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Coding Theory

Code 15635
Year 1
Semester S2
ECTS Credits 6
Workload TP(45H)
Scientific area Mathematics
Entry requirements Álgebra
Learning outcomes General objectives
1- Learn some fundamental concepts and examples in coding theory;
2- Apply the studied results and techniques in the analysis of several codes;
3- Analyse and understand proofs;
4- Communicate, written and orally, using mathematical language.

Specific objectives:
1-Explain the concepts of alphabet, word, code, transmission channel, and entropy
2- Apply the maximum likelihood decoding method;
3- Calculate the probability of incorrect decoding;
4- Calculate the Hamming distance between two words
5- Apply the minimum distance decoding method
6-Identify the parameters of a code;
7-Identify linear codes;
8-Distinguish the various linear codes studied;
9-Construct the generating matrix and parity matrix from a linear code
10-Apply decoding by Slepian Tables and by Syndrome.
Syllabus CHAPTER 1. Introduction
1.1 First examples and definitions
1.2 Broadcast channels
1.3 Maximum likelihood decoding
1.4 Hamming Distance
1.5 Decoding by nearest neighbor
1.6 Distance of a code
1.7 Main problem in code theory
1.8 Estimates

CHAPTER 2. Linear Codes
2.1 Vector spaces over finite fields
2.2 Parameters and Minimum Weight
2.3 Generating matrix and parity matrix;
2.4 Encoding and decoding
2.5 Equivalence of linear codes

CHAPTER 3. Examples of Linear Codes
3.1 Binary Hamming Codes
3.2 Q-ary Hamming Codes
3.3 Reed-Muller Codes
3.4 Minorant of Gilbert-Varshamov linear
3.5 Golay Codes
3.6 Maximum Separation Distance Codes

Main Bibliography Referências principais
- D. G. Hoffman, K.T. Phelps, D.A. Leonard, C. C. Lindner, C.A. Rodger, J.R. Wall, David Hoffman, Coding Theory: The Essentials (Pure and Applied Mathematics : a Series of Monographs and Textbooks, 150) Marcel Dekker Inc; First Edition (December 1, 1991)

Outras referências
- Cover, T. M., and Thomas, J. A. (2006), Elements of Information Theory (2.ª edição), Wiley
- R. Hill (1997), A First Course in Coding Theory, Oxford University Press
-Ling, S. & Xing, C. (2004). Coding theory: A first course. Cambridge, UK: Cambridge University Press.
- J.H. van Lint (1991), Introduction to Coding Theory, Graduate Texts in Mathematics (3.ª edição), Springer
Teaching Methodologies and Assessment Criteria The classes combine theory with practice. The teacher introduces the concepts, states and proves the fundamental results, provides examples and applications. The student is encouraged to participate in the classes, to interact with teacher and colleagues and to work independently, by solving exercises, guided reading, problem formulation and problem solving. Supporting materials are available to students on the Moodle course page and assistance is provided in accordance with a weekly schedule.

Regarding the assessment, the student can choose to take the final exam or/and the continuous assessment comprising two written tests, with a weight of 45% each for the final grade, and work presentations throughout the semester, with a weight of 10% for the final grade.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2024-03-04

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