The geometry of rank-metric codes
This work provides a new theoretical foundation for rank-metric codes, benefiting coding theorists by unifying and extending known results, but it is primarily theoretical and incremental.
The paper develops a geometric framework for matrix rank-metric codes using generator tensors and slice spaces, establishing a correspondence between code equivalence classes and systems, and deriving Delsarte-type identities. It introduces generalized weights, studies faithful and one-weight codes, and links additive Hamming-metric codes to rank-metric codes.
In this paper, we develop a geometric framework for matrix rank-metric codes based on generator tensors and their slice spaces. To every nondegenerate matrix rank-metric code, we associate two systems, which translate metric properties of the code into geometric conditions involving intersections with hyperplanes. This leads to a correspondence between equivalence classes of nondegenerate matrix rank-metric codes and equivalence classes of systems, as well as to Delsarte-type incidence identities relating the rank distribution of a code over a finite field to those of its associated systems. As an application, we introduce generalized weights through the notion of evasive systems, study faithful and one-weight codes over finite fields, and recover known bounds and results from the theory of semifields. Finally, we use this framework to associate additive Hamming-metric codes with matrix rank-metric codes and show that several metric properties are preserved under this correspondence.