
Modul: MAT076 Arbeitsgemeinschaft in Codierungstheorie und Kryptographie
MRD codes: further properties and standard form of Gabidulin codes
Alessandro Neri's talk Date: 22.11.17 Time: 16.00  17.00 Room: UNINE, A317 Generalized ReedSolomon codes were introduced in 1960 in the classical framework of block codes with the Hamming metric. The canonical generator matrix for these codes is given by a weighted Vandemonde matrix, that is the product of a Vandermonde together with a diagonal matrix. In 1985 Roth and Seroussi showed that these codes are in onetoone correspondence with Generalized Cauchy matrices, that are obtained by computing their generator matrix in standard form.
In the framework of rank distance, it is wellknown that generalized Gabidulin codes, introduced by Delsarte first, and Gabidulin later, represent the rankanalog of Generalized ReedSolomon codes. The canoncial generator matrix in this case is given by the socalled Moore matrix, that corresponds to the natural rankanalog of the weighted Vandermonde matrix.
In this work we study the generator matrix of a Gabidulin code in standard form in order to obtain a rankanalog for Generalized Cauchy matrices. The mathematical tools needed for this attempt are based on a deep understanding of the trace function over finite fields and its duality properties, as well as the study of generalized ArtinSchreier polynomials and their roots, that define an equivalence relation on the set of all Gabidulin codes.

