CIMPA course on Algebraic and Geometric Methods in Coding Theory

The Institute for Research in Fundamental Sciences (IPM), in Iran, will host the CIMPA course on Algebraic and Geometric Methods in Coding Theory on January 27-30, 2024. To register and for more information, please see the event website. This CIMPA course will be followed by a conference, the 19th Seminar on Commutative Algebra and Related Topics (January 31-February 1, 2024).

Lecturers:

• Sudhir Ghorpade (Indian Institute of Technology Bombay, India)
• Mesut Şahin (Hacettepe University, Turkey)

The aim of this course is to introduce some new research directions in coding theory and especially its interactions with commutative algebra and algebraic geometry. The course is meant for young researchers and PhD students, and it is intended for them to be acquainted with some topics of contemporary research interest. Considering the short duration of this course, we will assume familiarity with the basics of coding theory, and also some rudimentary knowledge of commutative algebra and algebraic geometry. We will focus mainly on the following topics:

• Toric Varieties and Toric Codes

The main theme of this part will be the linear codes defined on zero-dimensional subvarieties of a toric variety over a finite field. We briefly discuss local structure of an abstract toric variety, namely we review basics of (normal) affine toric varieties, (saturated) affine semigroups, toric ideals and their relations to strongly convex rational polyhedral cones. We discuss certain properties of projective toric varieties and give several examples. We exhibit the precise relation between the combinatorics of a fan and the geometry of the corresponding normal toric variety. We also discuss the GIT quotient representation of a simplicial normal toric variety and introduce homogeneous coordinates as done for a projective space. These will enable us to introduce and study the main parameters of a toric code, i.e. an evaluation code defined on some subset of rational points on a normal toric variety over a finite field.

• Linear Codes associated to higher dimensional varieties

Geometric approach to linear codes via the language of projective systems. The following specific classes of linear codes and some of their fundamental properties including determination of basic parameters will be discussed.

– Codes associated to Veronese varieties (Projective Reed-Muller Codes)
– Grassmann codes
– Schubert codes

• Betti numbers of linear codes and matroids

Matroids and simplicial complexes associated to linear codes. Graded minimal free resolutions and Betti numbers of Stanley-Reisner rings of these simplicial complexes. Relations with generalized Hamming weights and generalized weight enumerator polynomials of linear codes.

Organizers:

• Sara Faridi (Dalhousie University, Canada)
• Hassan Haghighi (K. N. Toosi University of Technology, Iran)
• Raheleh Jafari (IPM and Kharazmi University, Iran)
• Abbas Nasrollah Nejad (IPM and Institute for Advanced Studies in Basic Sciences, Iran)