The Performance of Short Random Linear Fountain Codes under Maximum Likelihood Decoding

In this paper, two particular instances of LT codes with short message blocklength k and maximum likelihood (ML) decoding are investigated, i.e., random linear fountain (RLF) codes and (nearly) check-concentrated LT codes. Both show an almost equally good performance. The focus of this paper will be on RLF codes, a type of LT codes whose generator matrices are constructed from independent Bernoulli trials and have a binomial check node degree distribution. A new simple expression for an upper bound on the bit erasure probability under ML decoding is derived for RLF codes with density Δ = 0.5, i.e., with check node degree distribution Ω(x) = 2^-k(1+x)^k. It is shown that RLF codes with a minimum density far less than 0.5 are equally well suited to achieve a certain bit erasure probability for a given reception overhead. Furthermore, a characteristic term from a general upper bound on the bit erasure probability under ML decoding is identified that can be used to optimise check node degree distributions. Its implications on the performance of LT codes are qualitatively analysed.