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

Antonia Wachter; Vladimir Sidorenko; Martin Bossert

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arXiv · 1006.1743

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

Antonia Wachter·Vladimir Sidorenko·Martin Bossert

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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.

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Antonia Wachter, Vladimir Sidorenko, Martin Bossert. 2010-06-09. A Basis for all Solutions of the Key Equation for Gabidulin Codes. https://arxiv.org/abs/1006.1743

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