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In this paper, we put forward the model of multiple linear-combination security multicast network coding, where the wiretapper desires to obtain some information about a predefined set of multiple linear combinations of the source symbols by eavesdropping any one (but not more than one) channel subset up to a certain size r, referred to as the security level. For this model, the security capacity is defined as the maximum average number of source symbols that can be securely multicast to all sink nodes for one use of the network under the linear-combination security constraint. For any security level and any linear-combination security constraint, we fully characterize the security capacity in terms of the ratio of the rank of the linear-combination security constraint to the number of source symbols. Also, we develop a general construction of linear security network codes. Finally, we investigate the asymptotic behavior of the security capacity for a sequence of linear-combination security models and discuss the asymptotic optimality of our code construction. [ABSTRACT FROM AUTHOR]

This paper presents the first systematic study of the zero-error capacity regions of noisy networks. First, we consider two simple such networks, each consisting of a stationary memoryless multiple access channel with two binary inputs and one discrete output. There are two users in each network. Each of the two users transmits a message through the network, and the sink(s) of the network can decode both messages with zero error. A graph is used to represent the distinguishability of the inputs of the channel, and a graph set is used to represent the distinguishability of the inputs of the network. We show that for two networks represented by the same graph set, their zero-error capacity regions are the same. We list all the possible graph sets for the two networks and determine the zero-error capacity regions for some of these graph sets. Based on this result, we explore a relation between graph theory and set theory, and then redefine the cancellative pair of families of subsets. We further extend the problem formulation to a general network called the parallel network, which may consist of more than one channel with multiple inputs and multiple outputs. [ABSTRACT FROM AUTHOR]

Most existing network communication systems, such as the Internet, are built around the TCP/IP protocol stack. However, TCP/IP cannot effectively support the upcoming network communication systems with more wireless communication links and higher requirements in terms of throughput, latency, and reliability. In this article, we introduce a general network communication protocol design based on batched network coding (BNC), which includes the existing framework as a special case and whose design can be guided by channel coding theory. BNC-based protocols can have significant throughput gain for single-hop or multihop wireless communications and provide better trade-off among throughput, latency and reliability. [ABSTRACT FROM AUTHOR]

Multilevel diversity coding is a classical coding model where multiple mutually independent information messages are encoded, such that different reliability requirements can be afforded to different messages. It is well known that superposition coding, namely separately encoding the independent messages, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999). In the current paper, we consider weakly secure SMDC where security constraints are injected on each individual message, and provide a complete characterization of the conditions under which superposition coding is sum-rate optimal. Two joint coding strategies, which lead to rate savings compared to superposition coding, are proposed, where some coding components for one message can be used as the encryption key for another. By applying different variants of Han’s inequality, we show that the lack of opportunity to apply these two coding strategies directly implies the optimality of superposition coding. It is further shown that under a set of particular security constraints, one of the proposed joint coding strategies can be used to construct a code that achieves the optimal rate region. [ABSTRACT FROM AUTHOR]

Information-theoretic security is considered in the paradigm of network coding in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, also referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this paper, we consider the problem of secure network coding when the information rate and the security level can change over time. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. We first consider the design of a family of local-encoding-preserving SLNCs for a fixed security level and a flexible rate. A simple approach is presented for efficiently constructing upon an SLNC that exists a local-encoding-preserving SLNC with the same security level and the rate reduced by one. By applying this approach repeatedly, we can obtain a family of local-encoding-preserving SLNCs with a fixed security level and multiple rates. We further consider the design of a family of local-encoding-preserving SLNCs for a fixed rate and a flexible security level. We present a novel and efficient approach for constructing upon an SLNC that exists a local-encoding-preserving SLNC with the same rate and the security level increased by one. Next, we consider the design of a family of local-encoding-preserving SLNCs for a fixed dimension (equal to the sum of rate and security level) and a flexible pair of rate and security level. We propose another novel approach for designing an SLNC such that the same SLNC can be applied for all the rate and security-level pairs with the fixed dimension. Also, two polynomial-time algorithms are developed for efficient implementations of the later two proposed approaches, respectively. Furthermore, we prove that all our three approaches do not incur any penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. Finally, we consider the ultimate problem of designing a family of local-encoding-preserving SLNCs that can be applied to all possible pairs of rate and security level. By combining the constructions of the three families of local-encoding-preserving SLNCs in the paper in suitable ways, we can obtain a family of local-encoding-preserving SLNCs that can be applied for all possible pairs of rate and security level. Three possible such constructions are presented. [ABSTRACT FROM AUTHOR]

