Videos

Invariants of Error-Correcting Codes: Theory and Applications

Presenter
August 28, 2025
Abstract
Error-correcting codes are essential in modern information theory, ensuring reliable transmission and storage of digital data in the presence of noise or interference. Their ability to detect and correct errors supports technologies such as communication systems, data storage, secure protocols, and emerging applications in post-quantum cryptography. Designing high-performance codes requires a deep understanding of their combinatorial and algebraic structure. A central goal in coding theory is to establish bounds on key parameters (such as length, dimension, and minimum distance) and to identify invariants that capture the structural properties of a code. These invariants play a fundamental role in the classification of codes and in the construction of codes with strong rigidity and extremality properties. This mini-course focuses on invariants of codes, with an emphasis on tensor codes in the rank metric. We will present a general framework for analyzing these invariants using the theory of anticodes, leading to bounds on code parameters and a generalized form of the MacWilliams identities. Applications will include code equivalence, data storage, encoding strategies, and quantum error correction. The course will also highlight open problems and directions for future research in the area.