Efficiently spheredecodable physical layer transmission schemes for wireless storage networks
 Hsiaofeng (Francis) Lu^{1},
 Amaro Barreal^{2},
 David Karpuk^{2} and
 Camilla Hollanti^{2}Email author
https://doi.org/10.1186/s1363401603342
© Lu et al. 2016
Received: 3 July 2015
Accepted: 9 March 2016
Published: 5 May 2016
Abstract
Three transmission schemes over a new type of multipleaccess channel (MAC) model with intersource communication links are proposed and investigated in this paper. This new channel model is well motivated by, e.g., wireless distributed storage networks, where communication to repair a lost node takes place from helper nodes to a repairing node over a wireless channel. Since in many wireless networks nodes can come and go in an arbitrary manner, there must be an inherent capability of internode communication between every pair of nodes.
Assuming that communication is possible between every pair of helper nodes, the newly proposed schemes are based on various smart timesharing and relaying strategies. In other words, certain helper nodes will be regarded as relays, thereby converting the conventional uncooperative multipleaccess channel to a multipleaccess relay channel (MARC). The diversitymultiplexing gain tradeoff (DMT) of the system together with efficient spheredecodability and low structural complexity in terms of the number of antennas required at each end is used as the main design objectives. While the optimal DMT for the new channel model is fully open, it is shown that the proposed schemes outperform the DMT of the simple timesharing protocol and, in some cases, even the optimal uncooperative MAC DMT.
While using a wireless distributed storage network as a motivating example throughout the paper, the MAC transmission techniques proposed here are completely general and as such applicable to any MAC communication with intersource communication links.
Keywords
Distributed communications Distributed storage systems Diversitymultiplexing gain tradeoff MIMO Multipleaccess channel Relay channel Sphere decoding Wireless networks1 Introduction
The amount of data in cloud storage systems and worldwide data traffic have reached incredible numbers. It was estimated that in 2011, 1.8·10^{21} bytes of data needed to be stored worldwide [1], a number that grew to an astonishing 4.4·10^{21} bytes in 2013, and which is further expected to grow tenfold by 2020 [2]. The availability of such an astronomical amount of data and rapid progress in (wireless) communications engineering explain the observed growth of mobile data traffic, which increased by 69 % in 2014, reaching 2.5·10^{18} bytes per month at the end of the year. This amount of data traffic, which is expected to increase tenfold until 2019, corresponds to nearly 30 times the size of the entire internet in 2000. In addition, about 497 million mobile devices and connections were added globally in 2014, of which smart phones account for 88 %, so that—as foreseen—the number of mobileconnected devices exceeded the number of people on earth by the end of 2014 [3].
The massive amount of available data demands that data no longer be stored on a single device, but rather distributed among several storage nodes in a network, hence usually referred to as distributed storage systems (DSSs) (see [4] for a nice introduction). One of the main advantages of storing information in a distributed manner is that the storage system can be made robust against failures by introducing some level of redundancy. Some examples of reallife distributed storage systems are Apache Cassandra [5], which is a DSS initially developed at Facebook, and Windows Azure [6], created by Microsoft.
More formally, a DSS consists of n storage nodes over which a file is stored in a redundant manner by dividing it into fragments and distributing the fragments among n nodes using, for instance, a (n,k) maximum distance separable (MDS) erasure code [7]. MDS codes satisfy the Singleton bound and have the convenient property of being able to reconstruct the file by contacting any k of the nodes. Another key feature of a DSS is the ability to repair, meaning that when a node fails (that is a device breaks down or leaves the network), the failed node can be repaired or replaced. If an (n,k) storage code further satisfies the condition that any failed node can be repaired or replaced by contacting any K of the remaining nodes, termed helpers, the code is called an (n,k,K) storage code, and the node replacing the failed one is called a repairing node, or a newcomer [8], if it is not one of the already existing nodes in the network.
Sophisticated storage protocols have been developed, always giving a tradeoff between the amount of data that needs to be stored in any of the storage nodes, and the amount of data that needs to be retrieved for repairing a lost node, also called repair bandwidth (see e.g. [8, 9]. for details), and codes lying on the storagerepair bandwidth tradeoff curve [8, 10] are called regenerating codes. Explicit, tradeoff achieving regenerating codes can be found in the literature, see [9, 11], among others.
