Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (2019), we define and study LRPC codes over Galois rings - a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.'s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.

中文翻译：

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伽罗瓦环上的低秩奇偶校验码

低秩奇偶校验 (LRPC) 是有限域上的秩度量代码，由 Gaborit 等人提出。(2013) 用于加密应用程序。受到 Kamche 等人最近将 Gabidulin 代码适应某些有限环的启发。(2019)，我们定义并研究了 Galois 环上的 LRPC 代码 - 一类广泛的有限交换环。我们给出了一个类似于 Gaborit 等人的解码器的解码算法，基于简单的线性代数运算。我们推导出解码器失败概率的上限，这比有限域的情况要复杂得多。界限仅取决于错误的等级，即独立于其自由等级。此外，我们分析了解码器的复杂性。我们得到在伽罗瓦环上存在一类 LRPC 代码，它可以解码与具有相同代码参数的 Gabidulin 代码大致相同数量的错误，但比目前最好的 Gabidulin 代码解码器更快。然而，一个人需要付出的代价是一个小的失败概率，我们可以从上面绑定。