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

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 (\$\mathcal{G}\$-) codes up to a given radius \$\tau\$. This algorithm is based on a symbolic equivalent of the \textit{Euclidean Algorithm} (EA) and can be applied for decoding of \$\mathcal{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 \$\mathcal O(\tau^2)\$ and is a generalization of the modified EA by Bossert and Bezzateev for Reed-Solomon codes.     «

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 (\$\mathcal{G}\$-) codes up to a given radius \$\tau\$. This algorithm is based on a symbolic equivalent of the \textit{Euclidean Algorithm} (EA) and can be applied for decoding of \$\mathcal{G}\$-codes beyond half the minimum rank distance. If the key equation has a unique solution, our algorithm reduces to Gabidulin's decoding algorithm...     »