Whether it comes to constructing new codes or developing new encoding or decoding methods, algebra has been one of the most fundamental tools
for coding theorists. Several different aspects of algebra have played an important role in developing major breakthroughs.
In this edition of the summer school, we delve into four such important algebraic aspects of error-correcting codes, namely q-combinatorics, algebraic geometry, commutative algebra, and association schemes. These will be disseminated in form of mini-courses.
SOME q-ANALOGUES IN COMBINATORICS
University College Dublin, Ireland
In this mini-course we will introduce three different q-analogues of combinatorics that have gained prominence in recent years, namely rank-metric codes, subspace designs over finite fields, and q-(poly)matroids. While each is a topic of independent interest, as with their classical counterparts, there are many connections between them. Rank metric codes are linear spaces of matrices where the distance between a pair of codewords is measured as the rank of their difference. They arise in applications such as error-correction in networks, criss-cross error correction, and in code-based cryptography. A t-subspace design over GF(q) is a collection of subspaces (called blocks) of a fixed dimension k with the property that every t-dimensional subspace is contained in the same number of blocks. A special subclass of these, the q-Steiner systems, are also optimal when viewed as subspace codes, which arise in network coding. The q-matroids and q-polymatroids correspond to submodular functions on the lattice of subspaces of a fixed vector space. A subclass of these, can be represented via rank-metric codes. Furthermore, several invariants of rank-metric codes are actually evaluations of functions associated with the underlying q-polymatroid. In special circumstances, these invariants indicate whether or not a subspace design can be identified in a q-polymatroid or a rank-metric code. In the first part of this course, we will introduce these three different objects with some comparison to their classical counterparts. In the second part of the course we will outline connections between them and how in some cases, one structure can be used to construct the other.
CODES AND MODULAR CURVES
INRIA Saclay-École Polytechnique, France
This lecture aims to introduce algebraic geometry codes and in particular the seminal result, dating back to 1982 which made them so famous. Namely, Tsfasman Vladut and Zink and independently by Ihara proved that some infinite families of modular curves have a ratio number of rational points divided by their genus going to √q - 1. This result lead to a breakthrough in coding theory, proving that some algebraic geometry codes are asymptotically better than random codes. The lecture will start by some basic notions on algebraic geometry codes and the way to estimate their parameters. Then, we will move to elliptic and modular curves in order to give a proof of Tsfasman-Vladut-Zink Theorem.
TO HIT OR TO BE HIT, THAT'S THE ISSUE
Aalborg University, Denmark
Algebraic methods play a crucial role in the theory of error-correcting codes. This is due to the fact, that applying algebraic structures with certain nice properties, one can estimate the corresponding code parameters (length, dimension, and minimum distance) and often also describe efficient decoding algorithms. In this mini course we treat the different levels of abstraction that can be used when describing algebraic codes. At the most basic level we have a pure linear algebra description in combination with information on how the component wise inner product behaves. Here, the phrase “How much can a given word hit” refers to a method for estimating the minimum distance of primary codes, and similarly “How much can hit a given word” to a method for dual codes. Higher order abstractions may involve commutative algebra, Gröbner basis theory, and the theory of algebraic function fields. Besides discussing the relationship between the different descriptions (levels of abstraction) we apply the methods to construct, not only classical error-correcting codes, but also secret sharing schemes and quantum error-correcting codes.
ALGEBRAIC COMBINATORICS AND CODING THEORY
University of Paderborn, Germany
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.
|May 31, 2022||Abstract submission deadline for a contributed talk|
|July 4-8, 2022||Summer School|
|08:45 - 09:00||Opening|
|09:00 - 11:00||Eimear Byrne||Alain Couvreur||ACT Graduate
|11:00 - 11:30||Coffee Break|
|11:30 - 12:30||Eimear Byrne||Alain Couvreur||ACT Graduate
|12:30 - 13:30||Lunch Break|
|13:30 - 15:30||Kai-Uwe Schmidt||Olav Geil||ACT Graduate
|15:30 - 16:00||Coffee Break|
|16:00 - 17:00||Kai-Uwe Schmidt||Olav Geil||ACT Graduate
|17:00 - 18:00||Open problem session|
The time indicated in the schedule is referred to Central European Summer Time (CEST).
Registrations are closed.
The summer school will take place at the University of Zurich (Irchel Campus).
All the talks will take place in the Math Department 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.
If you are interested in single/double sharing rooms in a student hostel, please send us your request at firstname.lastname@example.org
For general queries please contact:
Gianira AlfaranoInstitute for Mathematics