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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 Reed-Solomon 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 one-to-one correspondence with Generalized Cauchy matrices, that are obtained by computing their generator matrix in standard form.
In the framework of rank distance, it is well-known that generalized Gabidulin codes, introduced by Delsarte first, and Gabidulin later, represent the rank-analog of Generalized Reed-Solomon codes. The canoncial generator matrix in this case is given by the so-called Moore matrix, that corresponds to the natural rank-analog 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 rank-analog 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 Artin-Schreier polynomials and their roots, that define an equivalence relation on the set of all Gabidulin codes.