In a real-time transmission scenario, messages are transmitted through a channel that is subject to packet loss. The destination must recover the messages within the required deadline. In this paper, we consider a setup where two different types of messages with distinct decoding deadlines are transmitted through a channel that can introduce burst erasures of a length at most $B$, or $N$ random erasures. The message with a short decoding deadline $T_u$ is referred to as an urgent message, while the other one with a decoding deadline $T_v$ ($T_v > T_u$) is referred to as a less urgent message. We propose a merging method to encode two message streams of different urgency levels into a single flow. We consider the scenario where $T_v > T_u + B$. We establish that any coding strategy based on this merging approach has a closed-form upper limit on its achievable sum rate. Moreover, we present explicit constructions within a finite field that scales quadratically with the imposed delay, ensuring adherence to the upper bound. In a given parameter configuration, we rigorously demonstrate that the sum rate of our proposed streaming codes consistently surpasses that of separate encoding, which serves as a baseline for comparison.

翻译：在实时传输场景中，消息通过可能发生数据包丢失的信道进行传输。接收端必须在规定时限内恢复消息。本文考虑一种配置：两种具有不同解码截止期限的消息通过同一信道传输，该信道可能引入长度至多为$B$的突发擦除或$N$个随机擦除。解码截止期较短的消息$T_u$称为紧急消息，另一种解码截止期$T_v$（$T_v > T_u$）的消息称为非紧急消息。我们提出一种合并方法，将两种不同紧急程度的消息流编码为单一数据流。考虑$T_v > T_u + B$的场景，我们证明基于该合并方法的任何编码策略的可达和速率存在闭合形式的上界。此外，我们在与所施加延迟呈二次方关系的大小的有限域内给出显式构造，确保满足该上界。在给定参数配置下，我们严格证明所提出的流编码的和速率始终优于作为对比基线的独立编码方案。