On the Generalizations of the Rank Metric over Finite Chain Rings

The rank metric over finite fields has received a lot of attention these last decades. Several works propose generalizations of this metric to finite rings, each one using a particular notion of module theory. The first work that generalizes the rank metric to finite rings defines a new metric over finite principal ideal rings by replacing the notion of dimension of vector spaces by the minimum number of generators of modules. A second work also defines a new metric over Galois rings by using the notion of cardinal of modules, while another idea is to use the length of modules as a generalization of the dimension. In this paper, we study these three generalizations of the rank metric from fields to finite chain rings. We show that the generalizations using the length and the cardinal of modules are decoding equivalent, and give connections between the minimum distances and the packing radii of the three metrics. These links make it possible to show that up to the packing radii, the generalization using the minimum number of generators of modules corrects more errors than the metric using the length and the one defined by the cardinal of modules. Finally, we show that the use of linear codes with the metric based on the minimum number of generators in a McEliece type encryption scheme results in a cryptosystem with smaller public key sizes.