Intended Learning Outcomes:
After successful completion of the course, the student is able to
1. Apply the general basic theory about vector spaces and linear images.
2. recognise the situations in which vector spaces can be used and the actual application of the theory
3. To explain the main problems in the coding theory
4. Apply the basic theory of finite fields to linear code
5. Apply the theory of vector spaces in linear codes
6. Draft linear codes that satisfy the given specifications and decoding these codes
7. recognise situations in which (linear) codes can be used
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.