MATH30180 An Intro to Coding Theory

Academic Year 2021/2022

Error correcting codes play a central role in all areas of communications technology such as deep space communication, mobile telephony and digital storage.

This is intended as an introduction to algebraic coding theory. The approach uses principles of linear algebra, groups, rings and finite fields and applied them to the study of error-correcting codes. Questions of code constructions, existence and fundamental theorems will be addressed, all in respect of the Hamming metric. Several well-known families of codes will be studied.

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Curricular information is subject to change

Learning Outcomes:

The structure of finite fields;
Knowledge of coding theoretic bounds & an ability to apply bounds to code existence;
An understanding of codes as vector spaces, the dual code, generator and parity check matrices;
Knowledge of the principles of error correction, working knowledge of syndrome decoding and its complexity;
An understanding of code optimality and extremality;
Knowledge of defining properties of classical algebraic codes, such as cyclic, Reed-Solomon and BCH codes;
Structure of cyclic codes as ideals;
Knowledge of the weight enumerator and MacWilliams duality theorem.

Indicative Module Content:

Principles of Error Correction: Hamming distance, sphere-packing, the main coding problem;
Linear Codes: codes as vector spaces, generator matrices, parity check matrices, the dual code, syndrome decoding;
Code Optimailty: existence bounds, operations on codes, extremal codes;
Cyclic Codes: codes as ideals in polynomial rings, generator polynomials, parity check polynomials, cyclotomic cosets;
BCH Codes: constructions from vandermonde-like matrices, connection to cyclic codes, BCH bound, Reed-Solomon codes.

Student Effort Hours: 
Student Effort Type Hours
Lectures

24
Tutorial

10
Total

34
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
Pre-requisite:
MATH20300 - Linear Algebra 2 (MathSci), MATH20310 - Groups, Rings and Fields, MST20050 - Linear Algebra II, MST30010 - Group Theory and Applications

Additional Information:
Students should have completed a module in abstract algebra, such as MST30010 or MATH20310 as a pre-requisite. Students should have completed an intermediate module in linear algebra, such as MATH20300 or MST20050 as a pre-requisite.


 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Continuous Assessment: mid-term exam or regular assignment Unspecified n/a Standard conversion grade scale 40% No

25
Examination: End of Semester Exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

75

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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