Algebraic Channel Coding

Even though iterative coding based on probabilistic methods is nowadays applied in several technical applications, coding schemes based on algebraic methods still find a variety of applications, for example in data storage systems and also in concatenated code designs. Hence, this lecture focuses on these algebraic coding schemes.
In the first part of the lecture, the fundamentals of block codes and Galois theory are introduced in order to constitute the framework for algebraic code designs. Then some common algebraic code constructions are presented and analyzed. In the third part of the lecture, we focus on algebraic decoding methods, starting with the classical approaches, before modern list decoding algorithms and other concepts are considered.

In particular, the lecture focuses on the following topics:

Fundamentals of algebraic coding schemes

• Basics of Galois field theory
• Prime fields and extension fields
• Arithmetic over finite fields
• Fundamental algorithms such as the (extended) Euclidean Algorith

Algebraic code constructions

• Reed-Solomon Codes
• BCH codes
• Other code classes such a Golay- and RM-codes

Algebraic decoding approache

• Fundamental ideas of algebraic decoding
• Classical syndrome based approaches such as the Peterson algorithm
• Efficient syndrome decoding based on the Berlekamp-Massey algorithm or the Sugiyama-Kasahara-Hirasawa-Namekawa algorithm
• List decoding based on interpolation such as the Sudan algorithm or the Guruswami-Sudan algorithm
• Further algebraic approaches to decode beyond half the minimum distance