ALGEBRA SUPERIORE
Academic Year 2026/2027 - Teacher: MARCO D'ANNAExpected Learning Outcomes
The aim of this course is to deepen the study of the theory of linear codes. One of the goals of the course is to show how theoretical notions allow to develop concrete applications.
Course Structure
Lessons at the blackboard and discussions of the exercises.
Required Prerequisites
Basics on rings, fields and vector spaces. In the course some basic notions on algebraic curves will be used; on this subject some recalls will be given.
Attendance of Lessons
Highly recommended.
Detailed Course Content
Channels. Block codes.
Finite fileds and their construction. Irreducible polynomials on finite fields.
Linear codes. Hamming distance. Generator matrix and parity check matrix. Syndrome decoding. Dual code.
Codes constructions. MDS codes and their decoding.
Hamming codes. Reed-Muller codes. Cyclic codes.
Goppa codes. Algebraic geometry codes.
Code-based cryptography.
Textbook Information
1. Notes form the lecturer.
2. J.I.Hall. Notes on coding theory. Department of Mathematics. Michigan State University
3. Tom Høholdt, Jacobus H. van Lint and Ruud Pellikaan. Algebraic geometry codes. In Handbook of Coding Theory, vol 1, pp. 871-961 (V.S. Pless, W.C. Huffman and R.A. Brualdi Eds.), Elsevier, Amsterdam 1998.
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Channel. Block codes. | 1 o 2 |
| 2 | Finite fileds | 1 |
| 3 | Linear Codes. | 1 o 2 |
| 4 | Costruzione di codici | 1 o 2 |
| 5 | Codici MDS | 2 |
| 6 | Hamming codes. Reed-Muller codes | 1 o 2 |
| 7 | Cyclic codes | 1 o 2 |
| 8 | Goppa and algebraic geometry codes. | 3 |
| 9 | Code-based criptography | 1 |
Learning Assessment
Learning Assessment Procedures
Exercises will be given to the students, that will be discussed during in the classroom and that will contribute to the final valuation. At the end of the course there will be an oral exam.
Examples of frequently asked questions and / or exercises
The usual question is the request to talk organically about a topic of the program, giving details and examples. It will be also possible to discuss again the exercises given during the course.