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 MAIS is a lower bound on the broadcast rate. For IC problems with MAIS = n− 1 and MAIS = n− 2, prior work has shown that binary (over F2) linear index codes achieve the MAIS lower bound for the broadcast rate and thus are optimal. In this work, we use 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 MAIS 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.