1. Course Instructor: Prof. Trygve Johnsen, UiT - The Arctic University of Norway.
Number of Lectures and Tutorial Sessions: 4 + 2
Title: Algebraic methods for studying codes and matroids
Abstract: The lecture series will treat various ways in which error-correcting codes can be produced or studied using techniques from algebra and combinatorics. We will treat relations between matroids and linear codes. We will also introduce and study the larger class of almost affine codes and show how some of its properties are determined by matroids. In particular, we will show how Stanley-Reisner rings of simplicial complexes, and resolutions of them, various kinds of Betti numbers, determine important properties of the codes. We will also sketch how such techniques can be used to study the properties of q-matroids and Gabidulin rank-metric codes.
2. Course Instructor: Prof. Sudhir Ghorpade, IIT Bombay.
Number of Lectures and Tutorial Sessions: 4 + 2
Title: Basics of error-correcting codes.
Abstract: (a) Review of Classical Coding Theory: Basic notions including linear codes, Hamming weights, generator and parity check matrices, weight distributions, duality, equivalence and automorphisms, generalized Hamming weights. Basic results including several bounds, McWilliams identity, and Wei duality.
(b) Reed-Muller Codes and Grassmann Codes: Generalized Reed-Muller codes and their basic parameters. Projective Reed-Muller codes, Grassmann codes and their relatives.
3. Course instructor: Prof. Ilaria Cardinali, University of Siena, Italy
Number of Lectures and Tutorial Sessions: 4 + 2
Title: Finite geometry: Basic Notions
Abstract: (a) Projective spaces from an axiomatic point of view; Desarguesian projective spaces; coordinatization and algebraic presentation.
(b) Projective systems; the geometric language for projective codes.
(c) Combinatorial configurations in projective planes: arcs, ovals, unitals, blocking sets.
(d) Projective Grassmannians and related codes.
4. Course instructor: Prof. Luca Giuzzi, University of Brescia, Italy
Number of Lectures and Tutorial Sessions: 4 + 2
Title: Finite geometry: Advanced topics.
Abstract: (a) Polar spaces from an axiomatic and an algebraic point of view and their embeddings.
(b) Polar Grassmannians and related codes.
(c) Some classical varieties and constructions: quadrics, Hermitian varieties; Veronese and Segre varieties.
(d) Combinatorial configurations in higher dimension; caps, ovoids and spreads; sets with few intersection numbers; minimal word codes.
5. Course Instructor: Prof. Alp Bassa, Boğaziçi University, Istanbul, Turkey
Number of Lectures and Tutorial Sessions: 4 + 2
Title: Algebraic curves over finite fields
Abstract: Algebraic geometry and in particular algebraic curves have made a prominent appearance in coding theory and cryptography in the last decades. Powerful results from algebraic geometry have been used to improve and generalize long standing results in these fields. We will give a concise introduction to the relevant part of the theory of algebraic curves over finite fields, including the Riemann-Roch Theorem, zeta functions and the Hasse-Weil Theorem.
6. Course Instructor: Prof. Peter Beelen, Technical University of Denmark, Copenhagen
Number of Lectures and Tutorial Sessions: 4 + 2
Title: Algebraic Geometry codes
Abstract: Algebraic geometry (AG) codes are error-correcting codes constructed from algebraic curves defined over a finite field using Goppa's construction from the 1980's. If the algebraic curves are chosen carefully, the resulting codes can have excellent parameters in terms of their rate and relative minimum distance. In the first part of this course, Goppa's construction will be reviewed. Then the basic parameters (length, dimension, and minimum distance) of AG codes will be studied. Examples coming from maximal curves and asymptotically good towers will be given. Finally, the problem of decoding AG codes will be addressed.
7. Course Instructor: Prof. Frederique Oggier, Nanyang Technological University, Singapore
Number of Lectures and Tutorial Sessions: 4 + 2
Title: Lattices and codes
Abstract: (a) introduction to lattices (including Minkowsky's theorem and the notion of geometry of numbers)
(b) construction of lattice/lattice codes
(c) closest lattice point and decoding
8. Course Instructor: Prof. Anna-Lena Horlemann-Trautmann, University of St. Gallen, Switzerland
Number of Lectures and Tutorial Sessions: 4 + 2
Title: Rank Metric codes
Abstract: (a) Motivation and Applications of the Rank Metric: We will describe the coherent network coding channel and how the rank metric arises in this setting as the suitable metric to approximate maximal likelihood decoding with closest distance
decoding. Moreover, we will describe some other applications of the rank metric in criss-cross error correction and code-based cryptography.
(b) Constructions: We will explain the main construction for optimal rank-metric codes, called the Gabidulin code construction, which is based on the evaluation of linearized polynomials, Moreover, we will show several possible generalizations of this construction, including generalized and/or twisted Gabidulin codes. As an outlook, we furthermore present some results on the density of optimal codes in the rank metric, and on these constructions, to show that there are many (yet unknown) optimal rank metric codes that are not equivalent to (generalized twisted) Gabidulin codes. This can be seen as an outlook for open problems to work on in the future.
(c) Decoding Algorithms: Lastly we will describe several decoding algorithms for Gabidulin codes, based again on their structure as evaluations of linearized polynomials. We will show that they can be very efficiently decoded inside the unique decoding radius. Afterwards, we will present several results on the list decoding problem for these codes, and show that the previous statement does not necessarily hold beyond the unique decoding radius anymore. We will again conclude with some open problems in the area.