Linear Index Codes Research Articles

Hardness of Linear Index Coding on Perturbed Instances

Hardness of Linear Index Coding on Perturbed Instances

On the Optimality of Linear Index Coding Over the Fields With Characteristic Three

It has been known that the insufficiency of linear coding in achieving the optimal rate of the general index coding problem is rooted in its rate’s dependency on the field size. However, this dependency has been described only through the two well-known matroid instances, namely the Fano and non-Fano matroids, which, in turn, limits its scope only to the fields with characteristic two. In this paper, we extend this scope to demonstrate the reliance of linear index coding rate on fields with characteristic three. By constructing two index coding instances of size 29, we prove that for the first instance, linear coding is optimal only over the fields with characteristic three, and for the second instance, linear coding over any field with characteristic three can never be optimal. Then, a variation of the second instance is designed as the third index coding instance of size 58. For this instance, it is proved that while linear coding over any field with characteristic three cannot be optimal, there exists a nonlinear code over the fields with characteristic three, which achieves its optimal rate. Connecting the first and third index coding instances in two specific ways, called no-way and two-way connections, will lead to two new index coding instances of size 87 and 91, for which linear coding is outperformed by nonlinear codes. Another main contribution of this paper is the reduction of the key constraints on the space of the linear coding for the first and second index coding instances, each of size 29, into a matroid instance with the ground set of size 9, whose linear representability is dependent on the fields with characteristic three. The proofs and discussions provided in this paper through using these two relatively small matroid instances will shed light on the underlying reason causing the linear coding to become insufficient for the general index coding problem.

Embedded Index Coding

Motivated by applications in distributed storage and distributed computation, we introduce embedded index coding (EIC). EIC is a type of distributed index coding in which nodes in a distributed system act as both broadcast senders and receivers of information. We show how linear embedded index coding is related to linear index coding in general, and give characterizations and bounds on the communication costs of optimal embedded index codes. We also define task-based EIC, in which there is only one sender node responsible for transmitting a block to a particular receiving node. Task-based EIC is more computationally tractable and has advantages in applications such as distributed storage, in which senders may complete their broadcasts at different times. Finally, we give heuristic algorithms for approximating optimal linear embedded index codes, and demonstrate empirically that these algorithms perform well.

Task-Based Solutions to Embedded Index Coding

In the index coding problem a sender holds a message $x \in \{0,1\}^{n}$ and wishes to broadcast information to $n$ receivers in a way that enables the $i$ th receiver to retrieve the $i$ th bit $x_{i}$ . Every receiver has prior side information comprising a subset of the bits of $x$ , and the goal is to minimize the length of the information sent via the broadcast channel. Porter and Wootters have recently introduced the model of embedded index coding , where the receivers also play the role of the sender and the goal is to minimize the total length of their broadcast information. An embedded index code is said to be task-based if every receiver retrieves its bit based only on the information provided by one of the receivers. This paper studies the effect of the task-based restriction on linear embedded index coding. It is shown that for certain side information maps there exists a linear embedded index code of length quadratically smaller than that of any task-based embedded index code. The result attains, up to a multiplicative constant, the largest possible gap between the two quantities. The proof is by an explicit construction and the analysis involves spectral techniques.

Generalized Index Coding Problem and Discrete Polymatroids.

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, were used. The index coding problem has been generalized recently to accommodate receivers that demand functions of messages and possess functions of messages. 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, an index coding problem with coded side information is constructed and it is shown that if the index coding problem has a certain optimal length solution then the discrete polymatroid is representable. If the generalized index coding problem is constructed from a matroid, it is shown that the index coding problem has a binary scalar linear solution of optimal length if and only if the matroid is binary representable.

User-Cache Aided Transmission With Index Coding in $K$ -User Downlink Channels

This paper considers a single-input single-output broadcast channel with receiver side information. We find the optimal clique cover index code and the optimal transmission time allocation that minimize outage probability when total transmission time is limited. As our problem is NP-hard, we first find the optimal time allocation for a given index code. Then, we describe a brute-force algorithm that finds the set of all decodable index codes and chooses the optimal one adopting the optimal time allocation. To reduce the computational complexity of the brute-force algorithm, we propose a pruning algorithm which solves the same problem using the Hasse diagram but does not harm the optimality. Our analysis reveals that the optimal index code is dependent on the channel conditions, not simply on the number of required transmissions, which implies that the index coding-channel coupling improves the outage performance. It is also shown that this claim is still valid for general scalar linear index coding. Our simulation results verify that our proposed schemes effectively reduce the outage probability compared to other reference schemes, and our pruning algorithm considerably reduces the computational complexity required for the brute-force algorithm.

