On the Genericity of Maximum Rank Distance and Gabidulin Codes
Talk by Prof. Dr. Alessandro Neri
Date: 28.09.16 Time: 16.00 - 17.00 Room:
Codes in the rank-metric have been studied for the last four decades. For linear codes a Singleton-type bound can be derived for these codes. In analogy to MDS codes in the Hamming metric, we call rank-metric codes that achieve the Singleton-type bound MRD (maximum rank distance) codes. Since the works of Delsarte and Gabidulin we know that linear MRD codes exist for any set of parameters. The codes they describe are called Gabidulin codes. The question, if there are other general constructions of MRD codes that are not equivalent to Gabidulin codes, has been of large interest recently. Some constructions of non-Gabidulin MRD codes are known, but many of the derived codes are not linear over the underlying field but only linear over some subfield of it. In general, it remains an open question for which parameters non- Gabidulin MRD codes exist, and if so, how many such codes there are. We show that the properties of being MRD and non-Gabidulin are generic. This implies that over a large field extension degree a randomly chosen generator matrix generates an MRD and a non- Gabidulin code with high probability. Moreover, we give an upper bound on the respective probabilities in dependence on the extension degree.Joint work with Anna-Lena Horlemann-Trautmann, Tovohery Randrianarisoa and Joachim Rosenthal.