One important aspect of future DSSs lies in the ability to communicate over wireless channels, making it possible to store or retrieve a file using a wireless connection, even if the storage cloud itself might be wired. This is a feature related to the more general concept of wireless edge [12–15]. The mobility of a user has become crucial in everyday life, and wireless channels are used for data transmission for increased flexibility. However, it is wellknown that communicating over a fading channel in a wireless DSS [16, 17] makes repair transmissions prone to physical layer errors.
Yet another example to motivate such interhelper links is as follows. Notice that each of the n wireless storage nodes in the network consists of a wireless component and a memory component. In case of the wireless component of a node failing, we simply replace it with a new wireless component, and there is no need to contact the helpers to reconstruct the data. On the other hand, if the memory component fails, the “repair” of the node (hence the name of repairing node) happens by repairing the memory component of the node with the aid of helper nodes. That is, the node is not replaced by a completely new node. In this sense, the failed node is still one of the original n storage nodes, and the repairing process is done by contacting any K of the remaining nodes. In other words, the failed node and the repairing node are the same node with the same wireless component. This justifies the requirement that the interhelper link must exist between every pair of nodes.
When communicating over a wireless channel between terminals equipped with multiple antennas, spacetime codes [22–26] are often employed to protect the transmitted information from adverse channel effects such as fading and noise. The asymptotic errorperformance of spacetime codes is commonly dictated by the diversitymultiplexing gain tradeoff (DMT) [27]. Assume each of the K helper nodes has n _{ t } transmit antennas and transmits simultaneously at the same rate of R=r log2SNR in bits per channel use to the repairing node with n _{ r } receive antennas, where SNR is the signaltonoise power ratio, and r is commonly referred to as the multiplexing gain [27]. The optimal MIMOMAC DMT was given by Tse et al. in [28] and characterizes the maximal diversity gain, or equivalently the smallest error probability, that can possibly be achieved by any spacetime code. A general construction of MIMOMAC DMT optimal spacetime codes was proposed in [26] for any triple (n _{ t },n _{ r },K) and multiplexing gain r. These MIMOMAC codes are constructed from cyclic division algebras [23, 29] and have a lineardispersion form [30]. Therefore, they can be decoded in the maximallikelihood (ML) sense by a sphere decoder [31]. While sphere decoding is known to be an efficient implementation of ML decoding, it is unfortunate that when K n _{ t }>n _{ r }, decoding these DMToptimal codes requires at least partial bruteforce decoding before the sphere decoder begins to function [32], or alternatively one has to allow for suboptimal decoding methods, incurring degraded performance [33]. The former approach, though achieving the optimal performance, results in an exponential increase of decoding complexity when K n _{ t } becomes large and n _{ t } remains fixed [32].
Motivated by these realistic problems in wireless DSSs, new transmission schemes based on various strategies are proposed in this paper. The DMT, together with efficient spheredecodability and low transmitter and receiver structural complexity in terms of the number of antennas required at each end, are used as the main design objectives, thus naturally establishing a DMTcomplexity tradeoff.
1.1 Contributions and related work
In most of the storage and networkcoding related research the focus is on the (logical) network layer, while the physical layer functionality is usually ignored or assumed perfect. An exception is [16], where a socalled partial downloading scheme is proposed, which allows for data reconstruction with limited bandwidth by downloading only parts of the content of helper nodes. This is relevant in the fading channel scenario, and the idea can potentially be combined with the present work. In [17], optimal storage codes are constructed for the erroranderasure scenario, but fading is not addressed. Isolated from the storage point of view, a lot of research has been carried out in physical layer wireless communications, see e.g. [34] and the references therein.
Remark 1.
An obvious, but naïve attempt would be to try to simply combine an optimal storage code on the network layer and an optimal spacetime code on the physical layer. The reason for avoiding this approach is both its structural (many antennas) and computational (decoding subject to partial bruteforce) complexity.