A Discrete Polymatroidal Framework for Differential Error-Correcting Index Codes

In the conventional index coding problem, the messages transmitted by the source to satisfy the demands of the receivers are error-free. However, in practice, the messages may be error prone which led to the introduction of error-correcting index codes. An error-correcting index code is an encoding scheme which enables every receiver to decode its demanded message even in the presence of certain number of transmission errors. Formally, an index code capable of correcting at most $\delta $ errors at all its receivers is defined as a $\delta $ -error-correcting index code. In this paper, we explore the connections between vector linear error-correcting index codes and discrete polymatroids. This is motivated from the connections between linear network error-correcting codes and matroids. It is shown that a vector linear error-correcting index code exists if and only if there exists a representable discrete polymatroid satisfying certain conditions. Differential error-correcting index codes which allow different receivers to have different error-correcting capabilities are also studied. Note that the $\delta $ -error-correcting code is a special case of a differential error-correcting index code. Error correction at a subset of receivers is another special case which is discussed.

Multi-Sender Index Coding for Collaborative Broadcasting: A Rank-Minimization Approach

We consider a Multi-Sender Unicast Index-Coding (MSUIC) problem, where in a broadcast network, multiple senders collaboratively send distinct messages to multiple receivers, each having some subset of the messages a priori. The aim is to find the shortest index code that minimizes the total number of coded bits sent by the senders. In this paper, built on the classic single-sender minrank concept, we develop a new rank-minimization framework for MSUIC that explicitly takes into account the sender message constraints and minimizes the sum of the ranks of encoding matrices subject to the receiver decoding requirements. This framework provides a systematic way to construct multi-sender linear index codes and to study their capability in achieving the shortest index codelength per message length (i.e., the optimal broadcast rate). In particular, we establish the optimal broadcast rate for all critical MSUIC instances with up to four receivers and show that a binary linear index code is optimal for all, except 15 instances with four receivers. We also propose a heuristic algorithm (in lieu of exhaustive search) to solve the rank-minimization problem. The effectiveness of the algorithm is validated by numerical studies of MSUIC instances with four or more receivers.

Optimal Error Correcting Index Codes for Some Generalized Index Coding Problems

The index coding with side-information problem introduced by Birk and Kol has been generalized to the case, where the transmissions are subjected to errors by Dau et al.. Lower and upper bounds on the optimal number of transmissions required to correct a specified number δ of errors were established. In another generalization of index coding problem, namely, generalized index coding (GIC) problem introduced by Dai et al., linear combination of the messages can be demanded and held as side information by the receivers. Error correction for GIC problems was studied and the bounds for optimal number of transmissions required for δ-error correcting generalized index codes were established by Byrne and Calderini. In this paper, for a particular class of GIC problems construction of optimal scalar linear index codes are discussed. For this class of GIC problems, the optimal linear δ-error correcting index codes are also constructed. As special cases, optimal linear error correcting codes are obtained for two classes of original index coding problems, namely, single-prior index coding problems and single unicast index coding problems with symmetric neighboring consecutive side information.

Linear Index Coding With Multiple Senders and Extension to a Cellular Network

In this paper, linear index codes with multiple senders are studied, where every receiver receives encoded messages from all senders. A new fitting matrix for the multiple senders is proposed and it is proved that the minimum rank of the proposed fitting matrices is the optimal codelength of linear index codes for the multiple senders. In addition, a new type of a side information graph related with the optimal codelength is proposed and whether given side information is critical or not is studied. Furthermore, linear index codes for the cellular network scenario are studied, where each receiver can receive a subset of sub-codewords. Since some receivers cannot receive the entire codeword in the cellular network scenario, the encoding method based on the fitting matrix has to be modified. In the cellular network scenario, we propose another fitting matrix and prove that an optimal generator matrix can be found based on these fitting matrices. In addition, some properties on the optimal codelength of linear index codes for the cellular network case are studied.

