Error-correcting functional index codes, generalized exclusive laws and graph coloring

Gupta, Anindya; Rajan, B. Sundar

doi:10.1109/icc.2016.7511555

Cited by 8 publications

(14 citation statements)

References 34 publications

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“…In this section we discuss a variation of the classical index coding problem where each user demands a coded version of the information symbols present at the transmitter and already knows a subset of the (uncoded) information symbols as side information. This is a special case of the Generalized Index Coding problem [7], [8] and the Functional Index Coding problem [9]. The authors of [7], [8] generalized the classical index coding problem where each receiver knows some linearly coded information symbols as side-information and demands some linearly coded information symbols.…”

Section: Equivalence With a Functional Index Coding Problemmentioning

confidence: 99%

“…Additionally the authors of [7] assume that the information symbols present in the transmitter are also linearly coded information symbols. In [9], authors generalized the index coding problem, where the side-information as well as demanded messages can be arbitrary functions of information symbols, called functional index coding problem. Here we consider a special case of generalized index coding problem and functional index coding problem and then we introduce the relation between function update problem and this family of functional index coding problems.…”

Section: Equivalence With a Functional Index Coding Problemmentioning

confidence: 99%

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Codes for Updating Linear Functions over Small Fields

Ghosh

Natarajan

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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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 Médard 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 Médard 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.The authors are with the

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Section: Equivalence With a Functional Index Coding Problemmentioning

confidence: 99%

Section: Equivalence With a Functional Index Coding Problemmentioning

confidence: 99%

Codes for Updating Linear Functions over Small Fields

Ghosh

Natarajan

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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“…The source is aware of the messages possessed by each receiver and it aims to reduce the number of transmissions required to satisfy the demands of all the receivers. The conventional index coding has been generalized to functional index coding in [5]. In a functional index coding problem, the Has-set and the Want-set of users contain functions of messages rather than subsets of messages.…”

Section: Introductionmentioning

confidence: 99%

Generalized index coding problem and discrete polymatroids

Thomas

Rajan

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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The index coding problem has been generalized recently to accommodate receivers which demand functions of messages and which possess functions of messages. The connections between index coding and matroid theory have been well studied in the recent past. Index coding solutions were first connected to multi linear representation of matroids. For vector linear index codes discrete polymatroids which can be viewed as a generalization of the matroids was used. It was shown that a vector linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions. In this work we explore the connections between generalized index coding and discrete polymatroids. The conditions that need to be satisfied by a representable discrete polymatroid for a generalized index coding problem to have a vector linear solution is established. From a discrete polymatroid we construct an index coding problem with coded side information and shows that if the index coding problem has a certain optimal length solution then the discrete polymatroid satisfies certain properties. From a matroid we construct a similar generalized index coding problem and shows that the index coding problem has a binary scalar linear solution of optimal length if and only if the matroid is binary representable.

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“…The functional index coding problem was recently proposed in [20] as a generalization of the conventional index coding problem. In a functional index coding problem, the Has-and Want-sets of users may contain functions of messages rather than only a subset of messages as is the case in a conventional index coding problem.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the conventional index coding problem and the scenarios studied in [18] and [19] are special cases of the functional index coding problem. In [20], bounds on codebook size were obtained and a graph coloring approach to obtaining a functional index code was given.…”

Section: Introductionmentioning

confidence: 99%

A Relation Between Network Computation and Functional Index Coding Problems

Gupta

Rajan

2017

IEEE Trans. Commun.

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Abstract-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.

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Error-correcting functional index codes, generalized exclusive laws and graph coloring

Cited by 8 publications

References 34 publications

Codes for Updating Linear Functions over Small Fields

Codes for Updating Linear Functions over Small Fields

Generalized index coding problem and discrete polymatroids

A Relation Between Network Computation and Functional Index Coding Problems

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