MATH32031 - 2011/2012

 

Specification

Aims

To introduce students to a subject of convincing practical relevance that relies heavily on results and techniques from Pure Mathematics.

Brief Description of the unit

Coding theory plays a crucial role in the transmission of information. Due to the effect of noise and interference, the received message may differ somewhat from the original message which is transmitted. The main goal of Coding Theory is the study of techniques which permit the detection of errors and which, if necessary, provide methods to reconstruct the original message. The subject involves some elegant algebra and has become an important tool in banking and commerce.

Learning Outcomes

On successful completion of this course unit students will

• have a theoretical understanding of how methods of linear and polynomial algebra are applied in design of error correcting codes,
• and be able to analyse and compare error detecting/correcting facilities of simple linear and cyclic codes for the symmetric binary channel;
• be able to design simple cyclic codes with given properties.

Future topics requiring this course unit

None.

Syllabus

1. Introduction to the Main Problem of Coding Theory. [1 lecture]
2. Hamming Distance. Code detection. Code correction. ISBN code. [2]
3. Length and weight of a code. Perfect codes. [3]
4. Linear codes. Generator matrices and standard forms. Encoding. Nearest neighbour decoding. [4]
5. Dual code. Parity check matrix. Syndrome decoding. Incomplete decoding. [4]
6. Hamming Codes and Decoding. [4]
7. Finite fields. Cyclic codes. [4]
8. Reed-Muller codes.

Textbooks

Recommended text

• R Hill, A First Course in Coding Theory, 1986, OUP.

Teaching and learning methods

Two lectures and one examples class each week. In addition students should expect to spend at least four hours each week on private study for this course unit.

Assessment

Coursework: weighting 20%

End of semester examination: two hours weighting 80%

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Arrangements

Coursework Mode: Problem sheets handed out in Weeks 5 and 10 with deadlines in Weeks 7 and 11.