PhD ceremony: F.L. Tsang, 16.15 uur, Academiegebouw, Broerstraat 5, Groningen
Thesis: Skew rings, convolutional codes and discrete systems
Promotor(s): prof. M. van der Put
Faculty: Mathematics and Natural Sciences
In this thesis, a number of problems from coding theory and discrete systems theory are studied.
A discrete problem, naturally related to cyclic convolutional code - a class of convolutional codes with certain cyclicity, is the main subject discussed in Chapter 1. As a convolutional code, cyclic convolutional code has a degree concept inherited from a polynomial ring. The problem determines which Forney sequence, i.e., the minimal degrees for the generators of the code, may occur. Using vector bundles on the projective line, we come up with several strategies that lead to partial solutions. Chapter 2 is the continuation of the investigation of a certain matrix ring appeared in Chapter 1.
Chapter 3 is an article regarding a discussion with H. Gluesing-Luerssen, where the same problem is seen from another point of view. In fact, it contains a translation of the above problem into a combinatorial chessboard problem. The presumption is that every Forney sequence indeed occurs for a cyclic convolutional code.
Chapter 5 is to solve difference equations using homological methods. The solution spaces are usually chosen to be sitting in injective co-generators and are frequently called the signal spaces. A problem regarding sandpile models in physics, posed by K. Schmidt and E. Verbitskiy, is studied with these methods. The conclusion is that the bounded space is too large and the periodic space is too small for the sandpiles.
Finally, Chapter 6 is about two problems in polynomial matrices from systems theory, which are solved with the help of vector bundles on the projective line introduced in Chapter 1.
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