144,212 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
Broadcast Rate Requires Nonlinear Coding in a Unicast Index Coding Instance of Size 36
Insufficiency of linear coding for the network coding problem was first
proved by providing an instance which is solvable only by nonlinear network
coding (Dougherty et al., 2005).Based on the work of Effros, et al., 2015, this
specific network coding instance can be modeled as a groupcast index coding
(GIC)instance with 74 messages and 80 users (where a message can be requested
by multiple users). This proves the insufficiency of linear coding for the GIC
problem. Using the systematic approach proposed by Maleki et al., 2014, the
aforementioned GIC instance can be cast into a unicast index coding (UIC)
instance with more than 200 users, each wanting a unique message. This confirms
the necessity of nonlinear coding for the UIC problem, but only for achieving
the entire capacity region. Nevertheless, the question of whether nonlinear
coding is required to achieve the symmetric capacity (broadcast rate) of the
UIC problem remained open. In this paper, we settle this question and prove the
insufficiency of linear coding, by directly building a UIC instance with only
36users for which there exists a nonlinear index code outperforming the optimal
linear code in terms of the broadcast rate
Generalized Interlinked Cycle Cover for Index Coding
A source coding problem over a noiseless broadcast channel where the source
is pre-informed about the contents of the cache of all receivers, is an index
coding problem. Furthermore, if each message is requested by one receiver, then
we call this an index coding problem with a unicast message setting. This
problem can be represented by a directed graph. In this paper, we first define
a structure (we call generalized interlinked cycles (GIC)) in directed graphs.
A GIC consists of cycles which are interlinked in some manner (i.e., not
disjoint), and it turns out that the GIC is a generalization of cliques and
cycles. We then propose a simple scalar linear encoding scheme with linear time
encoding complexity. This scheme exploits GICs in the digraph. We prove that
our scheme is optimal for a class of digraphs with message packets of any
length. Moreover, we show that our scheme can outperform existing techniques,
e.g., partial clique cover, local chromatic number, composite-coding, and
interlinked cycle cover.Comment: Extended version of the paper which is to be presented at the IEEE
Information Theory Workshop (ITW), 2015 Jej
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