  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.
