Algebraic Coding Theory

Code:

M4081

Acronym:

M4081

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2019/2020 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:ENM	4	Official Study Plan since 2013-2014	1	-	6	56	162
			2

Teaching language

Suitable for English-speaking students
Obs.: As aulas serão dadas em português, podendo haver esclarecimentos individuais em inglês.

Objectives

Know classical examples of error correcting codes.
Reproduce key results of the theory and give rigorous and detailed proofs of them.
Construct new codes from old ones and examine their basic properties.
Apply the basic techniques, results and concepts of the course to concrete examples.

Learning outcomes and competences

The student should reach a number of proposed objectives.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Linear algebra over fields; theory of finite fields.

Program

1 Introduction to Coding Theory
Introductory Examples
Important Code Parameters
Correcting and Detecting Errors
Bounds on codes

2 Linear Codes
Binary Linear Codes
Fields, Vector Spaces, and General Linear Codes
Encoding Linear Codes: Generator Matrices
Introduction to Parity Check Matrices
Parity Check Matrices and Linear Decoding
Parity Check Matrices, Minimum Distance, and the Singleton Bound
Hamming Codes

3 Cyclic Codes, Rings, and Ideals
Introduction to Cyclic Codes
Rings and Ideals
Ideals and Cyclic Codes
Generator and Parity Check Polynomials

4 Classes of Powerful Cyclic Codes
Special Cases of BCH and RS Codes
Minimal Polynomials
BCH and Reed-Solomon codes

5 Special Topics
Dual Codes
Group of a Code
Other topics (if time permits)

Mandatory literature

Sarah Spence Adams; Introduction to Algebraic Coding Theory, 2008 (www.math.niu.edu/~beachy/courses/523/coding_theory.pdf)

Comments from the literature

Other bibliographic references will be given

Teaching methods and learning activities

Presentation of the course material and of examples by the teacher; solution of exercises by the students with the advice of the teacher.
There will be regular office hours for student advice and clarification of doubts.

Software

GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra
GUAVA, a GAP package for computing with error-correcting codes

keywords

Physical sciences > Mathematics > Algebra
Technological sciences > Technology > Information technology

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	50,00
Frequência das aulas	50,00
Total:	100,00

Eligibility for exams

Course registration is the only requirement.

Calculation formula of final grade

There will a number N of optional midterm sets of exercises, of which count the N-1 best classified. The exercises will take place in the TP classes, on dates to be announced.

The final exam consists of two parts, the first corresponding to the sets of exercises, with weight three quarters. The remaining quarter is the weight of the second part. At the student option, for the classification of the first part it may be used the classification obtained through the sets of exercises.

The classification obtained through the sets of exercises can not be used in the remaining exams.

Special assessment (TE, DA, ...)

Any type of special student evaluation takes the form of written examination.

Classification improvement

Grade improvement can be attempted only through examination.