Cut-set Bound Research Articles

Asymmetric Successive Compute-and-Forward and the Capacity Gap for the Gaussian Two-way Relay Channel

Background: The compute-and-forward strategy is one of the outstanding methods which is used for interference management in wireless relay networks where decoding linear combinations of code words is required. Recently, many efforts have been made for decoding integer and noninteger combinations. The difference between the methods is the manner of handling different conditions of networks, such as equal or unequal power constraints and equal or unequal channel gains. Objective: In this work, we present a modified n-step asymmetric successive compute-and-forward strategy for the communication network where we have both unequal power constraints and unequal channel gains conditions. Methods: In the proposed method, we scale channel gains and coefficients with the square root of power constraints. In this way, despite previous methods, without the need for scaling factors in our formulation, it is still able to solve the problem of general Gaussian relay networks with unequal power constraints and unequal channel gains. We also use scaling factors in our method in order to have the ability to divide the rates between users fairly. Results and Conclusion: We evaluate the ability of the modified strategy for the uplink communication of the two-way relay channel, where one relay can help communication between the two users. At the relay, we decode the linear combinations of the messages of the two users and obtain 1/2 bit/sec/Hz per user capacity gap from the cut-set bound. Through some theoretical and simulation results, we show that by appropriately adjusting parameters, different points and areas of rate regions are achievable.

On Distributed Computing With Heterogeneous Communication Constraints

We consider a distributed computing framework where the distributed nodes have different communication capabilities, motivated by the heterogeneous networks in data centers and mobile edge computing systems. Following the structure of MapReduce, this framework consists of Map computation phase, Shuffle phase, and Reduce computation phase. The Shuffle phase allows distributed nodes to exchange intermediate values, in the presence of heterogeneous communication bottlenecks for different nodes (heterogeneous communication load constraints). For this setting, we characterize the minimum total computation load and the minimum worst-case computation load in some cases, under the heterogeneous communication load constraints. While the total computation load depends on the sum of the computation loads of all the nodes, the worst-case computation load depends on the computation load of a node with the heaviest job. We show an interesting insight that, for some cases, there is a tradeoff between the minimum total computation load and the minimum worst-case computation load, in the sense that both cannot be achieved at the same time. The achievability schemes are proposed with careful design on the file assignment and the data shuffling. Beyond the cut-set bound, a novel converse is proposed using the proof by contradiction. For the general case, we identify two extreme regimes in which both the scheme with coding and the scheme without coding are optimal, respectively.

Nonlinear Repair of Reed-Solomon Codes

The problem of repairing linear codes and, in particular, Reed Solomon (RS) codes has attracted a lot of attention in recent years due to their extreme importance to distributed storage systems. In this problem, a failed code symbol (node) needs to be repaired by downloading as little information as possible from a subset of the remaining nodes. By now, there are examples of RS codes that have efficient repair schemes, and some even attain the cut-set bound. However, these schemes fall short in several aspects; they require a considerable field extension degree. They do not provide any nontrivial repair scheme over prime fields. Lastly, they are all linear repairs, i.e., the computed functions are linear over the base field. Motivated by these and by a question raised by Guruswami and Wootters, 2017, on the power of nonlinear repair schemes, we study the problem of nonlinear repair schemes of RS codes. Our main results are the first nonlinear repair scheme of RS codes with asymptotically optimal repair bandwidth (asymptotically matching the cut-set bound). This is the first example of a nonlinear repair scheme of any code and also the first example that a nonlinear repair scheme can outperform all linear ones. Lastly, we show that the cut-set bound for RS codes is not tight over prime fields by proving a tighter bound, using additive combinatorics ideas.

Network Decoding Against Restricted Adversaries

We initiate the study of the one-shot capacity of communication (coded) networks with an adversary having access only to a proper subset of the network edges. We introduce the Diamond Network as a minimal example to show that known cut-set bounds are not sharp in general, and that their non-sharpness comes precisely from restricting the action of the adversary to a region of the network. We give a capacity-achieving scheme for the Diamond Network that implements an adversary detection strategy. We also show that linear network coding does not suffice in general to achieve capacity, proving a strong separation result between the one-shot capacity and its linear version. We then give a sufficient condition for tightness of the Singleton Cut-Set Bound in a family of two-level networks. Finally, we discuss how the presence of nodes that do not allow local encoding and decoding does or does not affect the one-shot capacity.

