Notes on "Exact Decoding Probability Under Random Linear Network Coding"

In a recent letter, Trullols-Cruces et al. derived the exact decoding probability P_ns(K,N) that a receiver node obtained N linearly independent coded packets among K (K≥ N) received coded packets, when the source node uses a random linear coding scheme over a Galois Field GF(q). They discussed P_ns(K,N) both in the case of the source node sending the random linear combination of the original packets, including all-zero combinations, as well as the case of excluding all-zero combinations. However, their derivations are a bit complicated. In this note, we demonstrate simple derivations of the exact decoding probability P_ns(K,N) in both cases. Furthermore, we point out that there is a more succinct expression of P_ns(K,N) for the latter case.