Rank-metric codes and their duality theory

We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter. We also give an elementary proof of MacWilliams identities for the general case of Delsarte rank-metric codes, in a form that never appeared in the literature. The identities which we derive are very easy to handle, and allow us to re-establish in a very concise way the main results of the theory of rank-metric codes. We study how the minimum and maximum rank of a rank-metric code relate to the minimum and maximum rank of the dual code, giving some bounds and characterizing the codes attaining them. We also study optimal anticodes in the rank metric, describing them in terms of MRD codes. In particular, we prove that the dual of an optimal anticode is an optimal anticode. Finally, as an application of our results to a classical problem in enumerative combinatorics, we derive a recursive formula for the number of k ×m matrices over a finite field with given rank and h-trace. Introduction In [6] Delsarte defines rank-metric codes as sets of matrices of given size over a finite field Fq. The distance between two matrices is given by the rank of their difference. Interpreting matrices as bilinear forms, Delsarte studies rank-metric codes as association schemes, whose adjacency algebra yields the so-called MacWilliams transform of distance enumerators of codes. The results of [6] are based on the general theory of designs and codesigns in regular semilattices developed in [5]. In coding theory, a MacWilliams identity establishes a relation between metric properties of a code and metric properties of the dual code. More generally, in the sequel by “duality theory” we mean a series of results which relate a code to the dual code. MacWilliams identities exist for several types of codes and metrics. As Gluesing-Luerssen observed in [11], association schemes provide the most general approach to MacWilliams identities, and apply to both linear and non-linear codes (see [4], [3] and [7]). On the other side, the machinery of association schemes and of the related Bose-Mesner algebras is a very elaborated mathematical tool. Several authors proved independently the MacWilliams identities for the various types of codes in less sophisticated ways. The author was partially supported by the Swiss National Science Foundation through grant no. 200021 150207.