TY - JOUR
T1 - Index coding with side information
AU - Bar-Yossef, Ziv
AU - Birk, Yitzhak
AU - Jayram, T. S.
AU - Kol, Tomer
N1 - Funding Information:
Manuscript received August 25, 2007; revised July 12, 2010; accepted July 22, 2010. Date of current version February 18, 2011. The work of Z. Bar-Yossef was supported by the European Commission Marie Curie International Re-integration Grant. The material in this paper was presented at the 47th IEEE Annual Symposium on Foundations of Computer Science (FOCS), Berkeley, CA, October 2006.
PY - 2011/3
Y1 - 2011/3
N2 - Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1, ⋯ Rn. He holds an input x ∈ {0,1}n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph G; on n nodes; Ri knows the bits of x in the positions {j Ι (i,j) is an edge of G}. G is known to the sender and to the receivers. We call encoding schemes that achieve this goal indexcodes for { 0,1 }n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear index codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd anti-holes, minrank is the optimal length of arbitrary index codes. For arbitrary index codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but has been shown not to be tight.
AB - Motivated by a problem of transmitting supplemental data over broadcast channels (Birk and Kol, INFOCOM 1998), we study the following coding problem: a sender communicates with n receivers R1, ⋯ Rn. He holds an input x ∈ {0,1}n and wishes to broadcast a single message so that each receiver Ri can recover the bit xi. Each Ri has prior side information about x, induced by a directed graph G; on n nodes; Ri knows the bits of x in the positions {j Ι (i,j) is an edge of G}. G is known to the sender and to the receivers. We call encoding schemes that achieve this goal indexcodes for { 0,1 }n with side information graph G. In this paper we identify a measure on graphs, the minrank, which exactly characterizes the minimum length of linear and certain types of nonlinear index codes. We show that for natural classes of side information graphs, including directed acyclic graphs, perfect graphs, odd holes, and odd anti-holes, minrank is the optimal length of arbitrary index codes. For arbitrary index codes and arbitrary graphs, we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph. This bound holds even for randomized codes, but has been shown not to be tight.
KW - Broadcast channels
KW - code length
KW - error correction coding
KW - information cost
UR - http://www.scopus.com/inward/record.url?scp=79951927053&partnerID=8YFLogxK
U2 - 10.1109/TIT.2010.2103753
DO - 10.1109/TIT.2010.2103753
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AN - SCOPUS:79951927053
SN - 0018-9448
VL - 57
SP - 1479
EP - 1494
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 3
M1 - 5714242
ER -