Improved Schemes for Asymptotically Optimal Repair of MDS Codes

We consider (n, k , l) MDS codes of length n, dimension k, and subpacketization l over a finite field <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> . A codeword of such a code consists of n column-vectors of length l over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> , with the property that any k of them suffice to recover the entire codeword. Each of these n vectors may be stored on a separate node in a network. If one of the n nodes fails, we can recover its content by downloading symbols from the surviving nodes, and the total number of symbols downloaded in the worst case is called the repair bandwidth of the code. By the cut-set bound, the repair bandwidth of an (n, k , l) MDS code is at least l(n-1)/(n- k). There are several constructions of MDS codes whose repair bandwidth meets or asymptotically meets the cut-set bound. For example, letting r=n-k denote the number of parities, Ye and Barg constructed (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) Reed-Solomon codes that asymptotically meet the cut-set bound, Ye and Barg also constructed optimal-bandwidth and optimal-update (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) MDS codes. Wang, Tamo, and Bruck constructed optimal-bandwidth (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> n/(r+1)</sup> ) MDS codes, and these codes have the smallest known subpacketization for optimal-bandwidth MDS codes. A key idea in all these constructions is to represent certain integers in base r. We show how this technique can be refined to improve the subpacketization of the two MDS code constructions by Ye and Barg, while achieving asymptotically optimal repair bandwidth. Specifically, when r=s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> for an integer s, we obtain an (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+n-1</sup> ) Reed-Solomon code and an optimal-update (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+ n-1</sup> ) MDS code, both having asymptotically optimal repair bandwidth. Thus for r = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , for example, we achieve the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> m+n-1</sup> rather than 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>mn</i></sup> in the original constructions by Ye and Barg. When r is not an integral power, we can still obtain (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> m+n-1</sup> ) Reed-Solomon codes and optimal-update ( n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+ n-1</sup> ) MDS codes by choosing positive integers s and m such that s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m </sup> ≤ r. In this case, however, the resulting codes have bandwidth that is near-optimal rather than asymptotically optimal. We also present an extension of this idea to reduce the subpacketization of the Wang-Tamo-Bruck construction while achieving a repair-by-transfer scheme with asymptotically optimal repair bandwidth. For example, for r = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> we achieve the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k/r+ m-1</sup> , which significantly improves upon the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>mn</i>/(r+1)</sup> in the Wang-Tamo-Bruck construction. Based on the foregoing examples, we believe our approach may be generally useful in reducing the subpacketization of MDS code constructions that utilize r-ary expansion.

Nonlinear Forwarding Strategy for Firefly Ultra Dense Networks With mmWave Fronthaul Links

We consider the uplink of firefly ultra dense networks which combine the promising features of ultra dense deployment and centralized processing. In these networks, a large number of remote radio units which we denote as firefly nodes (FNs) are spatially distributed over an area. The mobile devices (MDs) in the coverage area are simultaneously connected via sub-6 GHz radio frequency links to all FNs. Unlike the cloud radio access network (C-RAN) architecture, in firefly ultra dense networks, the FNs forward the MDs’ data through multi-hop millimeter-wave (mmWave) links to one or multiple root nodes since the coverage radius of each mmWave link is limited. These root nodes then forward the data via optical fiber links further to a central unit (CU), where the MDs’ signals are decoded. The amount of data that is received at each FN is potentially huge, and hence, efficient signal processing is needed at each FN before the received signals can be forwarded to other FNs. Therefore, we propose a nonlinear processing strategy, which quantizes the received signals at each FN. In particular, we formulate an optimization problem for a local design strategy for the nonlinear forwarding at the FNs, and present an optimal solution by exploiting strong duality and using the Lagrangian method to convert the optimization problem into an unconstrained problem via its dual formulation. A closed-form solution for the primal variables and a bisection algorithm for finding the optimal dual variables are presented. Moreover, based on the cut-set bound, we develop an upper bound on the achievable sum rate of the considered firefly network. The proposed nonlinear forwarding strategy is shown to outperform a benchmark linear forwarding strategy and to approach the performance upper bound in relevant transmit power regimes at the expense of a higher computational complexity. Our results reveal that having more root nodes in the topology improves the performance of linear and nonlinear forwarding but requires additional optical fiber links to the CU.

