The landscape of algebraic coding theory has undergone major changes in the last years. In particular, the theory of error-correcting codes has inspired many mathematicians who were interested in applying techniques from algebra, algebraic geometry and discrete mathematics in order to progress on questions in information processing. Indeed, coding theory lies at the intersection of several disciplines in pure and applied mathematics such as algebra, number theory, probability theory, statistics, combinatorics, complexity theory, and statistical physics, which all have helped in the past to increase our knowledge in communication theory.
This summer school aims to focus the attention on different algebraic aspects of the theory of error-correcting codes. The summer school is structured in the following four mini-courses, each of which is intended to deepen a specific aspect of coding theory.
SUBSPACE AND MATRICES IN CODING THEORY
The purpose of this course is to motivate and introduce the theory of matrix codes. We will start with a brief introduction to error-correction in multicast networks during which we will describe different error models and show how the concept of subspace codes and rank metric codes can be applied in these models. We will give some basic coding theoretic bounds and introduce the maximum rank distance codes. We will define the Delsarte-Gabidulin and twisted Gabidulin codes, and describe their properties. Rank-metric codes will be defined more generally in the context of matrix codes. We will give structural properties of matrix codes and discuss the notion of support for this class of codes. We will hence prove the MacWilliams duality theorem for matrix codes, applying Moebius inversion. We will close the course by describing recent advances in the theory of matrix codes and outline a number of new research directions.
ALGEBRAIC GEOMETRY AND CODES
The objective of this course is to present a proof of Tsfasman-Vladut-Zink Theorem asserting the existence of sequences of codes whose asymptotic parameters beat those of random codes. To emphasize the spectacular aspect of such a result we will start with some probabilistic considerations of codes to understand the typical behaviour of codes whose length go to infinity. Then, after a short introduction to algebraic curves, we will introduce Goppa's original construction of the so-called Algebraic Geometry (AG) codes, study some basic features of these codes and then move to the proof of the main theorem. The proof involves many properties of elliptic and modular curves that will be detailed.
COMMUTATIVE ALGEBRA AND CODES
With the aim of estimating the minimum distance of algebraic and algebraic geometric codes we study methods from Gröbner basis theory on how to upper bound the size of varieties over finite fields. As a special case we obtain a simplified proof of the Goppa bound for one-point algebraic geometric codes. Furthermore, we explain the Feng-Rao bounds which in the context of one-point algebraic geometric codes can be viewed as improvements of the Goppa bound, but which can be applied to a much broader class of algebraic and algebraic geometric codes. The presented machinery allows us to establish good secret sharing schemes and quantum codes.
COMBINATORICS AND CODING THEORY
In this course, we consider coding theory from a broader perspective using association schemes. In general, this concept brings in algebraic methods to study problems in extremal combinatorics. This course provides a gentle introduction to association schemes and explains how coding-theoretic problems in various spaces fit into this concept. We will also study the notion of a design in an association scheme, which can be viewed as an object dual to a code. We will then explore how linear programming leads to powerful bounds on codes and designs and sometimes to classification results.
In addition to the mini-courses, open problem sessions and contributed talk sessions are planned during the week, according to the detailed program schedule. The event is dedicated to junior researchers, doctorate students and Postdocs, as well as other established researchers that are interested in the topic.
|March 15, 2020||Registration deadline|
|April 15, 2020||Notification of acceptance of participation|
|June 8-12, 2020||Summer School|
Will be published soon...
Click here to register. Registration is open until March 15, 2020.
The summer school will take place at the University of Zurich (Irchel Campus).
All the talks will be given in room H12 of the building Y27.
Information to reach the venue with public transportation is available here.
Information to reach the venue by car is available here.
For public transportation, tickets should be bought BEFORE boarding train or tram or bus; there are machines at all stations and tram/bus stops. If you are travelling more than once, it may well be cheaper to buy a 'Tageskarte' (24 hour ticket).
Information about some hotels and hostels can be found here.
For general queries please contact:
Institute for Mathematics
University of Zurich
Phone: +41 44 635 60 58