The Minrank of Random Graphs

The minrank of a directed graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding (Bar-Yossef et al. ), network coding (Effros et al. ), and distributed storage (Mazumdar, ISIT, 2014). We prove tight bounds on the minrank of directed Erdős–Renyi random graphs $G(n,p)$ for all regimes of $p\in [{0,1}]$ . In particular, for any constant $p$ , we show that $\mathsf {minrk}(G) = \Theta (n/\log n)$ with high probability, where $G$ is chosen from $G(n,p)$ . This bound gives a near quadratic improvement over the previous best lower bound of $\Omega (\sqrt {n})$ (Haviv and Langberg), and partially settles an open problem raised by Lubetzky and Stav. Our lower bound matches the well-known upper bound obtained by the “clique covering” solution and settles the linear index coding problem for random knowledge graphs.

Optimal Index Codes for a New Class of Interlinked Cycle Structure

The interlinked cycle (IC) structure that generalizes cycles and cliques was defined by Thapa, Ong, and Johnson. Interlinked-cycle-cover (ICC) scheme that leverages IC structures in digraphs to construct scalar linear index codes was proposed. A class of infinitely many digraphs, where the proposed scalar linear codes based on ICC scheme are optimal were characterized by Thapa et. al . In this paper, we provide an addition to this class by providing optimal length index codes for IC structures with one cycle among non-inner vertex set.

A Polynomial-Time Algorithm for Pliable Index Coding

In pliable index coding, we consider a server with $m$ messages and $n$ clients, where each client has as side information a subset of the messages. We seek to minimize the number of broadcast transmissions, so that each client can recover any one unknown message she does not already have. Previous work has shown that the pliable index coding problem is NP-hard and requires at most $\mathcal {O}(\log ^{2}(n))$ broadcast transmissions, which indicates exponential savings over the conventional index coding that requires in the worst case $\mathcal {O}(n)$ transmissions. In this paper, building on a decoding criterion that we propose, we first design a deterministic polynomial-time algorithm that can realize the exponential benefits, by achieving, in the worst case, a performance upper bounded by $\mathcal {O}(\log ^{2}(n))$ broadcast transmissions. We extend our algorithm to the $t$ -requests case, where each client requires $t$ unknown messages that she does not have, and show that our algorithm requires at most $\mathcal {O}(t\log (n)+\log ^{2}(n))$ broadcast transmissions. We construct lower bound instances that require at least $\Omega (\log (n))$ transmissions for linear pliable index coding and at least $\Omega (t+\log (n))$ transmissions for the $t$ -requests case, indicating that both our upper and lower bounds are polynomials of $\log (n)$ and differ within a factor of $\mathcal {O}(\log (n))$ . We provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely $\Theta (\log (n))$ , as compared with the $\Theta (n/\log (n))$ for index coding. In addition, we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances, implying that vector coding does not provide benefits in terms of these bounds. Our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50% less transmissions.

Efficient Quantization with Linear Index Coding for Deep‐Space Images

Due to inevitable propagation delay involved in deep‐space communication systems, very high cost is associated with the retransmission of erroneous segments. Quantization with linear index coding (QLIC) scheme is known to provide compression along with robust transmission of deep‐space images, and thus the likelihood of retransmissions is significantly reduced. This paper aims to improve its spectral efficiency as well as robustness. First, multiple quantization refinement levels per transmitted source block of QLIC are proposed to increase spectral efficiency. Then, iterative multipass decoding is introduced to jointly decode the subsource symbol‐planes. It achieves better PSNR of the reconstructed image as compared to the baseline one‐pass decoding approach of QLIC.

Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques

We consider a graphical approach to index coding. While cycles have been shown to provide coding gain, only disjoint cycles and cliques (a specific type of overlapping cycles) have been exploited in existing literature. In this paper, we define a more general form of overlapping cycles, called the interlinked-cycle (IC) structure, that generalizes cycles and cliques. We propose a scheme, called the interlinked-cycle-cover (ICC) scheme, that leverages IC structures in digraphs to construct scalar linear index codes. We characterize a class of infinitely many digraphs where our proposed scheme is optimal over all linear and non-linear index codes. Consequently, for this class of digraphs, we indirectly prove that scalar linear index codes are optimal. Furthermore, we show that the ICC scheme can outperform all existing graph-based schemes (including partial-clique-cover and fractional-local-chromatic number schemes), and a random-coding scheme (namely, composite coding) for certain graphs.