Capacity Limits of Full-Duplex Cellular Network

This paper aims to characterize the capacity limits of a wireless cellular network with a full-duplex (FD) base-station (BS) and half-duplex user terminals, in which three independent messages are communicated: the uplink message $m_1$ from the uplink user to the BS, the downlink message $m_2$ from the BS to the downlink user, and the device-to-device (D2D) message $m_3$ from the uplink user to the downlink user. From an information theoretical perspective, the overall network can be viewed as a generalization of the FD relay broadcast channel with a side message transmitted from the relay to the destination. We begin with a simpler case that involves the uplink and downlink transmissions of $(m_1,m_2)$ only, and propose an achievable rate region based on a novel strategy that uses the BS as a FD relay to facilitate the interference cancellation at the downlink user. We also prove a new converse, which is strictly tighter than the cut-set bound, and characterize the capacity region of the scalar Gaussian FD network without a D2D message to within a constant gap. This paper further studies a general setup wherein $(m_1,m_2,m_3)$ are communicated simultaneously. To account for the D2D message, we incorporate Marton's broadcast coding into the previous scheme to obtain a larger achievable rate region than the existing ones in the literature. We also improve the cut-set bound by means of genie and show that by using one of the two simple rate-splitting schemes, the capacity region of the scalar Gaussian FD network with a D2D message can already be reached to within a constant gap. Finally, a generalization to the vector Gaussian channel case is discussed. Simulation results demonstrate the advantage of using the BS as relay in enhancing the throughput of the FD cellular network.

Two-way relaying channel with lattice forwarding and microwave power transfer

In this paper, we consider a two-way relaying channel with simultaneous wireless information and power transfer. The intermediate relay station forwards lattice codes under transmit power constraint. Meanwhile, it charges user terminals for static circuit consumption and uplink transmission power. We novelly propose an iterative algorithm to maximize the sum-rate with joint beam-forming and power splitting design in MPT assisted TWRC. The method achieves the highest achievable rate compared with separate optimization designs and time switching method. Moreover, an extremely small gap between the proposed method and the outer cut-set bound is observed.

Degrees-Of-Freedom in Multi-Cloud Based Sectored Cellular Networks

This paper investigates the achievable per-user degrees-of-freedom (DoF) in multi-cloud based sectored hexagonal cellular networks (M-CRAN) at uplink. The network consists of N base stations (BS) and base band unit pools (BBUP), which function as independent cloud centers. The communication between BSs and BBUPs occurs by means of finite-capacity fronthaul links of capacities with P denoting transmit power. In the system model, BBUPs have limited processing capacity . We propose two different achievability schemes based on dividing the network into non-interfering parallelogram and hexagonal clusters, respectively. The minimum number of users in a cluster is determined by the ratio of BBUPs to BSs, . Both of the parallelogram and hexagonal schemes are based on practically implementable beamforming and adapt the way of forming clusters to the sectorization of the cells. Proposed coding schemes improve the sum-rate over naive approaches that ignore cell sectorization, both at finite signal-to-noise ratio (SNR) and in the high-SNR limit. We derive a lower bound on per-user DoF which is a function of , , and r. We show that cut-set bound are attained for several cases, the achievability gap between lower and cut-set bounds decreases with the inverse of BBUP-BS ratio for irrespective of , and that per-user DoF achieved through hexagonal clustering can not exceed the per-user DoF of parallelogram clustering for any value of and r as long as . Since the achievability gap decreases with inverse of the BBUP-BS ratio for small and moderate fronthaul capacities, the cut-set bound is almost achieved even for small cluster sizes for this range of fronthaul capacities. For higher fronthaul capacities, the achievability gap is not always tight but decreases with processing capacity. However, the cut-set bound, e.g., at , can be achieved with a moderate clustering size.

Capacity of the Gaussian Two-Pair Two-Way Relay Channel to Within $\frac{1}{2}$ Bit

This paper studies the transceiver design of the Gaussian two-pair two-way relay channel (TWRC), where two pairs of users exchange information through a common relay in a pairwise manner. Our main contribution is to show that the capacity of the Gaussian two-pair TWRC is achievable to within $\frac{1}{ 2}$ bit for arbitrary channel conditions. For the outer bound, we derive a genie-aided bound of the Gaussian two-pair TWRC, which is tighter than the cut-set bound. For the inner bound, we develop a hybrid coding scheme involving Gaussian random coding, nested lattice coding, superposition coding, and network-coded decoding. We further present a message-reassembling strategy to decouple the coding design for the user-to-relay and relay-to-user links, so as to provide flexibility to fully exploit the channel randomness. We show that judicious power allocation at the users and at the relay is necessary to approach the channel capacity under various channel conditions.

