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Gabidulin codes, originally defined over finite fields, are an importantclass of rank metric codes with various applications. Recently, theirdefinition was generalized to certain fields of characteristic zero and aWelch--Berlekamp like algorithm with complexity $O(n^3)$ was given. We proposea new application of Gabidulin codes over infinite fields: low-rank matrixrecovery. Also, an alternative decoding approach is presented based on a Gaotype key equation, reducing the complexity to at least $O(n^2)$. This methodimmediately connects the decoding problem to well-studied problems, which havebeen investigated in terms of coefficient growth and numerical stability.

An Alternative Decoding Method for Gabidulin Codes in Characteristic Zero