6 research outputs found
A Relation Between Network Computation and Functional Index Coding Problems
In contrast to the network coding problem wherein the sinks in a network
demand subsets of the source messages, in a network computation problem the
sinks demand functions of the source messages. Similarly, in the functional
index coding problem, the side information and demands of the clients include
disjoint sets of functions of the information messages held by the transmitter
instead of disjoint subsets of the messages, as is the case in the conventional
index coding problem. It is known that any network coding problem can be
transformed into an index coding problem and vice versa. In this work, we
establish a similar relationship between network computation problems and a
class of functional index coding problems, viz., those in which only the
demands of the clients include functions of messages. We show that any network
computation problem can be converted into a functional index coding problem
wherein some clients demand functions of messages and vice versa. We prove that
a solution for a network computation problem exists if and only if a functional
index code (of a specific length determined by the network computation problem)
for a suitably constructed functional index coding problem exists. And, that a
functional index coding problem admits a solution of a specified length if and
only if a suitably constructed network computation problem admits a solution.Comment: 3 figures, 7 tables and 9 page
Codes for Updating Linear Functions over Small Fields
We consider a point-to-point communication scenario where the receiver
maintains a specific linear function of a message vector over a finite field.
When the value of the message vector undergoes a sparse update, the transmitter
broadcasts a coded version of the modified message while the receiver uses this
codeword and the current value of the linear function to update its contents.
It is assumed that the transmitter has access to the modified message but is
unaware of the exact difference vector between the original and modified
messages. Under the assumption that the difference vector is sparse and that
its Hamming weight is at the most a known constant, the objective is to design
a linear code with as small a codelength as possible that allows successful
update of the linear function at the receiver. This problem is motivated by
applications to distributed data storage systems. Recently, Prakash and Medard
derived a lower bound on the codelength, which is independent of the size of
the underlying finite field, and provided constructions that achieve this bound
if the size of the finite field is sufficiently large. However, this
requirement on the field size can be prohibitive for even moderate values of
the system parameters. In this paper, we provide a field-size aware analysis of
the function update problem, including a tighter lower bound on the codelength,
and design codes that trade-off the codelength for a smaller field size
requirement. We first characterize the family of function update problems where
linear coding can provide reduction in codelength compared to a naive
transmission scheme. We then provide field-size dependent bounds on the optimal
codelength, and construct coding schemes when the receiver maintains linear
functions of striped message vector. Finally, we show that every function
update problem is equivalent to a generalized index coding problem.Comment: Keywords: distributed storage systems, function update problem, index
coding, side informatio
Error-Correcting Functional Index Codes, Generalized Exclusive Laws and Graph Coloring
We consider the functional index coding problem over an error-free broadcast network in which a source generates a set of messages and there are multiple receivers, each holding a set of functions of source messages in its cache, called the Has-set, and demands to know another set of functions of messages, called the Want-set. Cognizant of the receivers' Hassets, the source aims to satisfy the demands of each receiver by making coded transmissions, called a functional index code. The objective is to minimize the number of such transmissions required. The restriction a receiver's demands pose on the code is represented via a constraint called the generalized exclusive law and obtain a code using the confusion graph constructed using these constraints. Bounds on the size of an optimal code based on the parameters of the confusion graph are presented. Next, we consider the case of erroneous transmissions and provide a necessary and sufficient condition that an FIC must satisfy for correct decoding of desired functions at each receiver and obtain a lower bound on the length of an error-correcting FIC
Codes for Updating Linear Functions over Small Fields
We consider a point-to-point communication scenario where the receiver intends to maintain a specific linear
function of a message vector over a finite field. When the value of the message vector changes, which is modelled
as a sparse update, the transmitter broadcasts a coded version of the modified message while the receiver uses
this codeword and the current value of the linear function to update its contents. It is assumed that the transmitter
has access to only the modified message and is unaware of the exact difference vector between the original and
modified messages. Under the assumption that the difference vector is sparse and that its Hamming weight is at the
most a known constant, the objective is to design a linear code with as small a codelength as possible that allows
successful update of the linear function at the receiver. This problem is motivated by applications to distributed data
storage systems. Recently, Prakash and Medard derived a lower bound on the codelength, which is independent of ´
the size of the underlying finite field, and provided constructions that achieve this bound if the size of the finite field
is sufficiently large. However, this requirement on the field size can be prohibitive for even moderate values of the
system parameters. In this paper, we provide a field-size aware analysis of the function update problem, including
a tighter lower bound on the codelength, and design codes that trade-off the codelength for a smaller field size
requirement. We also show that the problem of designing codes for updating linear functions is related to functional
index coding or generalized index coding. We first characterize the family of function update problems where linear
coding can provide reduction in codelength compared to a naive transmission scheme. We then provide field-size
dependent bounds on the optimal codelength, and construct coding schemes based on error correcting codes and
subspace codes when the receiver maintains linear functions of striped message vector. These codes provide a
trade-off between the codelength and the size of the operating finite field, and whenever the achieved codelengths
equal those reported by Prakash and Medard the requirements on the size of the finite field are matched as well. ´
Finally, for any given function update problem, we construct an equivalent functional index coding or generalized
index coding problem such that any linear coding scheme is valid for the function update problem if and only if
it is valid for the constructed functional index coding problem