Algorithm and analysis of double-set online fountain code with limited feedback in underwater acoustic networks

Underwater acoustic networks (UANs) are characterized by long propagation latencys, high bit error rates, and limited energy. These characteristics often cause data loss and transmission errors, posing significant challenges to reliable underwater data transmission. Online fountain code (OFC) provides a promising solution for reliable transmission in UANs because of its rateless property, real-time feedback, and low overhead. However, the Sequential Recursive Online Fountain Code with Limited Feedback (SROFC-LF) shceme exhibits two essential limitations as verified by random graph theory analysis. First, a high bit error rate slows the formation of the largest connected component during the build-up phase. Second, the fully random encoding and feedback strategies used in the completion phase may cause unnecessary waiting and increase transmission latencys. To address these issues, this paper proposes a Double-Set Online Fountain Code with Limited Feedback (DSOFC-LF). During the build-up phase, DSOFC-LF employs a double-set encoding strategy to accelerate the formation of the largest connected component. During the completion phase, it introduces a state feedback strategy. This strategy adjusts the generation of encoded packets and the feedback process according to the decoding state at the receiver, thereby reducing unnecessary waiting. Furthermore, mathematical induction and the basic properties of trees are used to theoretically analyze the number of encoded packets required for complete decoding. Simulation experiments are then conducted to evaluate the performance of DSOFC-LF. The results show that DSOFC-LF outperforms most existing OFC schemes in terms of both overhead and the number of feedback packets. When the channel erasure rate is 0.2, DSOFC-LF requires 5.36 fewer feedback packets and reduces the overhead by 13.7% compared with SROFC-LF. Therefore, DSOFC-LF is more suitable for UANs with high error rates and limited resources.