# Lattices and Index Coding

## Abstract

A wireless channel is characterized by its broadcast nature: a signal transmitted by a source is received not only by the intended recipient but also by all the terminals within the transmission range. This can create a scenario where a passive listener becomes aware of the message originally intended for another node in the broadcast network, while the intended recipient himself is yet to receive the message packet successfully. This might occur, for example, if the main recipient is out of transmission range or if the packet is lost due to fading or channel noise, while the channel gain at the passive listener is strong enough for decoding to be successful. As a result, we have a broadcast channel where the source is required to transmit a finite set of messages intended to be delivered to finitely many receivers, each receiver desires to decode a subset of the transmitted messages while having prior knowledge of the values of a different subset of messages. This prior knowledge at the receivers, called *side information* , can arise when receivers overhear the previous transmissions as information propagates through a communication network in multiple hops or through multiple rounds of transmission.

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