Left degree distribution shaping for LT codes over the binary erasure channel

Fountain codes were introduced to provide higher reliability, lower complexities, and more scalability for networks such as the Internet. Luby-Transform (LT) codes, which are the first realization of Fountain codes, achieve the capacity of the binary erasure channel (BEC) asymptotically and universally. For finite lengths, the search is continued to find codes closer to the capacity limits at even lower encoding and decoding complexities. Most previous work on single-layer Fountain coding targets the design via the right degree distribution. The left degree distribution of an LT code is left as Poisson to protect the universality. For finite lengths, this is no longer an issue; thus, we focus on the design of better codes for the BEC at practical lengths. Our left degree shaping provides codes outperforming LT and all other competing schemes in the literature. At a bit error rate of 10^-7 and packet length k = 256, our scheme provides a realized rate of 0.6 which is 23.5% higher than Sorensen et al.´s scheme [1].