Establishing reliable and efficient multicast services in heterogeneous networks under an erasure channel is one of the current priorities in coding theory, particularly in Network Coding (NC) over data networks. Additionally, the increasing number of clients with mobile devices with high processing capabilities and the prevalence of non-delay tolerant traffic have led to the demand for non-feedback multicast schemes regarding distributed resource management. Current communication platforms lack gradual and dynamic coding control based on the type of data being transmitted. This paper proposes a reliable transmission scheme based on a concatenation of systematic coding and Random Linear Network Coding (RLNC) called Fulcrum coding. This hybrid rateless coding scheme with distributed characteristics allows the implementation of an adaptive resource management system to increase the decoding probability during data reception. The method ultimately results in a higher network throughput and shorter transmission times (RTT) per packet by implementing efficient Forward Error Correction (FEC).

In this article, we introduce a new type of pseudo-differential equations naturally connected with non-archimedean pseudo-differential operators and whose symbols are new classes of negative definite functions in the p-adic context and in arbitrary dimension. These equations are proposed as a mathematical models to study the spreading of infectious diseases (say COVID-19) through a random walk on a complex energy landscape and taking into account social clusters in a situation of extreme social isolation.

We formalize and evaluate an adaptation of the Middleware Fulcrum coding scheme, developed under a Fog computing architecture, for 5G-IoT mobile devices. This adaptation includes a D2D extension with distributed coding, specifically designed to address the challenges posed by a dynamic topology that involves heterogeneous devices in terms of processing capabilities. The proposed method is established at the application layer level to allow for the mapping of packet segments to a finite field scheme (Galois field), which allows for exploiting the algebraic properties of the system to exercise data retrieval with encoded information. In addition, it will enable dynamic adjustment of the coding parameters according to the processing capacity of the communication devices, wireless channel conditions, network topology, and type of stream in the system. In this sense, high-performance coding is distributed to nodes that can offer higher coding complexity in the design, effectively improving the wireless network’s performance in many aspects, such as delay, energy consumption, and goodput. By characterizing the system as an algebraic system, it is possible to exploit the advantages of FEC through decoding to reduce the signaling in the system.