Ripple-Oriented Degree Distribution of Rateless Codes: A Backward Design Approach for Finite Block Lengths

Rateless codes are a class of error correction codes designed for efficient and reliable data transmission. Luby transform (LT) codes are the first realization of rateless codes. The degree distribution is crucial for the performance of LT codes, but the existing degree distributions do not perform well in finite block lengths, especially in short block lengths. Different from the forward design, such as robust soliton distribution (RSD), we provide a backward design of degree distribution for LT codes in finite block lengths, which starts from the detailed design of ripple size evolution and then arrives at the corresponding degree distribution. This design pattern addresses the limitation of the traditional forward design that cannot guarantee a high probability of successful decoding under finite block lengths. Specifically, we argue that the ripple should increase and then decrease during the decoding process by analyzing the trade-off between redundancy and ripple disappearance. Therefore, a new random walk model is proposed based on adjustable release probability to describe the ideal ripple size evolution. Furthermore, we present a novel design framework to achieve the desired ripple size evolution and the resulting degree distribution is called ripple-oriented degree distribution (RODD). Simulation results show that RODD is superior to other degree distributions in decoding success rate, block error rate, complexity, etc.