Index Coding with Side Information

Motivated by a problem of transmitting data over broadcast channels (BirkandKol, INFOCOM1998), we study the following coding problem: a sender communicates with n receivers R_l,.., R_n. He holds an input x isin {0, 1}_n and wishes to broadcast a single message so that each receiver R_i can recover the bit x_i. Each R_i has prior side information about x, induced by a directed graph G on n nodes; R_i knows the bits of x in the positions {j | (i, j) is anedge of G}. We call encoding schemes that achieve this goal INDEX codes for {0, 1} ⁿ with side information graph G. In this paper we identify a measure on graphs, the minrank, which we conjecture to exactly characterize the minimum length of INDEX codes. We resolve the conjecture for certain natural classes of graphs. For arbitrary graphs, we show that the minrank bound is tight for both linear codes and certain classes of non-linear codes. For the general problem, we obtain a (weaker) lower bound that the length of an INDEX code for any graph G is at least the size of the maximum acyclic induced subgraph of G