It is well known that superposition coding, namely separately encoding the independent sources, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999) for any $L\geq 2$ , where $L$ is the number of levels of the coding system. However, the characterization of the coding rate region therein involves uncountably many linear inequalities and the constant term (i.e., the lower bound) in each inequality is given in terms of the solution of a linear optimization problem. Thus this implicit characterization of the coding rate region does not enable the determination of the achievability of a given rate tuple. In principle, the achievability of a given rate tuple can be determined by direct computation, but the complexity is prohibitive even for $L=5$. In this paper, for any fixed $L$ , we obtain in closed form a finite set of linear inequalities for characterizing the coding rate region. We further show by the symmetry of the problem that only a much smaller subset of this finite set of inequalities needs to be verified in determining the achievability of a given rate tuple. Yet, the cardinality of this smaller set grows at least exponentially fast with $L$. We also present a subset entropy inequality, which together with our explicit characterization of the coding rate region, is sufficient for proving the optimality of superposition coding. [ABSTRACT FROM AUTHOR]

The problem of network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a target function, of which the inputs are generated at multiple source nodes. The edges in the network are assumed to be error-free and have limited capacity. The nodes in the network are assumed to have unbounded computing capability and be able to perform network coding. The computing rate of a network code that can compute the target function over the network is the average number of times that the target function is computed with zero error for one use of the network. In this paper, we obtain an improved upper bound on the computing capacity, which is applicable to arbitrary target functions and arbitrary network topologies. This improved upper bound not only is an enhancement of the previous upper bounds but also is the first tight upper bound on the computing capacity for computing an arithmetic sum over a certain non-tree network, which has been widely studied in the literature. We also introduce a multi-dimensional array approach that facilitates evaluation of the improved upper bound. Furthermore, we apply this bound to the problem of computing a vector-linear function over a network. With this bound, we are not only able to enhance a previous result on computing a vector-linear function over a network but also simplify the proof significantly. Finally, we prove that for computing the binary maximum function over the reverse butterfly network, our improved upper bound is not achievable. This result establishes that in general our improved upper bound is non-achievable, but whether it is asymptotically achievable or not remains open. [ABSTRACT FROM AUTHOR]

We consider a communication network where there exist wiretappers who can access a subset of channels, called a wiretap set, which is chosen from a given collection of wiretap sets. The collection of wiretap sets can be arbitrary. Secure network coding is applied to prevent the source information from being leaked to the wiretappers. In secure network coding, the required alphabet size is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of such coding schemes in terms of computational complexity and storage requirement. In this paper, we develop a systematic graph-theoretic approach for improving Cai and Yeung’s lower bound on the required alphabet size for the existence of secure network codes. The new lower bound thus obtained, which depends only on the network topology and the collection of wiretap sets, can be significantly smaller than Cai and Yeung’s lower bound. A polynomial-time algorithm is devised for efficient computation of the new lower bound. [ABSTRACT FROM PUBLISHER]

We study the degrees of freedom (DoFs) of the multiple input multiple output (MIMO) multipair two-way distributed relay channel (mTDRC), where K pairs of users, each equipped with M antennas, exchange messages in a pairwise manner with the help of two geographically separated relay nodes, each with N antennas. We establish a general framework for the DoF analysis by combining the ideas of signal space alignment and interference neutralization. The proposed framework involves a joint design of user transmit beamformers, relay precoders, and user receive beamformers. Novel signal alignment techniques are proposed to reduce the number of linearly independent constraints for interference neutralization. Based on that, the original joint transceiver and relay design problem boils down to a problem solely on the design of the relay precoders. This problem is then solved utilizing recent development on the solvability of linear matrix equations. As a result, we derive an achievable DoF for the MIMO mTDRC with an arbitrary configuration of (K,M,N) . For K=2 by showing that the obtained achievable DoF meets a DoF upper bound. This result beats the state of the art by establishing the optimal DoF of the considered network in an extra range of (M/N)\in (1,2) . In addition, the achievable DoF obtained for K=2 is much higher than the existing result in the range of (M/N)\in (11/16,1) . Furthermore, we show that the optimal DoF of the considered network with $K\geq 3$ is derived for ({M}/{N})\in (0, ({2K+\sqrt {2K}})/({4K^{2}-2K})) \cup ({3}/{2},\infty ) . [ABSTRACT FROM AUTHOR]

