MAS2706 : Linear Algebra & Coding Theory

• Offered for Year: 2020/21
• Module Leader(s): Dr David Kimsey
• Lecturer: Professor Sarah Rees
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
Semesters
 Semester 1 Credit Value: 10 Semester 2 Credit Value: 10 ECTS Credits: 10

Aims

To provide students with an introduction to modern abstract linear algebra. Building on their existing knowledge of matrix methods, students will experience the benefits of an abstract and rigorous mathematical theory for the deeper understanding of a mathematical subject. 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
Linear algebra is a fundamental subject that pervades many areas of modern mathematics. On the one hand it is often convenient to replace a complicated problem by a linear approximation which is easier to solve. On the other hand, linear algebra has beautiful applications in coding theory, projective geometry, and many other areas of mathematics and statistics.
Initially linear algebra aims to solve systems of linear equations. In the first year courses this led naturally to matrix algebra. In MAS2701 abstraction and generalisation are pushed one level further with the formal introduction of vector spaces and linear maps as a replacement for real n-dimensional space and matrices, respectively. This allows us to consider analogous problems in different settings simultaneously and eventually makes explanations easier and faster. We will need to introduce notions of dimension and basis in this general setting. A guiding question is how to transform matrices (or linear maps) to a simple form in which essential properties can be immediately read off.

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

Vector spaces, span and bases, linear maps and their properties, eigenvectors and eigenvalues, inner product spaces, change of basis, diagonalisation.

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

Teaching Methods

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

Teaching Activities
Category Activity Number Length Student Hours Comment
Structured Guided LearningLecture materials361:0036:00Non-Synchronous Activities
Guided Independent StudyAssessment preparation and completion301:0030:00N/A
Scheduled Learning And Teaching ActivitiesLecture91:009:00Present in Person
Scheduled Learning And Teaching ActivitiesLecture91:009:00Synchronous On-Line Material
Structured Guided LearningStructured non-synchronous discussion181:0018:00N/A
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Office Hour or Discussion Board Activity
Guided Independent StudyIndependent study941:0094:00N/A
Total200:00
Jointly Taught With
Code Title
MAS2701Linear Algebra
Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer:  a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.
Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

Assessment Methods

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A8024-hour exam
Exam Pairings
Module Code Module Title Semester Comment
MAS2701Linear Algebra1N/A
Other Assessment
Description Semester When Set Percentage Comment
Written exercise1M8written exercises
Written exercise2M12written exercises
Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will will 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; these assessments have a secondary formative purpose as well as their primary summative purpose.