The Repair Problem for Reed–Solomon Codes: Optimal Repair of Single and Multiple Erasures With Almost Optimal Node Size

The repair problem in distributed storage addresses recovery of the data encoded using an erasure code, for instance, a Reed–Solomon (RS) code. We consider the problem of repairing a single node or multiple nodes in RS-coded storage systems using the smallest possible amount of inter-nodal communication. According to the cut-set bound, communication cost of repairing $h\geqslant 1$ failed nodes for an $(n,k=n-r)$ maximum distance separable (MDS) code using $d$ helper nodes is at least $dhl/(d+h-k)$ , where $l$ is the size of the node. Guruswami and Wootters (2016) initiated the study of efficient repair of RS codes, showing that they can be repaired using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality. In this paper, we construct the families of RS codes that achieve the cut-set bound for repair of one or several nodes. In the single-node case, we present the RS codes of length $n$ over the field ${\mathbb F}_{q^{l}}, l=\exp ((1+o(1))n\log n)$ that meet the cut-set bound. We also prove an almost matching lower bound on $l$ , showing that super-exponential scaling is both necessary and sufficient for scalar MDS codes to achieve the cut-set bound using linear repair schemes. For the case of multiple nodes, we construct a family of RS codes that achieve the cut-set bound universally for the repair of any $h=1,2, {\dots },r$ failed nodes from any subset of $d$ helper nodes, $k\leqslant d\leqslant n-h$ . For a fixed number of parities $r$ , the node size of the constructed codes is close to the smallest possible node size for codes with such properties.

Strong Converse Theorems for Multimessage Networks with Tight Cut-Set Bound

This paper considers a multimessage network where each node may send a message to any other node in the network. Under the discrete memoryless model, we prove the strong converse theorem for any network whose cut-set bound is tight, i.e., achievable. Our result implies that for any fixed rate vector that resides outside the capacity region, the average error probability of any sequence of length-n codes operated at the rate vector must tend to 1 as n approaches infinity. The proof is based on the method of types and is inspired by the work of Csiszar and Korner in 1982 which fully characterized the reliability function of any discrete memoryless channel with feedback for rates above capacity. In addition, we generalize the strong converse theorem to the Gaussian model where each node is subject to an almost-sure power constraint. Important consequences of our results are new strong converses for the Gaussian multiple access channel with feedback and the following relay channels under both models: the degraded relay channel (RC), the RC with orthogonal sender components, and the general RC with feedback.

Secure Multi-Source Multicast

The principal mission of multi-source multicast (MSM) is to disseminate all messages from all sources in a network to all destinations. MSM is utilized in numerous applications. In many of them, securing the messages disseminated is critical. A common secure model is to consider a network where there is an eavesdropper which is able to observe a subset of the network links, and seeks a code which keeps the eavesdropper ignorant regarding all the messages . While this is solved when all messages are located at a single source, secure MSM (SMSM) is an open problem, and the rates required are hard to characterize in general. In this paper, we consider individual security , which promises that the eavesdropper has zero mutual information with each message individually , or, more generally, with sub sets of messages . We completely characterize the rate region for SMSM under individual security, and show that such a security level is achievable at the full capacity of the network, that is, the cut-set bound is the matching converse, similar to non-secure MSM. Moreover, we show that the field size is similar to non-secure MSM and does not have to be larger due to the security constraint.

Channel Decoding for Nonbinary Physical-Layer Network Coding in Two-Way Relay Systems

In this paper, we propose a generalized channel decoding scheme for nonbinary physical-layer network coding (CD-NC) in two-way relay channels (TWRCs), where two source nodes A and B exchange their nonbinary symbols via a relay. The two sources use the same nonbinary low-density parity-check (LDPC) channel code over the integer ring Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> and M-pulse-amplitude modulation, respectively. The existing channel decoding schemes for nonbinary network coding suffer severe rate loss compared with the cut-set bound of TWRC, especially in the low-to-medium signal-to-noise ratio regime. The proposed CD-NC can decrease the rate loss. Our contributions are as follows: 1) We develop a generalized nonbinary sum product algorithm (G-SPA) for CD-NC according to the principle of virtual encoding of the superimposed symbols. Simulation results show that our CD-NC can achieve significant performance gains over the conventional nonbinary network coding for both additive white Gaussian noise and fading channels; and 2) We exploit two-dimensional fast-Fourier-transform-based belief propagation (2-D-FFT-BP) and extended min-sum (EMS) decoding algorithms to reduce the decoding complexity of G-SPA. Simulation results show that the 2-D-FFT-BP has the same performance as G-SPA, while EMS can greatly reduce the decoding complexity of G-SPA at the cost of slight performance degradation.

