Optimal Multiplexed Erasure Codes for Streaming Messages With Different Decoding Delays

• Published in: IEEE transactions on information theory Vol. 66; no. 7; pp. 4007 - 4018
• Main Authors: Fong, Silas L., Khisti, Ashish, Li, Baochun, Tan, Wai-Tian, Zhu, Xiaoqing, Apostolopoulos, John
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Burst erasures; capacity region; Codes; Decoding; Delays; different decoding delays; Forward error correction; forward error correction (FEC); Internet; low-latency; Messages; Multiplexing; packet erasure channel; streaming codes; Streaming media
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2020.2977077

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 <inline-formula> <tex-math notation="LaTeX">{T}_{\mathrm{ v}} </tex-math></inline-formula> and every source message in the second sequence subject to a shorter decoding delay <inline-formula> <tex-math notation="LaTeX">{T}_{\mathrm{ u}}\le {T}_{\mathrm{ v}} </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">{T}_{\mathrm{ v}}\le {T}_{\mathrm{ u}}+{B} </tex-math></inline-formula> was previously solved. In this paper, we fully characterize the capacity region for the remaining case <inline-formula> <tex-math notation="LaTeX">{T}_{\mathrm{ v}}> {T}_{\mathrm{ u}}+{B} </tex-math></inline-formula>. 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 main idea in the converse proof is 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-<inline-formula> <tex-math notation="LaTeX">{T}_{\mathrm{ u}} </tex-math></inline-formula> noiseless duration.