Elena Berardini

Project title: CRIS&S (Codes in the Rank metric, Information Security & Storage)

Host Institution: Eindhoven University of Technology (TU/e)

Host Supervisor: Prof. Alberto Ravagnani

Co-host Institution: École Polytechnique (L’X)

Co-host Supervisor: Prof. Alain Couvreur

Summary project:

Information security and data integrity are, and will continue to be, some of the greatest challenges of the modern world. Strategies that have been recently proposed to guarantee these are based on error-correcting codes, and in particular on rank-metric codes. Having access to a large variety of rank-metric codes with different properties is crucial for the aforementioned strategies to be implemented and made robust. Quantum computing is a rapidly-emerging technology that uses quantum mechanics to solve problems too complex for classical computers. To build and operate quantum computers we must rise to the challenge of protecting quantum information from errors that occur due to quantum noise and interference from the environment.

These emerging applications raise new challenges in coding theory and provide an incentive to anticipate future issues by generating new research in coding theory and related topics. The goal of this project is twofold: to apply sophisticated methods from algebraic geometry to construct new families of classical codes in the rank metric for post-quantum cryptography, and to study and conceive families of error-correcting quantum codes.

Concerning the main goal, I introduced the first algebraic geometry construction of codes in the sum-rank metric, which I called linearized Algebraic Geometry codes (have a look here https://arxiv.org/abs/2303.08903). I also made progress on the theory behind the computation of Riemann-Roch spaces, a tool that is central to the implementation of algebraic geometry codes. In particular, I proposed a new autonomous and elementary proof of the Brill–Noether method for computing Riemann-Roch spaces (have a look here https://arxiv.org/abs/2208.12725).

Currently, I am focusing on the second goal, by studying the structural properties of quantum codes. Notably, I proved some density results on the existence of quantum codes, as long as some crucial bounds on the parameters of particular families of quantum codes.