Index Coding with Side Information

Ziv Bar-Yossef, Yitzhak Birk, T. S. Jayram, Tomer Kol

Abstract: 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> ,.., R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . He holds an input x isin {0, 1} <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> and wishes to broadcast a single message so that each receiver R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> can recover the bit x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> . Each R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> has prior side information about x, induced by a directed graph G on n nodes; R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i </sub> 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} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> 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