Optimal Multiplexed Erasure Codes for Streaming Messages with Different Decoding Delays
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This paper considers multiplexing two sequences of messages with two different decoding delays over a packet erasure channel. In each time slot, the source constructs a packet based on the current and previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. The destination must perfectly recover every source message in the first sequence subject to a decoding delay T_v and every source message in the second sequence subject to a shorter decoding delay T_u< T_v. We assume that the channel loss model introduces a burst erasure of a fixed length B on the discrete timeline. Under this channel loss assumption, the capacity region for the case where T_v< T_u+B was previously solved. In this paper, we fully characterize the capacity region for the remaining case T_v> T_u+B. The key step in the achievability proof is achieving the non-trivial corner point of the capacity region through using a multiplexed streaming code constructed by superimposing two single-stream codes. The converse is proved by obtaining a genie-aided bound when the channel is subject to a periodic erasure pattern where each period consists of a length-B burst erasure followed by a length-T_u noiseless duration.
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