Coded Caching for Hierarchical Networks with a Different Number of Layers

Coded caching is a technique to reduce network loads. Recent works have discussed single-layer caching systems, but many networks consist of multiple layers of cache. In this paper, we discuss coded caching problem for a hierarchical network that has a different number of layers of cache. The network has users who connect to an origin server via a mirror server and users who directly connect to the origin server. We provide lower bounds on this problem setting based on cut-set bound. In addition, we propose three basic caching schemes and characterize these schemes. Also, we propose a novel coded caching scheme by combining two basic schemes and provide achievable rates of the combination coded caching scheme. Finally, we show that proposed combination scheme demonstrates good performance by a numerical result.

Asymptotically Optimal Regenerating Codes Over Any Field

The study of regenerating codes has advanced tremendously in recent years. However, most known constructions require large field size and, hence, may be hard to implement in practice. By restructuring a code construction by Rashmi et al. , we obtain two explicit families regenerating codes. These codes approach the cut-set bound as the reconstruction degree increases and may be realized over any given finite field if the file size is large enough. Essentially, these codes constitute a constructive proof that the cut-set bound does not imply a field size restriction, unlike some known bounds for ordinary linear codes. The first construction attains the cut-set bound at the MBR point asymptotically for all parameters, whereas the second one attains the cut-set bound at the MSR point asymptotically for low-rate parameters. Even though these codes require a large file size, this restriction is trivially satisfied in most conceivable distributed storage scenarios, that are the prominent motivation for regenerating codes.

Comments on Cut-Set Bounds on Network Function Computation

A function computation problem in directed acyclic networks has been considered in the literature, where a sink node wants to compute a target function with the inputs generated at multiple source nodes. The network links are error-free but capacity-limited, and the intermediate network nodes perform network coding. The target function is required to be computed with zero error. The computing rate of a network code is measured by the average number of times that the target function can be computed for one use of the network, i.e., each link in the network is used at most once. In the papers [1], [2], two cut-set bounds were proposed on the computing rate. However, we show in this paper that these bounds are not valid for general network function computation problems. We analyze the arguments that lead to the invalidity of these bounds and fix the issue with a new cut-set bound, where a new equivalence relation associated with the inputs of the target function is used. Our bound is qualified for general target functions and network topologies. We also show that our bound is tight for some special cases where the computing capacity is known. Moreover, some results in [11], [12] were proved using the invalid upper bound in [1] and hence their correctness needs further justification. We also justify their validity in the paper.

A Repair-Efficient Coding for Distributed Storage Systems Under Piggybacking Framework

Piggybacking is an efficient framework to reduce the repair bandwidth of distributed storage systems, especially, when the systems meet the settings–maximum distance separable (MDS), high code rate, and a small number of substripes. Through an analysis on the repair ratio of a piggybacking construction, this paper reveals that the proportion $p_{p}$ of piggybacking protected stripes is the key to significantly decrease the repair bandwidth. Based on this analysis, this paper proposes a repair efficient coding under piggybacking framework (REPB) by considering various piggybacking protected stripes and MDS only protected stripes. The repair ratio of REPB tends to 0 and is close to theoretical cut-set bound, as the proportion $p_{p}$ is no longer fixed at 1/2. Furthermore, the proposed REPB codes enjoy low computational complexity in repair operation.

Cut-Set Bound is Loose for Gaussian Relay Networks

The cut-set bound developed by Cover and El Gamal in 1979 has since remained the best known upper bound on the capacity of the Gaussian relay channel. We develop a new upper bound on the capacity of the Gaussian primitive relay channel which is tighter than the cut-set bound. Our proof is based on typicality arguments and concentration of Gaussian measure. Combined with a simple tensorization argument proposed by Courtade and Ozgur in 2015, our result also implies that the current capacity approximations for Gaussian relay networks, which have linear gap to the cut-set bound in the number of nodes, are order-optimal and leads to a lower bound on the pre-constant.

Comments on “Cutset Bounds on the Capacity of MIMO Relay Channels”

We comment on the paper “Cutset Bounds on the Capacity of MIMO Relay Channels” by Jeong et al. and point out that, unlike what appears from a remark and some other contents by these authors, the matrix distribution for the sum of two complex random Wishart matrices has already been derived by Kumar for the general case of arbitrary covariance matrices and not only for the special case when one of them is assumed proportional to the identity matrix. The latter assumption has been made only for deriving the corresponding eigenvalue distribution. Furthermore, we draw attention to the result that when all covariance matrices are chosen proportional to the identity matrix, then it is possible to obtain exact and closed form expressions for the sum of an arbitrary number of Wishart matrices and not only for two as considered by Jeong et al.