Optional for: All MSci programmes in Mathematics, BSc Mathematical Sciences, BSc Mathematical Sciences with Study in Continental Europe, All programmes including a Major in Mathematics, All JH programmes
The first part of the module will give an introduction to several combinatorial structures, which have applications in different areas. Topics are likely to include combinatorial games, applications of counting principles to discrete probability and basic Ramsey theory. (Here Ramsey theory can be viewed as a formalization of the notion that `complete disorder is impossible’ – this surprising phenomenon will be investigated for graph colourings and arithmetic properties of the integers). The second part of the module consists of an introduction to information theory and coding theory. The aim here is to transmit information (i) efficiently and (ii) reliably over a noisy channel. For (i), the main result will be Shannon’s noiseless coding theorem, which relates coding efficiency to the entropy of a source. For (ii), we will discuss error correcting codes, including several linear codes, such as Hamming codes. Both parts of the module are linked by the methods and ideas that are used.
Learning Outcomes
By the end of the module the student should be able to:
Be familiar with basic combinatorial structures
Understand and be able to apply fundamental results on finite sets, codes and communication over a noisy channel
Apply the basic techniques introduced in the lectures
Assessment
19601-01 : Raw Module Mark : Coursework (100%)
Assessment Methods & Exceptions
3 hour Written Unseen Examination (80%); In-course Assessment (20%).
Other
Reading List
Welsh, D. 1988. Codes and Cryptography. Oxford University Press.
Jukna, S. 2001. Extremal Combinatorics. Springer-Verlag.
