Convolutional Codes - A generalized view on coding theory and applications

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Convolutional codes are natural generalizations of the classical block codes but many research questions have only been studied for block codes but not for convolutional codes yet. Convolutional codes are especially suitable for error-correction in data streaming systems as they are very efficient when it comes to sequential encoding and decoding with an upper bound on the tolerable time delay. The aim of this research project is to enhance the knowledge on convolutional codes by considering various facets of these codes and possible applications that have not been investigated yet.As a first research objective we will investigate connections between convolutional codes and combinatorics. While there are already many relations between block codes and combinatorics established, for convolutional codes there are many open questions related to this.As a second research objective we plan to investigate convolutional codes endowed with the rank-metric. The classical metric to measure the error-correcting capability of a code is the Hamming metric. However, motivated by applications in the area of network-coding, studying codes in the rank-metric became more and more popular. We will work on the construction and deeper understanding of rank-metric convolutional codes.As third research objective we will investigate the use of convolutional codes for code-based cryptography. For the design of cryptographic schemes of the McEliece type the main task is to develop error-correcting decoding algorithms and constructions of convolutional codes that are matched to each other.