Code distances: a new family of invariants of linear codes

In this paper, we introduce code distances, a new family of invariants for linear codes. We establish some properties and prove bounds on the code distances, and show that they are not invariants of the matroid (for a linear block code) or $q$-polymatroid (for a rank-metric code) associated to the code. By means of examples, we show that the code distances allow us to distinguish some inequivalent MDS or MRD codes with the same parameters. We also show that no duality holds, i.e., the sequence of code distances of a code does not determine the sequence of code distances of its dual. Further, we define a greedy and an asymptotic version of code distances. Finally, we relate these invariants to other invariants of linear codes, such as the maximality degree, the covering radius, and the partial distances of polar codes.