Probability of complete decoding of random codes for short messages

A random code is a rateless erasure code with a generator matrix of randomly distributed binary values. It encodes a message of k symbols into a potentially infinite number of coded symbols. For asymptotically large k, the tail bound in Kolchin´s theorem asserts that the high probability of complete decoding (PCD) is attained almost surely with k + 10 coded symbols. However, for small values of k (short messages) it is unclear if such asymptotics are useful. That the random codes achieve a high PCD with k + 10 coded symbols for small k is demonstrated. In particular, a set of lemmas is established and show that the PCD converges to five decimal digits after k = 30. A theorem extending Kolchin´s work is formulated and the theorem is used to explain the complete decoding probabilities of random codes in short messages.