Rank-Metric Codes and $q$-Polymatroids

We study some algebraic and combinatorial invariants of rank-metric codes, specifically generalized weights. We introduce q-polymatroids, the q-analogue of polymatroids, and develop their basic properties. We show that rank-metric codes give rise to q-polymatroids, and that several of their structural properties are captured by the associated combinatorial object. Introduction and Motivation Due to their application in network coding, rank-metric codes have received a lot of interest over the past years. Originally studied by Delsarte [4] and later independently by Gabidulin [5] and Roth [20], their practical application was made concrete by Kötter and Kschischang [12]. Studies have been not only into the practical application of rank-metric codes, but also into the algebra and combinatorics related these structures. This paper is a contribution to the latter line of study. Our contributions can be placed in two directions: the first is to study certain properties of rank-metric codes, the second is to study the link with other combinatorial objects. In this paper, we will see an example of both: we study generalized weights of rank-metric codes, and we show that rank-metric codes are examples of q-polymatroids, the q-analogue of a polymatroid, that we define here. We begin in Section 1 with defining rank-metric codes and vector rank-metric codes. We focus on notions of equivalence between the various objects. Generalized weights were studied previously by various researchers. Several definitions have been around. Two of the first were for vector rank-metric codes by Kurihara, Matsumoto and Uyematsu [13] and by Oggier and Sboui [14]. More definitions, plus equivalence between the definitions, are due to Jurrius and Pellikaan [7] and Mart́ınez-Peñas and Matsumoto [10]. Ravagnani [18] generalized the definitions and the duality theory to matrix rank-metric codes via rank-metric anticodes. In our paper, we develop further the study into generalized weights, tying together several previously known results on the subject. Since some of the previous definitions were for vector rank-metric codes and others for rank-metric codes in general, we start with discussing similarities and differences, illustrated with examples. This is the subject of Section 2. In Section 3, we prove a relation between generalized weights for codes in the Hamming metric, vector rank-metric codes, and matrix rank-metric codes in general.