In this paper^{2}, we propose a class of transmission schemes for MIMOMAC when communication links among the sources (helper nodes) do exist, which is generally true in many wireless storage networks. The proposed schemes allow for the design of efficiently spheredecodable spacetime codes^{3} with only one or two receive antennas. This is in contrast to the stateoftheart MIMOMAC codes [26] that have extremely good performance but require K n _{ t } receive antennas at the repairing node to enable efficient sphere decoding. This is of course unacceptable even for a relatively small value of K, since wireless networks are often heterogenous and might include nodes with only few or even just one antenna. At the moment, to the best of the authors’ knowledge, no such scheme exists for large value of K when the receiver has only 1 or 2 antennas, except for the trivial scheme of timesharing among K helper nodes.
This paper is organized as follows. In Section 2, we will present the channel model for DSS repair transmission, which can be seen as a MIMOMAC in the presence of communication links among helper nodes. A brief introduction on DMT will also be given therein. Section 2.2 briefly reviews the notion of complexity exponent, which was established by Jaldén and Elia [35] for measuring the minimal computational complexity required by sphere decoders in order to achieve a certain diversity performance. The complexity exponent of existing MIMOMAC DMT optimal codes [26, 32, 36] is also given to further motivate our design objectives that are presented in Section 3 and to serve as a baseline for comparing the complexity of the first proposed scheme given in Section 4. The first scheme is based on a simple time sharing among pairs of helpers, when n _{ t }=1, n _{ r }=2 and K≥2. The resulting DMT falls between the simple time sharing DMT and optimal MIMOMAC DMT [28].
We then present two more elaborate schemes extending the first one, achieving a higher DMT by taking advantage of the interhelper communication links and transforming the overall DSS network into a series of relay networks, where conventional halfduplex^{4} cooperativecommunication protocols such as the nonorthogonal amplifyandforward (NAF) strategy [37, 38] will be used. In particular, it will be seen that these schemes can outperform the MIMOMAC DMT at certain multiplexing gains, simply due to the use of interhelper communications in the DSS. Moreover, our results on the DMTs for NAFbased relay networks not only improve, but also extend the ones presented in related works, such as [39, 40].
2 Transmission model and preliminaries
Consider a wireless DSS with K helper nodes, equipped with n _{ t } transmit antennas each, and a repairing node with n _{ r } receive antennas. Let \(H_{i} \in \mathbb {C}^{n_{r} \times n_{t}}\) be the channel matrix, and \(X_{i} \in \mathbb {C}^{n_{t} \times T}\) the code matrix associated with the ith helper node, where T is the number of channel uses needed for transmitting X _{ i }. The received signal matrix at the repairing node is given by
where \(W \in \mathbb {C}^{n_{r} \times T}\) is a matrix modeling complex additive white Gaussian noise (AWGN). The entries of H _{ i } and W _{ i } are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero mean and unit variance, a distribution which we henceforth denote as \(\mathbb {C}{\cal {N}}(0,1)\). The code matrices X _{ i } are required to satisfy the average power constraint \(\mathbb {E}{X_{i}}^{2} \leq T \cdot \text {SNR}\). It is also assumed throughout the paper that the repairing node has a complete knowledge of channel state information {H _{ i }:i=1,…,K}.
2.1 The DMT
The outage probability P _{out}(r) is an asymptotic lower bound on the error probability of the scheme [27, 34] when SNR is large and the multiplexing gain r remains fixed.
where \(d^{*}_{m,n}(r)\) is the optimal DMT for an (n _{ t }×n _{ r }) pointtopoint MIMO channel and is given by the piecewise linear function connecting the points (r,(n _{ t }−r)(n _{ r }−r)) for r=0,1,…, min{n _{ t },n _{ r }} [27]. We must emphasize that (5) is known to be optimal for the noncooperative MIMOMAC, that is, the case when the K helper nodes share no common information, meaning that the interhelper channels (2) do not exist, and the file fragments stored at the K helper nodes are all statistically independent^{6}. Therefore, with a properly designed scheme, it is expected that a higher DMT performance than (5) can be achieved in DSS repair transmission.
