Aims

To explain the necessity for error correcting codes, to establish their general properties, to show how to construct linear and cyclic codes and to gain practice in their use.

Module Summary
Error-correcting codes are at the heart of the digital revolution. They are used to store music on CDs and video on DVDs; to send data across telecommunications networks; and to broadcast digital television. In practice, a digital signal may be degraded in transit by many factors - cosmic rays, fluctuations in power supplies, even (in the case of a CD) dust and scratches - so that some 0s are changed to 1s and vice versa; error-correcting codes are designed to rectify this. We work with words, binary strings of some standard length n. Certain words are designated as codewords, and the signal is converted to a sequence of codewords before transmission. At the receiving end, each word is examined as it arrives, and, if it turns out to be a non-codeword (indicating that the signal has been degraded), it is replaced by the nearest codeword. This explains why small imperfections on a CD do not affect the quality of the sound that you hear. We shall concentrate on a particularly nice class of codes called linear codes, a beautiful application of elementary linear algebra. Here errors can be corrected automatically by simple matrix operations. In particular, we shall investigate cyclic codes, linear codes based on polynomials.

Outline Of Syllabus

General properties of codes. Perfect codes. Linear codes. Parity-check matrices and syndrome decoding. Hamming codes. Extensions of codes. Cyclic codes.

Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) will provide an opportunity for more direct contact between individual students and the lecturer.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments are expected to consist of four written assignments of equal weight: the exact nature of assessment will be explained at the start of the module. These allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; this assessment has a secondary formative purpose as well as its primary summative purpose. The coursework assignments may be written assignments, computer based assessments or a combination of the two, and in the case of combined assessments the deadlines for the two parts will not necessarily be the same.