Fast decoding of Gabidulin codes

Antonia Wachter-Zeh, Valentin B. Afanassiev, Vladimir Sidorenko

Abstract: Gabidulin codes are the analogues of Reed–Solomon codes in rank metric and play an important role in various applications. In this contribution, a method for efficient decoding of Gabidulin codes up to their error correcting capability is shown. The new decoding algorithm for Gabidulin codes (defined over $${\mathbb{F}_{q^m}}$$ ) directly provides the evaluation polynomial of the transmitted codeword. This approach can be seen as a Gao-like algorithm and uses an equivalent of the Euclidean Algorithm. In order to achieve low complexity, a fast symbolic product and a fast symbolic division are presented. The complexity of the whole decoding algorithm for Gabidulin codes is $${\mathcal{O} (m^3 \, \log \, m)}$$ operations over the ground field $${\mathbb{F}_q}$$ .