MA322 Proofs and Numbers, MA323 Matrices and Probability or MA326 Matrices and Computing, MA553 Linear Algebra. Helpful MA565 Groups and Rings.
OverviewRecently some quite novel applications have been found for "Discrete Mathematics", as opposed to the Continuous Mathematics based on the Differential and Integral Calculus. Thus methods for the encoding of information in order to safeguard against eavesdropping or distortion by noise, for example in online banking and digital television, have involved using some basic results from abstract algebra. This module will provide a self-contained introduction to this general area and will cover most of the following topics: (a) Modular arithmetic, polynomials and finite fields: Applications to orthogonal Latin squares, cryptography, coin-tossing over a telephone, linear feedback shift registers and m-sequences. (b) Error correcting codes: Binary block, linear and cyclic codes including repetition, parity-check, Hamming, simplex, Reed-Muller, BCH, Golay codes; channel capacity; Maximum likelihood, nearest neighbour, syndrome and algebraic decoding.
This module appears in:
48 hours - 36 hours of lectures of formal exposition of the subject and 12 hours of examples classes.
Method of assessment
90% Examination, 10% Coursework
D Welsh, Codes and Cryptography, Oxford University Press, 1988
N L Biggs, Discrete Mathematics, Oxford University Press, 1989
The Intended Subject Specific Learning Outcomes. On successful completion of the module students will have:
a. improved their precision in logical argument and enhanced their skills in symbolic calculation with more complex discrete structures;
b. a reasonable knowledge of the definitions of terms used in the module and a reasonable understanding of the statements, proofs and implications of the basic theorems given in the course (sufficiently well to be able to construct simple proofs of related results);
c. revised modular arithmetic and polynomial algebra and obtained a reasonable understanding of the theory of finite fields (and related finite rings);
d. developed a critical appreciation as to how this material can be applied to concrete problems in a number of different areas relating to electronic communication systems (cryptography and, primarily, in the study of error correcting codes);
The Intended Generic Learning Outcomes. On successful completion of the Module, students will have:
(i) developed a logical, mathematical approach to solving problems and will be able to solve
problems and present solutions relevant to discrete structures and their applications to IT
(ii) furthered their ability to work with relatively little guidance on the subject matter and
exercises associated with the course;
(iii) obtained the basic mathematical background necessary to follow the rapidly changing developments in IT communications.
(iv) improved their key skills in written communication, numeracy and problem solving.