An Assmus--Mattson Theorem for Rank Metric Codes
Eimear Byrne, Alberto Ravagnani · SIAM Journal on Discrete Mathematics · 2019
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 $n \times m$ matrices over $\mathbb{F}_{q}$ is said to hold a $t$-design over $\mathbb{F}_{q}$ 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 $\mathbb{F}_{q}$. We show that for $\mathbb{F}_{q^m}$-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 $\mathbb{F}_{q}$-linear matrix MRD codes that are not linear over $\mathbb{F}_{q^m}$. 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.