Sparse random linear network coding (SRLNC) used as a class of erasure codes to ensure the reliability of multicast communications has been widely investigated. However, an exact expression for the decoding success probability of SRLNC is still unknown, and existing expressions are either asymptotic or approximate. In this paper, we derive an exact expression for the decoding success probability of SRLNC. The key to achieving this is to propose a criterion that a vector is contained in a subspace. To obtain this criterion, we construct a basis of a subspace, with respect to this basis, the coordinates of a vector are known, based on a maximal linearly independent set of the columns of a matrix. The exactness and the computation of the derived expression are demonstrated by a simple example.

翻译：使用随机线性网络编码(SRLNC)作为确保多信道通信可靠性的取消代码类别,这种随机线性网络编码(SRLNC)是用作确保多信道通信可靠性的加密代码类别,但对此进行了广泛调查,但目前还不清楚该代码解码成功概率的确切表达方式,现有表达方式不是无线表达方式,就是近似表达方式。在本文中,我们得出该代码解码成功概率的确切表达方式。实现这一点的关键在于提出一个标准,即矢量是否包含在一个子空间中。为获得这一标准,我们根据这一标准,我们根据一个矩阵列中最宽的线性独立数据集,构建了一个子空间的基础,矢量的坐标是已知的。衍生表达方式的准确性和计算用一个简单的例子来说明。