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In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. We apply the various bounds to the classes of maximal rank distance and quasi maximal rank distance codes.
We define the rank-metric zeta function of a code as a generating function of its normalized $q$-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metric codes. We
Four constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the se
Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM)
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
Let $d, n in mathbb{Z}^+$ such that $1leq d leq n$. A $d$-code $mathcal{C} subset mathbb{F}_q^{n times n}$ is a subset of order $n$ square matrices with the property that for all pairs of distinct elements in $mathcal{C}$, the rank of their differenc