Generalized Gabidulin codes over fields of any characteristic

TL;DR

This paper extends Gabidulin codes to infinite fields, including characteristic zero, and explores their decoding, reduction modulo prime ideals, and applications over integer rings, broadening their theoretical and practical scope.

Contribution

It generalizes Gabidulin codes to infinite fields with specific automorphisms and develops decoding algorithms, including over integer rings and modulo prime ideals.

Findings

Codes over number rings reduce to classical Gabidulin codes modulo primes

Decoding algorithms work for errors and erasures in generalized settings

Examples and timing results demonstrate practical feasibility

Abstract

We generalise Gabidulin codes to the case of infinite fields, eventually with characteristic zero. For this purpose, we consider an abstract field extension and any automorphism in the Galois group. We derive some conditions on the automorphism to be able to have a proper notion of rank metric which is in coherence with linearized polynomials. Under these conditions, we generalize Gabidulin codes and provide a decoding algorithm which decode both errors and erasures. Then, we focus on codes over integer rings and how to decode them. We are then faced with the problem of the exponential growth of intermediate values, and to circumvent the problem, it is natural to propose to do computations modulo a prime ideal. For this, we study the reduction of generalized Gabidulin codes over number ideals codes modulo a prime ideal, and show they are classical Gabidulin codes. As a consequence, knowing side information on the size of the errors or the message, we can reduce the decoding problem over the integer ring to a decoding problem over a finite field. We also give examples and timings.