Rank metric and Gabidulin codes in characteristic zero

TL;DR

This paper extends the theory of rank metric and Gabidulin codes from finite fields to characteristic zero fields, replacing the Frobenius automorphism with Galois group elements, and adapts decoding algorithms accordingly.

Contribution

It introduces a framework for rank metric and Gabidulin codes in characteristic zero fields, generalizing existing finite field results and providing new decoding methods.

Findings

Conditions on automorphisms for transposing results

Classical polynomial-time decoding algorithm adapted

Multiple definitions for the rank-metric in characteristic zero

Abstract

We transpose the theory of rank metric and Gabidulin codes to the case of fields of characteristic zero. The Frobenius automorphism is then replaced by any element of the Galois group. We derive some conditions on the automorphism to be able to easily transpose the results obtained by Gabidulin as well and a classical polynomial-time decoding algorithm. We also provide various definitions for the rank-metric.