Generalization of Gabidulin Codes over Fields of Rational Functions

TL;DR

This paper extends Gabidulin codes to fields of rational functions using Galois automorphisms, enabling new code constructions over algebraic function fields with potential applications in convolutional coding.

Contribution

It introduces a novel generalization of Gabidulin codes over rational function fields via Galois automorphisms, broadening their applicability beyond finite fields.

Findings

Defined Gabidulin codes over rational function fields.

Constructed examples using Kummer and Artin-Schreier extensions.

Linked the matrices to convolutional code generators.

Abstract

We transpose the theory of rank metric and Gabidulin codes to the case of fields which are not finite fields. The Frobenius automorphism is replaced by any element of the Galois group of a cyclic algebraic extension of a base field. We use our framework to define Gabidulin codes over the field of rational functions using algebraic function fields with a cyclic Galois group. This gives a linear subspace of matrices whose coefficients are rational function, such that the rank of each of this matrix is lower bounded, where the rank is comprised in term of linear combination with rational functions. We provide two examples based on Kummer and Artin-Schreier extensions.The matrices that we obtain may be interpreted as generating matrices of convolutional codes.