245,253 research outputs found
Error Correction for Index Coding With Coded Side Information
Index coding is a source coding problem in which a broadcaster seeks to meet
the different demands of several users, each of whom is assumed to have some
prior information on the data held by the sender. If the sender knows its
clients' requests and their side-information sets, then the number of packet
transmissions required to satisfy all users' demands can be greatly reduced if
the data is encoded before sending. The collection of side-information indices
as well as the indices of the requested data is described as an instance of the
index coding with side-information (ICSI) problem. The encoding function is
called the index code of the instance, and the number of transmissions employed
by the code is referred to as its length. The main ICSI problem is to determine
the optimal length of an index code for and instance. As this number is hard to
compute, bounds approximating it are sought, as are algorithms to compute
efficient index codes. Two interesting generalizations of the problem that have
appeared in the literature are the subject of this work. The first of these is
the case of index coding with coded side information, in which linear
combinations of the source data are both requested by and held as users'
side-information. The second is the introduction of error-correction in the
problem, in which the broadcast channel is subject to noise.
In this paper we characterize the optimal length of a scalar or vector linear
index code with coded side information (ICCSI) over a finite field in terms of
a generalized min-rank and give bounds on this number based on constructions of
random codes for an arbitrary instance. We furthermore consider the length of
an optimal error correcting code for an instance of the ICCSI problem and
obtain bounds on this number, both for the Hamming metric and for rank-metric
errors. We describe decoding algorithms for both categories of errors
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
The Single-Uniprior Index-Coding Problem: The Single-Sender Case and The Multi-Sender Extension
Index coding studies multiterminal source-coding problems where a set of
receivers are required to decode multiple (possibly different) messages from a
common broadcast, and they each know some messages a priori. In this paper, at
the receiver end, we consider a special setting where each receiver knows only
one message a priori, and each message is known to only one receiver. At the
broadcasting end, we consider a generalized setting where there could be
multiple senders, and each sender knows a subset of the messages. The senders
collaborate to transmit an index code. This work looks at minimizing the number
of total coded bits the senders are required to transmit. When there is only
one sender, we propose a pruning algorithm to find a lower bound on the optimal
(i.e., the shortest) index codelength, and show that it is achievable by linear
index codes. When there are two or more senders, we propose an appending
technique to be used in conjunction with the pruning technique to give a lower
bound on the optimal index codelength; we also derive an upper bound based on
cyclic codes. While the two bounds do not match in general, for the special
case where no two distinct senders know any message in common, the bounds
match, giving the optimal index codelength. The results are expressed in terms
of strongly connected components in directed graphs that represent the
index-coding problems.Comment: Author final manuscrip
A New Class of Index Coding Instances Where Linear Coding is Optimal
We study index-coding problems (one sender broadcasting messages to multiple
receivers) where each message is requested by one receiver, and each receiver
may know some messages a priori. This type of index-coding problems can be
fully described by directed graphs. The aim is to find the minimum codelength
that the sender needs to transmit in order to simultaneously satisfy all
receivers' requests. For any directed graph, we show that if a maximum acyclic
induced subgraph (MAIS) is obtained by removing two or fewer vertices from the
graph, then the minimum codelength (i.e., the solution to the index-coding
problem) equals the number of vertices in the MAIS, and linear codes are
optimal for this index-coding problem. Our result increases the set of
index-coding problems for which linear index codes are proven to be optimal.Comment: accepted and to be presented at the 2014 International Symposium on
Network Coding (NetCod
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
Locally Decodable Index Codes
An index code for broadcast channel with receiver side information is locally
decodable if each receiver can decode its demand by observing only a subset of
the transmitted codeword symbols instead of the entire codeword. Local
decodability in index coding is known to reduce receiver complexity, improve
user privacy and decrease decoding error probability in wireless fading
channels. Conventional index coding solutions assume that the receivers observe
the entire codeword, and as a result, for these codes the number of codeword
symbols queried by a user per decoded message symbol, which we refer to as
locality, could be large. In this paper, we pose the index coding problem as
that of minimizing the broadcast rate for a given value of locality (or vice
versa) and designing codes that achieve the optimal trade-off between locality
and rate. We identify the optimal broadcast rate corresponding to the minimum
possible value of locality for all single unicast problems. We present new
structural properties of index codes which allow us to characterize the optimal
trade-off achieved by: vector linear codes when the side information graph is a
directed cycle; and scalar linear codes when the minrank of the side
information graph is one less than the order of the problem. We also identify
the optimal trade-off among all codes, including non-linear codes, when the
side information graph is a directed 3-cycle. Finally, we present techniques to
design locally decodable index codes for arbitrary single unicast problems and
arbitrary values of locality.Comment: Accepted for publication in the IEEE Transactions on Information
Theory. Parts of this manuscript were presented at IEEE ISIT 2018 and IEEE
ISIT 2019. This arXiv manuscript subsumes the contents of arXiv:1801.03895
and arXiv:1901.0590
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