathbb{F}}_q$ is said to hold a subspace design 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 subspace design.