We present a new decoding algorithm based on error locating pairs and correcting an amount of errors exceeding half the minimum distance. When applied to Reed–Solomon or algebraic geometry codes, the algorithm is a reformulation of the so-called power decoding algorithm. Asymptotically, it corrects errors up to Sudan’s radius. In addition, this new framework applies to any code benefiting from an error locating pair. Similarly to Pellikaan’s and Kötter’s approach for unique algebraic decoding, our algorithm provides a unified point of view for decoding codes with an algebraic structure beyond the half minimum distance. It permits to get an abstract description of decoding using only codes and linear algebra and without involving the arithmetic of polynomial and rational function algebras used for the definition of the codes themselves. Such algorithms can be valuable for instance for cryptanalysis to construct a decoding algorithm of a code without having access to the hidden algebraic structure of the code.

中文翻译：

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电源错误定位对

我们提出了一种基于错误定位对和纠正超过最小距离一半的错误量的新解码算法。当应用于 Reed-Solomon 或代数几何代码时，该算法是所谓的幂解码算法的重新表述。渐近地，它纠正了苏丹半径的错误。此外，这个新框架适用于任何受益于错误定位对的代码。与 Pellikaan 和 Kötter 的独特代数解码方法类似，我们的算法为解码具有超过半最小距离的代数结构的代码提供了统一的观点。它允许仅使用代码和线性代数获得解码的抽象描述，而不涉及用于定义代码本身的多项式和有理函数代数的算术。