Module MATH3501: Codes and Knots III

Department: Mathematical Sciences

MATH3501: Codes and Knots III

Prerequisites

• Linear Algebra I (MATH1071) OR Linear Algebra I (Maths hons) (MATH1091)

Corequisites

• None

Excluded Combination of Modules

• If Cryptography and Codes III (MATH3401) has already been taken, then Codes and Knots III cannot be taken.

Aims

• Codes: To introduce the notion of (error-correcting) codes and provide examples of such codes some of which are used in real-life applications (data transmission and data storage).
• Knots: To give an introduction to topology in an intuitive, visual way by studying knots, links, and surfaces.

Content

• Coding Theory:
• Introduction to codes, Hamming distance, error detection and correction, equivalence of codes.
• Linear codes: rank, generator and check matrices. Equivalence. Dual codes.
• Hamming codes, sphere-packing bound and perfect codes.
• MDS codes, Reed-Solomon Codes.
• Construction of finite fields, linear Codes over finite fields.
• A selection of the following topics: Golay Codes, Cyclic codes, Reed-Muller codes.
• Knot Theory:
• Introduction to knots and links, diagrams, Reidemeister moves.
• Knot invariants: 3-colouring, linking number, Jones polynomial, Alexander polynomial, HOMFLY-polynomial, determinant.
• Surfaces, Euler characteristic, Classification Theorem of Surfaces.
• Knots bounded by surfaces, genus of a knot, signature of a knot.

Learning Outcomes

• By the end of the module students will:
• Be able to solve seen and unseen problems on the given topics.
• Have an awareness of the abstract concepts of theoretical mathematics in the fields of Codes and Knots.
• Have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas: Codes of different types (Hamming, Reed-Solomon etc), Bounds on codes, Knots and their invariants, Surfaces.

• In addition, students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: abstract reasoning.

• Students will have basic mathematical skills in the following areas: abstract reasoning, problem solving.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four assignments to be submitted.

■ Students who do not attend monitored activities shown under Teaching Methods and Learning Hours, or who fail to complete the summative or formative assessment(s) specified above, may be subject to the Academic Progress procedures defined in the University's General Regulation V, and may be required to leave the University.