Low-Rank Parity-Check (LRPC) codes are a class of rank metric codes that have many applications specifically in network coding and cryptography. Recently, LRPC codes have been extended to Galois rings which are a specific case of finite rings. In this paper, we first define LRPC codes over finite commutative local rings, which are bricks of finite rings, with an efficient decoder. We improve the theoretical bound of the failure probability of the decoder. Then, we extend the work to arbitrary finite commutative rings. Certain conditions are generally used to ensure the success of the decoder. Over finite fields, one of these conditions is to choose a prime number as the extension degree of the Galois field. We have shown that one can construct LRPC codes without this condition on the degree of Galois extension.

低秩奇偶校验 (LRPC) 代码是一类秩度量代码，特别在网络编码和密码学中具有许多应用。最近，LRPC 代码已扩展到伽罗瓦环，伽罗瓦环是有限环的一种特殊情况。在本文中，我们首先在有限交换局部环上定义 LRPC 代码，有限交换局部环是有限环的砖块，具有高效的解码器。我们改进了解码器失败概率的理论界限。然后，我们将工作扩展到任意有限交换环。通常使用一定的条件来确保解码器的成功。在有限域上，这些条件之一是选择一个素数作为伽罗瓦域的外延度。我们已经证明，无需伽罗瓦扩展度这一条件，就可以构造 LRPC 代码。