Tensor rank extremal rank-metric codes

Error-correcting codes are codes which are used to send messages over a channel in which many errors occur. Typically, the set of codewords of a code C could be a set of vectors over some finite field. To be able to correct errors which might have occurred during the transmission over the channel, it is desirable that two such vectors u and v from C are at a large enough distance, meaning that there are not too many zeros in their difference u-v. Rank-metric codes form are error-correcting codes where the codewords are matrices with entries in a finite field, and where the minimum distance between two codewords is defined as the rank of the difference of the two matrices. They are related to subspace codes and random network coding. Rank-metric codes can be represented by tensors in a 3-fold tensor product which allows to define the tensor rank of the code and various other interesting parameters and invariants. The aim of this project is to investigate the existence of tensor rank extremal rank-metric codes and minimal tensor rank codes, and to study their relation with MDS codes. We also aim to study the existing MRD codes from this perspective.