14 research outputs found
Graph Theory versus Minimum Rank for Index Coding
We obtain novel index coding schemes and show that they provably outperform
all previously known graph theoretic bounds proposed so far. Further, we
establish a rather strong negative result: all known graph theoretic bounds are
within a logarithmic factor from the chromatic number. This is in striking
contrast to minrank since prior work has shown that it can outperform the
chromatic number by a polynomial factor in some cases. The conclusion is that
all known graph theoretic bounds are not much stronger than the chromatic
number.Comment: 8 pages, 2 figures. Submitted to ISIT 201
A Matrix Completion Approach to Linear Index Coding Problem
In this paper, a general algorithm is proposed for rate analysis and code design of linear index coding problems. Specifically a solution for minimum rank matrix completion problem over finite fields representing the linear index coding problem is devised in order to find the optimum transmission rate given vector length and size of the field. The new approach can be applied to both scalar and vector linear index coding
Perfectly Secure Index Coding
In this paper, we investigate the index coding problem in the presence of an
eavesdropper. Messages are to be sent from one transmitter to a number of
legitimate receivers who have side information about the messages, and share a
set of secret keys with the transmitter. We assume perfect secrecy, meaning
that the eavesdropper should not be able to retrieve any information about the
message set. We study the minimum key lengths for zero-error and perfectly
secure index coding problem. On one hand, this problem is a generalization of
the index coding problem (and thus a difficult one). On the other hand, it is a
generalization of the Shannon's cipher system. We show that a generalization of
Shannon's one-time pad strategy is optimal up to a multiplicative constant,
meaning that it obtains the entire boundary of the cone formed by looking at
the secure rate region from the origin. Finally, we consider relaxation of the
perfect secrecy and zero-error constraints to weak secrecy and asymptotically
vanishing probability of error, and provide a secure version of the result,
obtained by Langberg and Effros, on the equivalence of zero-error and
-error regions in the conventional index coding problem.Comment: 25 pages, 5 figures, submitted to the IEEE Transactions on
Information Theor
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
A Rate-Distortion Approach to Index Coding
We approach index coding as a special case of rate-distortion with multiple
receivers, each with some side information about the source. Specifically,
using techniques developed for the rate-distortion problem, we provide two
upper bounds and one lower bound on the optimal index coding rate. The upper
bounds involve specific choices of the auxiliary random variables in the best
existing scheme for the rate-distortion problem. The lower bound is based on a
new lower bound for the general rate-distortion problem. The bounds are shown
to coincide for a number of (groupcast) index coding instances, including all
instances for which the number of decoders does not exceed three.Comment: Substantially extended version. Submitted to IEEE Transactions on
Information Theor
On Locally Decodable Index Codes
Index coding achieves bandwidth savings by jointly encoding the messages
demanded by all the clients in a broadcast channel. The encoding is performed
in such a way that each client can retrieve its demanded message from its side
information and the broadcast codeword. In general, in order to decode its
demanded message symbol, a receiver may have to observe the entire transmitted
codeword. Querying or downloading the codeword symbols might involve costs to a
client -- such as network utilization costs and storage requirements for the
queried symbols to perform decoding. In traditional index coding solutions,
this 'client aware' perspective is not considered during code design. As a
result, for these codes, the number of codeword symbols queried by a client per
decoded message symbol, which we refer to as 'locality', could be large. In
this paper, considering locality as a cost parameter, we view index coding as a
trade-off between the achievable broadcast rate (codeword length normalized by
the message length) and locality, where the objective is to minimize the
broadcast rate for a given value of locality and vice versa. We show that the
smallest possible locality for any index coding problem is 1, and that the
optimal index coding solution with locality 1 is the coding scheme based on
fractional coloring of the interference graph. We propose index coding schemes
with small locality by covering the side information graph using acyclic
subgraphs and subgraphs with small minrank. We also show how locality can be
accounted for in conventional partition multicast and cycle covering solutions
to index coding. Finally, applying these new techniques, we characterize the
locality-broadcast rate trade-off of the index coding problem whose side
information graph is the directed 3-cycle.Comment: 10 pages, 1 figur