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Sparse random linear network coding (SRLNC) used as a class of erasure codesto ensure the reliability of multicast communications has been widelyinvestigated. However, an exact expression for the decoding success probabilityof SRLNC is still unknown, and existing expressions are either asymptotic orapproximate. In this paper, we derive an exact expression for the decodingsuccess probability of SRLNC. The key to achieving this is to propose acriterion 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 coordinatesof a vector are known, based on a maximal linearly independent set of thecolumns of a matrix. The exactness and the computation of the derivedexpression are demonstrated by a simple example.

Exact Decoding Probability of Sparse Random Linear Network Coding for Reliable Multicast