Optimal Degree Distribution for LT Codes with Small Message Length

Fountain codes provide an efficient way to transfer information over erasure channels. We give an exact performance analysis of a specific type of fountain codes, called LT codes, when the message length N is small. Two different approaches are developed. In a Markov chain approach the state space explosion, even with reduction based on permutation isomorphism, limits the analysis to very short messages, N < 4. An alternative combinatorial method allows recursive calculation of the probability of decoding after N received packets. The recursion can be solved symbolically for values of N < 10 and numerically up to N ap30. Examples of optimization results give insight into the nature of the problem. In particular, we argue that a few conditions are sufficient to define an almost optimal LT encoding.