Course Details in 2025/26 Session

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Module Title Combinatorics and Communication Theory Mathematics Mathematics 06 19608 Deryk Osthus Masters Level 20 Semester 2 LC Algebra & Combinatorics 1 - (06 25659) Optional for all MSci programmes in Mathematics and MSci JH programmes including mathematics Lecture-46 hours Tutorial-10 hours Total: 56 hours 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. By the end of the module the student should be able to:Apply counting principles in various settings, including discrete probabilityBe familiar with basic combinatorial structures and arguments, be able to prove simple results in Ramsey theory.Understand and be able to apply fundamental results on codes and communication over a noisy channel. Apply the basic coding techniques such as Huffman codes and linear codes. Level M students will explore the subject beyond the taught syllabus. 19608-01 : Raw Module Mark : Coursework (100%) 3 hour Written Unseen Examination (80%); In-course Assessment (20%). Welsh, D. 1988. Codes and Cryptography. Oxford University Press. Jukna, S. 2001. Extremal Combinatorics. Springer-Verlag.