# LINEAR ALGEBRA AND CODING THEORY

### Content

This course focuses on Linear Structures and its application in coding theory. For example the theory about vector spaces and the linear images between them belong to Linear Structures. Often vectors are seen as columns with real numbers and linear maps are indentified with matrices. But even continuous functions can be interpreted as vectors, and so can matrices. By looking at vectors and linear maps in a general way more, seemingly different, cases can be treated and understood in one go. A really nice application of this general treatment can be found in the coding theory, namely the theory of linear codes. Coding theory and, in particular, linear codes have wide applications. In the most divergent situations where data has to be sent from a transmitter to a receiver via an unreliable channel, codes are used to correct errors that may occur in the data. This is done by incorporating redundancy in the data in a smart and efficient way. In that way erroneous data, that were received, can often be correctly. One of the most appealing applications is the compact disc. Due to scratches and fingerprints the signal read by the laserhead contains almost always errors, but by the use of coding these errors can be corrected, if not too numerous. As a result Beethoven’s ninth can still be enjoyed without the disturbing cracks that are characteristic for good old vinyl records.

The course starts with a general introduction to the basic notions in coding theory. Subsequently we shall focus on linear codes. As an intermezzo linear structures will be treated in a general context. There will be special attention to finite vector spaces Finally, we shall apply the theory of finite vector spaces linear coding theory. It will turn out that it is very handy and useful to think in a more abstract way about linear spaces and vectors.

Learning goals:

 · Understanding the general basic theory about vector spaces and linear images.
 · Recognizing the situations in which vector spaces can be used and the actual application of the theory
 · Knowing the main problems in the coding theory
 · The application of the theory of vector spaces in linear codes
 · Drafting linear codes that satisfy the given specifications and decoding these codes
 · Recognizing situations in which codes can be used

Prerequisites: basic linear algebra

### General information

This course will take place in the third quartile of your first year in the mathematics track. It will be taught by Jan Willem Polderman.

### Teachers

 Jan Willem Polderman Jan Willem Polderman graduated cum laude in 1983 at the university of Groningen on a topic in algebraic number theory. After his study Mathematics he was a research assistant at the Center for Mathematics and Computer Science in Amsterdam. In 1987 he defended his PhD thesis on adaptive control systems at the Rijkuniversiteit Groningen.. His promotor was prof. J.C. Willems, who sadly passed away recently. With professor Willems he published a book about mathematical system theory. With I. Mareels (Melbourne) he wrote an introductory book on adaptive systems. Since 1987 Jan Willem Polderman has been with the department of Applied Mathematics at the University of Twente, First as an assistant professor and from 2003 as an associate professor. His main research interest is adaptive systems and system theoretic approach tp coding and decoding. Recently his research interest has shifted towards stability and control of hybrid systems. Since 2008 he is the program director in the department Applied Mathematics and the 3TU master Systems and Control. Jan Willem Polderman likes to play tennis and ice skating, be it at a very modest level. He listens to classical music and plays the piano, again at a questionable level. Together with Ella de Jong he has two daughters, Lotte (1996) and Jorieke (1998) with whom he loves to go camping, for example on the beloved island of Vlieland. Back to programme