**MRD Rank Metric Convolutional Codes**

**Prof. Paolo Vettori **

University of Aveiro, Portugal

A rank metric code C is a set of n×m matrices over a finite field, equipped with a distance given by the rank of A-B, for any two codewords A and B. The code is called MRD when the minimum distance between different codewords reaches its upper bound (depending on m, n, and the size of the code).

This kind of codes, together with an MRD construction (also known as Gabidulin code), were introduced at the end of the 70s by Delsarte. Surprisingly, new MRD constructions were found by various authors only in 2015. The first works on rank metric convolutional codes were published in 2015 too: in this case, codewords are (finite) sequences of n×m matrices, usually represented as polynomial matrices.

In this talk we will generalize the notion of codeword distance to the polynomial case, establish the corresponding upper bound, construct a general family of MRD rank metric convolutional codes, and show how to decode a received message, detecting and correcting the errors that may have occurred. Finally, we will propose strategies for more efficient implementations.

Joint work with Raquel Pinto and Diego Napp (University of Aveiro) and Joachim Rosenthal (University of Zurich).

Paolo Vettori received the M.S. degree in Electronic Engineering from the University of Padua, Italy, in 1995, and the Ph.D. degree in (Mathematical) Systems Theory from the University of Bologna, Italy, in 1999. After completing one year of postdoctoral research at the Department of Mathematics of the University of Aveiro, Portugal, he joined the local group on Systems Theory as Invited Assistant Professor, becoming Assistant Professor in 2008.

His research activities mainly concern Linear Systems Theory, particularly in the Behavioral approach: among others, he investigated linear systems with delays, with symmetries, with fractional order and systems defined on time scales. He recently began to study Random Network Coding and linear rank metric convolutional codes.