A Basis for all Solutions of the Key Equation for Gabidulin Codes

TL;DR

This paper introduces an efficient algorithm based on a symbolic Euclidean Algorithm to find a basis for all solutions of the key equation in decoding Gabidulin codes, enabling decoding beyond half the minimum distance.

Contribution

It presents a novel algorithm that generalizes existing methods, providing a basis for all solutions of the key equation in Gabidulin code decoding, applicable beyond half the minimum distance.

Findings

Algorithm has time complexity O(tau^2)

Enables decoding beyond half the minimum distance

Reduces to Gabidulin's decoding when solution is unique

Abstract

We present and prove the correctness of an efficient algorithm that provides a basis for all solutions of a key equation in order to decode Gabidulin (G-) codes up to a given radius tau. This algorithm is based on a symbolic equivalent of the Euclidean Algorithm (EA) and can be applied for decoding of G-codes beyond half the minimum rank distance. If the key equation has a unique solution, our algorithm reduces to Gabidulin's decoding algorithm up to half the minimum distance. If the solution is not unique, we provide a basis for all solutions of the key equation. Our algorithm has time complexity O(tau^2) and is a generalization of the modified EA by Bossert and Bezzateev for Reed-Solomon codes.