In a network, a node is said to incur a delay if its encoding of each transmitted symbol involves only its received symbols obtained before the time slot in which the transmitted symbol is sent (hence the transmitted symbol sent in a time slot cannot depend on the received symbol obtained in the same time slot). A node is said to incur no delay if its received symbol obtained in a time slot is available for encoding its transmitted symbol sent in the same time slot. Under the classical model, every node in a discrete memoryless network (DMN) incurs a unit delay, and the capacity region of the DMN satisfies the well-known cut-set outer bound. In this paper, we propose a generalized model for the DMN where some nodes may incur no delay. Under our generalized model, we obtain a new cut-set outer bound, which is proved to be tight for some two-node DMN and is shown to subsume an existing cut-set bound for the causal relay network. In addition, we establish under the generalized model another cut-set outer bound on the positive-delay region—the set of achievable rate tuples under the constraint that every node incurs a delay. We use the cut-set bound on the positive-delay region to show that for some two-node DMN under the generalized model, the positive-delay region is strictly smaller than the capacity region. [ABSTRACT FROM PUBLISHER]

In a point-to-point communication system, which consists of a sender, a receiver, and a set of noiseless channels, the sender wishes to transmit a private message to the receiver through the channels, which may be eavesdropped by a wiretapper. The set of wiretap sets is arbitrary. The wiretapper can access any one but not more than one wiretap set. From each wiretap set, the wiretapper can obtain some partial information about the private message, which is measured by the equivocation of the message given the symbols obtained by the wiretapper. The security strategy is to encode the message with some random key at the sender. Only the message is required to be recovered at the receiver. Under this setting, we define an achievable rate tuple consisting of the size of the message, the size of the key, and the equivocation for each wiretap set. We first prove a tight rate region when both the message and the key are required to be recovered at the receiver. Then, we extend the result to the general case when only the message is required to be recovered at the receiver. Moreover, we show that even if stochastic encoding is employed at the sender, the message rate cannot be increased. [ABSTRACT FROM PUBLISHER]

Network coding can significantly improve the transmission rate of communication networks with packet loss compared with routing. However, using network coding usually incurs high computational and storage costs in the network devices and terminals. For example, some network coding schemes require the computational and/or storage capacities of an intermediate network node to increase linearly with the number of packets for transmission, making such schemes difficult to be implemented in a router-like device that has only constant computational and storage capacities. In this paper, we introduce batched sparse code (BATS code), which enables a digital fountain approach to resolve the above issue. BATS code is a coding scheme that consists of an outer code and inner code. The outer code is a matrix generation of a fountain code. It works with the inner code that comprises random linear coding at the intermediate network nodes. BATS codes preserve such desirable properties of fountain codes as ratelessness and low encoding/decoding complexity. The computational and storage capacities of the intermediate network nodes required for applying BATS codes are independent of the number of packets for transmission. Almost capacity-achieving BATS code schemes are devised for unicast networks and certain multicast networks. For general multicast networks, under different optimization criteria, guaranteed decoding rates for the destination nodes can be obtained. [ABSTRACT FROM AUTHOR]

Consider a communication network represented by a directed graph \cal G=(\cal V,\cal E) , where \cal V is the set of nodes and \cal E is the set of point-to-point channels in the network. On the network, a secure message M is transmitted, and there may exist wiretappers who want to obtain information about the message. In secure network coding, we aim to find a network code, which can protect the message against the wiretapper whose power is constrained. Cai and Yeung studied the model in which the wiretapper can access any one but not more than one set of channels, called a wiretap set, out of a collection {\cal A} of all possible wiretap sets. In order to protect the message, the message needs to be mixed with a random key K . They proved tight fundamental performance bounds when \cal A consists of all subsets of \cal E of a fixed size r . However, beyond this special case, obtaining such bounds is much more difficult. In this paper, we investigate the problem when \cal A consists of arbitrary subsets of \cal E and obtain the following results: 1) an upper bound on $H(M)$ and 2) a lower bound on H(K)$ in terms of H(M)$ . The upper bound on H(M)$ is explicit, while the lower bound on H(K) is fixed. The tightness of the lower bound for the point-to-point communication system is also proved. [ABSTRACT FROM PUBLISHER]

Internet of Things (IoT) connects billions of everyday objects to the Internet. The mobility of devices can be facilitated by means of employing multiple wireless links. However, packet loss is a common phenomenon in wireless communications, where the traditional forwarding strategy undergoes severe performance issues in a multi-hop wireless network. One solution is to apply batched sparse (BATS) codes. A fundamental difference from the traditional strategy is that BATS codes require the intermediate network nodes to perform recoding, which generates recoded packets by network coding operations. Literature showed that advanced recoding schemes and burst packet loss can enhance and diminish the performance of BATS codes respectively. However, the existing protocols for BATS codes cannot handle both of them at the same time. In this paper, we propose a paradigm of protocol design for BATS codes. Our design can be applied in different layers of the network stack and it is compatible to the existing network infrastructures. The modular nature of the protocol can support different recoding techniques and different ways to handle burst packet loss. We also give some examples to demonstrate how to use the protocol. [ABSTRACT FROM AUTHOR]