# NCMW - Finite Geometry and Coding Theory (2023)

## Speakers and Syllabus

 Name of the Speakers with their affiliation. No. of Lectures Detailed Syllabus Prof. Trygve Johnsen, UiT - The Arctic University of Norway 4+2 Algebraicmethodsforstudyingcodesandmatroids:Thelectureseries willtreatvariouswaysinwhicherror-correctingcodescanbeproduced 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-matroidsand Gabidulin rank-metric codes Prof. Sudhir R. Ghorpade, IIT Bombay. 4+2 Basics of error-correcting codes: (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. Prof. Ilaria Cardinali, University of Siena, Italy 4+2 Finite geometry: Basic Notions : (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. Prof. Luca Giuzzi, University of Brescia, Italy 4+2 Finite geometry: Advanced topics: (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. Prof. Alp Bassa,Boğaziçi University, Istanbul, Turkey 4+2 Algebraic curves over finite fields: 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. Prof. Peter Beelen, Technical University of Denmark, Copenhagen 4+2 Algebraic Geometry codes: 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 Prof. Frederique Oggier, Nanyang Technological University, Singapore 4+2 Lattices and codes: (a) introduction to lattices (including Minkowski's theorem and the notion of geometry of numbers)    (b) construction of lattice/lattice codes    (c) closest lattice point and decoding Prof. Anna-Lena Horlemann- Trautmann, University of St. Gallen, Switzerland 4+2 Rank Metric codes:    (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 distancedecoding. 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 concludewith some open problems in the area.

References:

1. V. Pless and W.C. Huffmann (Eds.), Handbook of Coding Theory, North Holland, Amsterdam, 1998.

2. J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford University Press, New York, 1998,

3. J. W. P. Hirschfeld, G. Korchmáros and F. Torres, Algebraic Curves over a Finite Field, Princeton University Press, Princeton, 2008.

4. (Lecture notes from the speakers, if available)

Names of the tutors with their affiliation:

1. Gianira Alfarano, Postdoc, Eindhoven University of Technology, NETHERLANDS (GA)

2. Alessandro Neri, Postdoc, University of Ghent, BELGIUM (AN)

3. Rakhi Pratihar, Postdoc, INRIA Saclay Centre, FRANCE (RP)

4. Tovohery H. Randrianarisoa, Postdoc, Umea University, SWEDEN (TH)

5. Prasant Singh, Assistant Professor, IIT Jammu, INDIA (PS)

## Time Table

 Day Date Lecture 1 (9.30to10.30) Lecture 2 (10.45to11.45) Lecture 3 (12.00to13.00) Lecture 4 (15.00to16.00) Lecture 5 (16.15to17.15 Open Slot (17.30to18.00) (name of the speaker) (name of the speaker) (name of the speaker) (name of the speaker/ speaker+tutor) (name of the speaker/ speaker+tutor) Participant Presentations Mon 20/11/2023 SG TJ IC AB SG+PS Tues 21/11/2023 SG TJ IC TJ+RP IC+GA Wed 22/11/2023 SG AB IC AB IC Thu 23/11/2023 SG TJ IC SG+PS TJ Fri 24/11/2023 AB LG TJ IC+GA AB+AN Sat 25/11/2023 AB LG AH Sunday off Mon 27/11/2023 FO LG AH AB LG+AN Tues 28/11/2023 PB LG AH PB AH+TR Wed 29/11/2023 FO PB AH FO+GA LG+AN Thu 30/11/2023 PB PB LG PB+PS AH+TR Fri 01/12/2023 FO PB FO+GA PB+PS

Speakers:
AB: Alp Bassa
PB: Peter Beelen
IC: Ilaria Cardinali
LG: Luca Giuzzi
AH: Anna-Lena Horlemann- Trautmann
TJ: Trygve Johnsen
FO: Frederique Oggier

Tutors:
GA: Gianira Alfareno
AN: Alessandro Neri
RP: Rakhi Pratihar
TR: Tovohery Randrianarisoa
PS: Prasant Singh

File Attachments: