2 research outputs found
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 (with number of vertices
equal to the number of source symbols). The size of the maximum acyclic induced
subgraph, denoted by is a lower bound on the \textit{broadcast rate}.
For IC problems with and , prior work has shown that
binary (over ) linear index codes achieve the 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 , binary linear index codes achieve the lower bound on
the broadcast rate. In contrast, it is known that there exists IC problems with
and optimal broadcast rate strictly greater than
Index Coding: Rank-Invariant Extensions
An index coding (IC) problem consisting of a server and multiple receivers
with different side-information and demand sets can be equivalently represented
using a fitting matrix. A scalar linear index code to a given IC problem is a
matrix representing the transmitted linear combinations of the message symbols.
The length of an index code is then the number of transmissions (or
equivalently, the number of rows in the index code). An IC problem is called an extension of another IC problem if the
fitting matrix of is a submatrix of the fitting matrix of . We first present a straightforward \textit{-order} extension
of an IC problem for which an index code is
obtained by concatenating copies of an index code of . The length
of the codes is the same for both and , and if the
index code for has optimal length then so does the extended code for
. More generally, an extended IC problem of having
the same optimal length as is said to be a \textit{rank-invariant}
extension of . We then focus on -order rank-invariant extensions
of , and present constructions of such extensions based on involutory
permutation matrices