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. A well-known application is satellite communications, as described in one of the earliest papers on the subject (Birk and Kol, 1998). It is readily seen that if the sender has knowledge of 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 are encoded before sending. The collection of side-information indices as well as the indices of the requested data is described as an instance ${ \mathcal I}$ of the index coding with side-information (ICSI) problem. The encoding function is called the index code of $ \mathcal I$ , and the number of transmissions, resulting from the encoding is referred to as its length . The main ICSI problem is to determine the optimal length of an index code and instance $ \mathcal I$ . As this number is hard to compute, bounds approximating it are sought, as are algorithms to compute efficient index codes. These questions have been addressed by several authors (e.g., see Alon et al. 2008, Bar-Yossef et al. 2011, Blasiak et al. 2013), often taking a graph-theoretic approach. Two interesting generalizations of the problem that have appeared in the literature are the subject of this paper. The first of these is the case of index coding with coded side information (Dai et al. 2014), in which linear combinations of the source data are both requested by and held as users’ side-information. This generalization has applications, for example, to relay channels and necessitates algebraic rather than combinatorial methods. The second is the introduction of error-correction in the problem, in which the broadcast channel is subject to noise (Dau et al. 2013). 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 $\delta $ -error correcting code for an instance of the ICCSI problem and obtain bounds analogous to those described in (Dau et al. 2013), both for the Hamming metric and for rank-metric errors. We describe decoding algorithms for both categories of errors.

Multihop Caching-Aided Coded Multicasting for the Next Generation of Cellular Networks

Next generation of cellular networks deploying wireless distributed femtocaching infrastructure proposed by Golrezaei et. al. are studied. By taking advantage of multihop communications in each cell, the number of required femtocaching helpers is significantly reduced. This reduction of femtocaches is achieved by using the underutilized storage and communication capabilities in the User Terminals (UTs), which results in reducing the deployment costs of distributed femtocaches. A multihop index coding technique is proposed to code the cached contents in helpers to achieve order optimal capacity gains. This can serve as an efficient content delivery algorithm for the solution provided by Golrezaei et. al. As an example, we consider a wireless cellular system in which contents have a popularity distribution. It has been shown that if the contents follow a high content reuse popularity distribution, our approach can replace many unicast communication with multicast communication. We will prove that simple linear index codes found by heuristics based on graph coloring algorithms can achieve order optimal capacity under Zipfian content popularity distribution.

Linear Network Coding, Linear Index Coding and Representable Discrete Polymatroids

Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding, and representable discrete polymatroids. We consider the vector linear solutions of networks over a field $\mathbb {F}_{q}$ , with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. It is well known that a scalar linear solution over $\mathbb {F}_{q}$ exists for a network if and only if the network is matroidal with respect to a matroid representable over $\mathbb {F}_{q}$ . We define a discrete polymatroidal network and show that a linear fractional solution over a field $\mathbb {F}_{q}$ exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over $\mathbb {F}_{q}$ . An algorithm to construct the networks starting from certain class of discrete polymatroids is provided. Every representation over $\mathbb {F}_{q}$ for the discrete polymatroid, results in a linear fractional solution over $\mathbb {F}_{q}$ for the constructed network. Next, we consider the index coding problem, which involves a sender, which generates a set of messages $X=\{x_{1},x_{2},\dotso x_{k}\}$ , and a set of receivers $\mathcal {R}$ , which demand messages. A receiver $R \in \mathcal {R}$ is specified by the tuple $(x,H)$ , where $x \in X$ is the message demanded by $R$ and $H \subseteq X \!\setminus \! \{x\}$ is the side information possessed by $R$ . We first show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid, satisfying certain conditions, which are determined by the index coding problem considered. Rouayheb et al. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that matroid. The multi-linear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid. We generalize the result of Rouayheb et al. , by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.

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.

Critical Graphs in Index Coding

In this paper, we define critical graphs as minimal graphs that support a given set of rates for the index coding problem and study them for both the one-shot and asymptotic setups. For the case of equal rates, we find the critical graph with minimum number of edges for both one-shot and asymptotic cases. For the general case of possibly distinct rates, we show that for one-shot and asymptotic linear index coding, as well as asymptotic nonlinear index coding, each critical graph is a union of disjoint strongly connected subgraphs. On the other hand, we identify a non-USCS critical graph for a one-shot nonlinear index coding problem. Next, we identify a few graph structures that are critical. In addition, we show that the capacity region of the index coding is additive for union of disjoint graphs.