# Optimal Multiplexed Erasure Codes for Streaming Messages with Different Decoding Delays

## TL;DR

This paper characterizes the capacity region for multiplexed streaming erasure codes with different decoding delays over burst erasure channels, extending previous results to a new parameter regime.

## Contribution

It fully characterizes the capacity region for the case where the decoding delay of the longer sequence exceeds the sum of the shorter delay and burst length, using superimposed coding strategies.

## Key findings

- Achieves the non-trivial corner point of the capacity region.
- Provides a genie-aided bound for periodic erasure patterns.
- Extends capacity characterization to the case T_v > T_u + B.

## Abstract

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 $T_\mathrm{v}$ and every source message in the second sequence subject to a shorter decoding delay $T_\mathrm{u}\le T_\mathrm{v}$. 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 $T_\mathrm{v}\le T_\mathrm{u}+B$ was previously solved. In this paper, we fully characterize the capacity region for the remaining case $T_\mathrm{v}> T_\mathrm{u}+B$. 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 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-$T_\mathrm{u}$ noiseless duration.