Unit CODING THEORY
- Course
- Mathematics
- Study-unit Code
- 55A00041
- Curriculum
- Matematica per la crittografia
- Teacher
- Giorgio Faina
- Teachers
-
- Giorgio Faina
- Hours
- 42 ore - Giorgio Faina
- CFU
- 6
- Course Regulation
- Coorte 2020
- Offered
- 2020/21
- Learning activities
- Caratterizzante
- Area
- Formazione teorica avanzata
- Academic discipline
- MAT/03
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Linear codes and projective codes. Basic inequalities. Algebraic curves over finite fields. Fields of rational functions, divisors. Rational maps. Reed-Solomon codes. Algebraic Geometric codes. One point Goppa codes. Hermitian codes. Outline on elliptic curve cryptography.
- Reference texts
- - L. Giuzzi, Codici Correttori, Milano, Springer, 2006.
- R. E. Klima, N. Sigmon, E. Stitzinger, Applications of abstract Algebra (with MAPLE), CRC Press, 2000 - Educational objectives
- eoria dei codici is a course of the degree in Mathematics addressed in a special way for students interested in the applications of algebra and geometry.
The main goal of the course is to provide students with advanced elements of algebra and geometry useful for dealing with concrete problems related to network communications.
The main knowledge gained will be:
-Familiarity with finite fields
-Familiarity with the concepts of encoding and decoding of information.
-Familiarity with the theory of algebraic curves flat and with coding systems associated with them.
The main skills will be:
- Evaluating the performance of a linear code
- Building linear codes appropriate to specific instances
- Building and evaluating codes defined from algebraic curves - Prerequisites
- In order to understand and know how to apply most of the techniques described in the course, the student must have successfully passed the exams of Algebra I-II and Geometry I-III of the first degree
- Teaching methods
- The course consists of classroom lectures on all topics of the course. In each lesson about half of the time will be devoted to solving problems and exercises
The lessons can also be held online. - Learning verification modality
- The exam consists of an oral interview. Three questions relating to three separate parts of the program will be submitted to the student. The interviews lasts about 30-40 minutes and is designed to ensure the level of knowledge and ability of understanding reached by the student on the theoretical and methodological implications listed in the program (finite fields, linear codes, plane algebraic curves, Goppa codes), The oral test will also allow to verify communication skills, appropriateness of language and autonomous organization of the exposure.
- Extended program
- Finite fields. The primitive element theorem. Group action on a set. Cyclotomic polynomials. Linear codes and projective codes. Basic inequalities and bounds: Singleton bound, Hamming bound, Plotkin bound, Gilbert-Varshamov bound, Griesmer bound. Algebraic curves over finite fields. Fields of rational functions, divisors, Riemann-Roch spaces.. Rational maps between algebraic curves. Algebraic Geometric codes as a generalization of the Reed-Solomon codes and the BCH codes. One point Goppa codes. Hermitian codes. Outline on elliptic curve cryptography.