Rank-metric codes and their duality theory

Abstract

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. 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 first proved by Delsarte employing the theory of association schemes and regular semilattices. We also show that our identities imply as a corollary the original MacWilliams identities established by Delsarte. We describe 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. Then we study optimal anticodes in the rank metric, describing them in terms of optimal codes (namely, 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 both a recursive and an explicit formula for the number of k × m matrices over a finite field with given rank and h-trace. * 1 special case of Delsarte codes. It is however not clear in general how the duality theories of these two families of codes relate to each other. This is one of the questions that we address in this work.Both linear Delsarte and Gabidulin codes have interesting applications in information theory. Recently it was shown how to employ them for error correction in non-coherent linear network coding and in coherent linear network coding under an adversarial channel model (see e.g. [21] and the references within). Rank-metric codes were also proposed to secure a network coding communication system against an eavesdropper in a universal way (see [22] for details).Motivated by these applications, in this paper we study the duality theories of linear Delsarte and Gabidulin codes, mainly focusing on their MacWilliams identities. In coding theory, a MacWilliams identity establishes a relation between metric properties of a code and metric properties of the dual code. MacWilliams identities exist for several types of codes and metrics. As Gluesing-Luerssen observed in [12], 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 hand, 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.A different viewpoint on MacWilliams identities for general additive codes was recently proposed by Gluesing-Luerssen in [12]. The approach is based on character theory and partitions of groups. See also [14] for a character-theoretic approach to MacWilliams identities for the rank and the Hamming metric.Both the theory of association sch...

Rank-metric codes, linear sets, and their duality

“…The (Delsarte) dual of an MRD code is again an MRD code. For more information about duality of rank-metric codes, we refer to [17].…”

“…We recall the result of Delsarte [8] extending the classical MacWilliams identities for linear codes to rank metric codes. We state instead the following more convenient recursion from [17]. Though in this paper we require only the case m = n, we state the more general result.…”

An Assmus--Mattson Theorem for Rank Metric Codes

“…The MacWilliams duality theorem for rank metric codes [16,28] then yields a system of d − t equations in at most d − t unknowns. In fact, using Theorem 4, for ℓ = 0, ..., d − t − 1 these equations are explicitly given by…”

“…The weight distributions of a linear matrix code and its dual are related by the rank metric MacWilliams identities [16]. We will use the following formulation from [28,Theorem 31].…”

Puncturing maximum rank distance codes

“…Although MRD codes are very interesting by their own and they caught the attention of many researchers in recent years [1,5,29,30], such codes also have practical applications in error-correction for random network coding [16,25,32], space-time coding [33] and cryptography [15,31].…”