A Homomorphism Transform Based Analysis of Sparse Random Linear Network Coding Over Erasure Channels

In this letter, we provide a method for deriving the first exact expression for the decoding success probability of sparse random linear network coding over erasure channels. The key idea of the proposed method is to transform the characterization of the full rank probability of a sparse random matrix into the characterization of the homomorphism transform of the sparse distribution and to obtain a closed-form exact expression for the homomorphism transform of the sparse distribution. The former is solved by employing Möbius inversion theorem and the property of the homomorphism transform, and the latter is solved by employing the feature of the sparse distribution and the property of the nontrivial character of a finite field.