We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sum-rank metric. The speed-ups are achieved by reducing the core of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over usual polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new faster algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials.

中文翻译：

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秩、子空间和和秩度量中代码的快速解码

我们为不同度量中的三个代码类加速了现有的解码算法：秩度量中的交错 Gabidulin 代码、子空间度量中的提升交错 Gabidulin 代码和和秩度量中的线性化 Reed-Solomon 代码。加速是通过将解码器的底层计算问题的核心简化为一种常用工具来实现的：计算偏斜多项式环上矩阵的左右近似基。为了实现这一点，我们描述了现有 PM-Basis 算法的偏斜模拟，用于通常多项式上的矩阵。这捕获了偏斜多项式乘法的大部分工作，并且复杂性优势来自现有算法比经典二次复杂性更快地执行此操作。