Encoding schemes for Gabidulin codes with applications
University of Zurich
Rank metric codes have been introduced in 1978 by Delsarte, an independently by Gabidulin in 1985 and Roth in 1991. These codes are linear subspaces of the space of n x m matrices over a finite field F_q, but they can also be seen as sets of vectors of length n over an extension field F_(q^m). Codes that are optimal in this metric are called Maximum Rank Distance (MRD) codes. The first and most studied family of MRD codes is given by the so-called generalized Gabidulin codes, and they represent the analogue of generalized Reed-Solomon codes for the rank metric.
In this talk we give an overview on the analogies between MRD codes and their counterpart in the classical Hammig metric, i.e. the so-called Maximum Distance Separable (MDS) codes. This will be done focusing in particular on the generator matrix of these two families of codes. In particular, we also examine the structure of generalized Gabidulin codes, that represent the analogue of Generalized Reed-Solomon codes in the rank metric. The study of these codes and their encoders leads to matrices with a deep structure in the context of finite fields and finite geometry. Using this characterization, we are able to design special Gabidulin codes where the generator matrix is a structured matrix, such as Toeplitz or Hankel. Moreover, a new efficient algorithm for determining wether a given code is equivalent to a generalized Gabidulin code is obtained.
Alessandro Neri received both his Bachelor's and Master's degree in Mathematics from University of Pisa (Italy). Since 2015, he is a PhD student in the Applied Algebra group at University of Zurich, under the supervision of Prof. Joachim Rosenthal.
His main research interests are Algebraic Coding Theory and Combinatorics. The main topic of his PhD project concerns rank metric codes, their algebraic structure and their invariants.