## 1 Introduction  
Multi-access single-relay networks are typical access networks frequently used in multi-user communications. They generally consist of multiple source nodes, a single relay node, and a single destination node, also known as distributed networks [1, 2, 3]. Each source node connects to a common relay node and communicates with the destination node through it. Fountain coding for distributed networks can be represented by two-layer concatenated fountain codes. Firstly, each source node performs the first-layer fountain coding on its original information symbols according to the degree distribution, and then sends them to the relay node. Secondly, the relay node performs the second-layer fountain coding by using the network coding method and sends the encoded symbols to the destination node, rather than simply amplifying and forwarding. Finally, at the destination node, the overall degree distribution of the received symbols can be considered a composite function of the degree distributions of the two-layer fountain codes [4, 5]. With a carefully designed overall degree distribution, the destination node is guaranteed to be able to decode efficiently using a belief propagation (BP) decoder [6, 7]. Distributed fountain coding is an important area in fountain code research. Many researchers have focused on the coding methods [8, 9], performance analysis [4, 10], and degree distribution design [11, 12, 10] of distributed fountain codes, aiming to improve the performance and apply them to more network types [13; 14; 15]. In [1], a distributed fountain coding scheme named DLT (distributed Luby transform) codes was proposed for multiple source networks. By using the deconvolution of robust soliton distribution (RSD) at each source, the resulting bit stream has a degree distribution approximating that of an LT code and has good performance on erasure channels. In [6; 16], two novel distributed rateless coding schemes and the corresponding analysis methods were proposed, respectively. Both of them can provide the unequal error protection (UEP) property. A novel generic fountain coding scheme named generalized distributed fountain codes (GDFC) for multiple access networks was proposed and developed in [5; 12], which can be applied to several instances and provide strong UEP property across sources. The authors in [17] proposed a new relay combining scheme for DLT codes, which improves the error floor performance of DLT codes over additive white Gaussian noise (AWGN) channels. In [18], the authors proposed an efficient fountain coding scheme with a memory-based strategy to enhance the performance of the Y-network topology in 5G and 6G networks, which can achieve better performance than regular LT codes. A rateless two-way multiply-and-forward relaying system on Rayleigh fading channels was proposed in [19], which outperforms the amplify-and-forward relay scheme with optimized degree distributions. In [20], the authors proposed an opportunistic decoding and recoding mechanism with a coordinative routing algorithm for fountain codes in multi-hop wireless sensor networks, which is superior to the existing opportunistic routing-based fountain coding schemes. The authors in [21] proposed an edge caching scheme in vehicular networks using fountain codes, which can improve the transmission efficiency and successful acquisition rate. Rateless codes have attracted much attention for their good channel adaptability and have been increasingly used in various communication systems.