20 November 2025 — Padova

Open problems in ring-linear coding theory

Giulia Cavicchioni, University of Trento

Abstract

Ring-linear coding theory extends classical coding theory by employing finite rings as alphabets rather than finite fields. Originally introduced by Assmus and Mattson in 1963, the theory gained renewed attention in the 1990s when Hammons et al. demonstrated that the well-known Kerdock and Preparata codes can be expressed as Gray images of cyclic codes over \(\mathbb{Z}_4\) equipped with the Lee metric. More recently, ring-linear codes have found promising applications in cryptography, where efforts to reduce public key sizes in code-based cryptosystems have motivated the exploration of alternative ambient spaces and metrics beyond vector spaces over finite fields with the Hamming metric.

Although its foundations were established decades ago, ring-linear coding theory has gained significant interest only recently, leaving many open problems in the field. In this talk, we will investigate linear codes over rings, focusing on their algebraic structure, the metrics with which the code is endowed and some main open problems in this area.