New Geometric and Algebraic Foundations of Coding Theory

01 October 2022 → 15 May 2024
Research Foundation - Flanders (FWO)
Research disciplines
  • Natural sciences
    • Field theory and polynomials
    • Linear and multilinear algebra, matrix theory
    • Information and communication, circuits
    • Geometry not elsewhere classified
    • Coding and information theory
Skew Polynomial Ring Algebraic Coding Theory Finite Geometry
Project description

Algebraic coding theory was created in order to design error-correcting codes for reliable data transmission through noisy channels. However, coding theory is also used when there is control on the channel, and errors are designed ad-hoc for privacy/security purposes. This is the case of secret sharing schemes: a secret is shared among several participants, who can access it only if enough of them agree. In this project, I will develop a more fundamental understanding of the mathematics underpinning these applications, by studying the algebra and geometry behind a variety of so-called "metrics". I will study various aspects of algebraic and geometric structures and their interactions, allowing me to prove new results which will guide future real-world applications. A well-known example of the interaction between geometry and coding theory is the correspondence between linear codes and sets of points in a projective space, which was recently extended to additive codes and to linear rank-metric codes. Furthermore, codes in the Hamming, rank and sum-rank metric can all be represented in an algebraic way via noncommutative polynomials. We believe that the correspondences observed so far are only a few special cases of a wider theory bridging three worlds: algebraic coding theory and the theory of supports, finite geometry and intersection theory, noncommutative polynomials and their space of roots. The goal of this project is to identify and develop such a theory.