Optimal Index Codes via a Duality between Index Coding and Network Coding

In Index Coding, the goal is to use a broadcast channel as efficiently as possible to communicate information from a source to multiple receivers which can possess some of the information symbols at the source as side-information. In this work, we present a duality relationship between index coding (IC) and multiple-unicast network coding (NC). It is known that the IC problem can be represented using a side-information graph $G$ (with number of vertices $n$ equal to the number of source symbols). The size of the maximum acyclic induced subgraph, denoted by MAI S is a lower bound on the broadcast rate. For IC problems with MAIS = n - 1 and MAI S = n - 2, prior work has shown that binary (over F 2) linear index codes achieve the MAI S lower bound for the broadcast rate and thus are optimal. In this work, we use the the duality relationship between NC and IC to show that for a class of IC problems with MAIS = n - 3, binary linear index codes achieve the MAI S lower bound on the broadcast rate. In contrast, it is known that there exists IC problems with MAIS = n - 3 and optimal broadcast rate strictly greater than MAIS.