Convolutional Codes - A generalized view on coding theory and applications

卷积码是经典分组码的自然推广，但许多研究问题只针对分组码而未针对卷积码进行研究。卷积码特别适用于数据流系统中的纠错，因为当涉及到可容忍时间延迟的上限的顺序编码和解码时，卷积码非常有效。本研究项目的目的是通过考虑卷积码的各个方面以及尚未研究的可能应用来提高对卷积码的认识。作为第一个研究目标，我们将研究卷积码和组合学之间的联系。虽然分组码和组合学之间已经建立了许多关系，但对于卷积码，还有许多与此相关的开放问题。作为第二个研究目标，我们计划研究赋予秩度量的卷积码。衡量码的纠错能力的经典度量是汉明度量。然而，在网络编码领域的应用的推动下，在秩度量中研究代码变得越来越流行。我们将致力于秩度量卷积码的构造和更深入的理解。作为第三个研究目标，我们将研究卷积码在基于代码的密码学中的应用。对于McEliece类型的密码方案的设计，主要任务是开发相互匹配的卷积码的纠错解码算法和构造。

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.