Subject: ALGEBRA AND CODING THEORY (A.A. 2020/2021)
Unit Algebra e teoria dei codici
Advanced Theoretical Studies (lesson)
The main objective of the course is to furnish the basic tools for the study of the theory of error correcting codes.
PRE-REQUISITES: The representation of integers with respect to different bases, in particular base 2. Algebraic structures and substructures. Homomorphisms and foundamental theorems of algebraic structures. Permutations and their properties. Groups. Rings, ideals, polynomial rings. The division algoritm for polynomials. The minimal polynomial. Fields, the characteristic of a field, field extensions. Finite fields. Integer classes modulo n. Vector spaces.
Basic algebraic concepts. Polynomials with coefficients in a finite field. Block-codes, Hamminh distance. Linear codes, generator matrices, parity-check matrices, syndrome-decoding. Cyclic codes. Bounds on the parameters of a code. Optimal codes with respect to a given bound.
Frontal lectures which include theory and exercises.
Oral interview on the course material.
- Knowledge and understanding:
at the end of the course a student should have learned the basic features of the Theory of Error-Correcting Codes, with focus on the main algebraic features
- Applying knowledge and understanding:
at the end of the course a student should be able to apply this knowledge to the description of some standard models of communication systems that involve sending messages along a noisy channel
- Making judgements:
at the end of the course a student should be able to recognize independently some approaches and solving techniques which are typical of Coding Theory
- Communicating skills:
at the end of the course a student should be able to describe the topics presented in the course with an appropriate technical language and a correct mathematical formalism
- Learning skills:
studying the subject matter should stimulate independent learning skills and the capability of treating connected topics in further detail
1) Lezioni del docente disponibili sul portale DOLLY del corso.
2) L. Bernardi, "Algebra e Teoria dei codici correttori", Milano, Franco Angeli, 1994.
3) W. Heise, P. Quattrocchi,
"Informations- und Codierungstheorie".
Collana Springer-Lehrbuch, Springer, Berlin, 1995.
4) L. Giuzzi, "Codici correttori".
Collana Unitext, Springer-Verlag Italia, Milano, 2006.
5) F.J. MacWilliams, N.J.A. Sloane,
"The Theory of Error-Correcting Codes".
Collana North-Holland Mathematical Library,
North-Holland, Amsterdam, 2006.