MATH30180 An Intro to Coding Theory

Academic Year 2020/2021

*** Not available in the academic year indicated above ***

This course is an introduction to the mathematical theory of linear block codes. Codes play a role of growing importance in all areas of communications technology such as deep space communication, mobile telephony and magnetic storage. The module prepares the mathematical methods and ingredients of the theory and may be considered as an impressing example of applied algebra suitable for students from engineering, computer science and mathematics.From the contents: 1) Linear block codes: block codes, Hamming distance, Hamming weight, vector spaces and finite fields, construction of linear codes, decoding problem, sphere-packing bound.2) Generator matrices, check matrices minimum weight in terms of check matrix properties, Gilbert-Varshamov bound for linear codes, general Gilbert-Varshamov bound.3) Cylic codes: polynomial rings as principal ideal domains, division algorithm, unique factorization, generator polynomials, check polynomials and their relation to generator matrices and check matrices, determination of all cyclic linear codes of a given length over a finite field.4) Perfect codes: Hamming codes and Golay codes, parameters of perfect codes.5) Reed-Muller codes: Plotkin construction, properties of the Plotkin sum, recursive definition of the Reed-Muller codes, parameters, a decoding algorithm that exploits the recursive structure.6) Code-duality and MacWilliams' theorem: duality, weight enumerators, character theory, fourier transform, weight enumerator theorem, examples.7) Reed-Solomon codes: definition via van-der-Monde determinants, Goppa description, decoding using the Euclidean algorithm.8) Further bounds on the parameters of block codes.Structure of the module: 2 lectures per week; 1 tutorial per fortnight; homework assignments; final examination.

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Curricular information is subject to change

Learning Outcomes:

computations in finite fields,
knowledge of coding theortic bounds & an ability to apply bounds to code existence,
understand principles of error correction, working knowledge of syndrome decoding and its complexity,
algebraic constructions of codes, constructions and decoding of cyclic, Reed-Solomon & BCH codes, constructions and decoding of Reed-Muller codes, constructions of optimal codes
codes as ideals in quotient polynomial rings, evaluation codes

Student Effort Hours: 
Student Effort Type Hours
Lectures

24
Tutorial

6
Total

30
Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning 
Requirements, Exclusions and Recommendations

Not applicable to this module.


Module Requisites and Incompatibles
Not applicable to this module.
 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Examination: End of Semester Exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

70
Examination: Mid term Exam Unspecified No Standard conversion grade scale 40% No

30

Carry forward of passed components
No
 
Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Group/class feedback, post-assessment

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