Rank distribution of Delsarte codes

In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for Delsarte rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a result by Delsarte on the rank distribution of MRD codes. In the general case of codes of positive defect, we show that the rank distribution is determined by the parameters of the code, together the number of codewords of small rank. Moreover, we prove that if the rank defect of a code and its dual are both one, and the dimension satisfies a divisibility condition, then the number of minimum-rank codewords and dual minimum-rank codewords is the same. Finally, we discuss how our results specialize to Gabidulin codes. Introduction Rank-metric codes were first introduced in coding theory by Delsarte in [5]. They are sets of matrices of fixed size, endowed with the rank distance. Rank-metric codes are of interest within network coding, public-key cryptography, and distributed storage, where they stimulated a series of works aimed at better understanding their properties. In this paper, we study the rank distribution of rank-metric codes. We always assume that the codes are linear, and often refer to them as Delsarte codes. The study of the weight distribution of a code is a topic of current interest in coding theory, where several authors have studied the case of linear codes endowed with the Hamming distance. In particular, it is a classical result that the weight distribution of an MDS code is determined by its parameters. The so-called MRD codes are the analogue of MDS codes in the context of Delsarte codes. They were introduced in [5] by Delsarte, who also showed that Part of the work was done while J. de la Cruz was visiting the University of Zurich. The first author thanks Joachim Rosenthal for the invitation. E. Gorla and A. Ravagnani were partially supported by the Swiss National Science Foundation through grant no. 200021 150207 and by the ESF COST Action IC1104. H. López was partially supported by CONACyT and by the Swiss Confederation through the Swiss Government Excellence Scholarship no. 2014.0432. Email addresses: [email protected], [email protected], [email protected], [email protected].