2.2 Sphere decoding complexity of stateoftheart MIMOMAC codes
A general construction of MIMOMAC spacetime codes was proposed in [26] and was shown to achieve the optimal DMT (5) for any triple (n _{ t },n _{ r },K) and multiplexing gain r. More specifically, given n _{ t } and K, the component code \({{\mathcal {C}}}_{i}\) of the ith helper node is taken from an algebraic lattice of lattice rank^{7} \(2n_{t}{K_{o}^{2}}\) in [26], and \({{\mathcal {C}}}_{i}\) consists of (n _{ t }×T) matrices with \({{\mathcal {C}}_{i}} \doteq \text {SNR}^{rT}\) and T=n _{ t } K _{ o }, where K _{ o } is the smallest odd integer ≥K.
To estimate the complexity of decoding the overall code \({\mathcal {C}}_{1} \times \cdots \times {\mathcal {C}}_{K}\) using a joint sphere decoder, we follow [35] by using the notion of complexity exponent as a complexity measure.
Definition 1 (Complexity exponent [35]).
Given the multiplexing gain r, let \({\mathcal {C}}_{r,k}\) be a lattice code consisting of (n _{ t }×T) codeword matrices with \({{\mathcal {C}}_{r,k}} \doteq \text {SNR}^{rT}\), k=1,…,K. Let \({\mathcal {D}}_{r}\) be a decoder for the overall code \({\mathcal {C}}_{r} = {\mathcal {C}}_{r,1} \times \cdots \times {\mathcal {C}}_{r,K}\), subject to a computational constraint N _{max}(r), in floating point operations (flops) per T channel uses, in the sense that after N _{max}(r) flops, the decoder \({\mathcal {D}}_{r}\) must simply terminate, potentially prematurely and before completing the task, thus declaring an error. We then say \({\mathcal {D}}_{r}\) achieves diversity order d(r) with complexity exponent c(r) if \({\mathcal {D}}_{r}\) achieves error probability \(P_{e} \doteq \text {SNR}^{d(r)}\) using at most \(N_{\max }(r) \doteq \text {SNR}^{c(r)}\) (cf. (3),(4)) flops of computational reserves. □
The above definition means that in order to decode the code \({\mathcal {C}}_{r}\) using a joint sphere decoder, one does not have to decode every received signal matrix, especially when the communication channel is deeply faded. Instead, one can enforce a complexity constraint (also called a halting policy) at the sphere decoder, say at most N _{max}(r) flops of computational reserves. By choosing N _{max}(r) large enough such that the probability of any premature termination of the sphere decoder is asymptotically no larger than \(\text {SNR}^{d^{*}_{n_{t},n_{r},K}(r)}\), the overall error probability at most \(2 \cdot \text {SNR}^{d^{*}_{n_{t},n_{r},K}(r)}\), thereby achieving the same diversity \(d^{*}_{n_{t},n_{r},K}(r)\).
Remark 2.

Damens’ definition focuses on a code with a fixed rate, and Definition 1 concerns more with the theoretical asymptote at high SNR regime when the rate scales linearly with log2SNR.

Definition 1 considers the possibility of having a halting policy, while Damens’ definition requires the sphere decoder to complete its task at all channel realizations.
Remark 3.
In [42] Damen et al. proposed to decode (9) by using GDFEMMSE preprocessing followed by the sphere decoder when K n _{ t }>n _{ r }, in hope of making the matrix R uppertriangular and avoiding the need of resolving the ambiguities. However, it can be seen from [32, 36] that at high SNR regime the matrix R – after MMSEGDFE preprocessing – is illconditioned with K _{ o } n _{ t }(K n _{ t }−n _{ r }) number of singular values arbitrarily close to zero. This also explains the appearance of the first term in (6).
On the other hand, when the code has a fixed rate and operates in the low or moderate SNR regime, the MMSEGDFE approach does offer a certain complexity reduction with a negligible performance loss, as the singular values of R are numerically wellbehaved in general. Other approaches for further complexity reductions under such premises are also available in the literature. For instance, Barbero and Thompson [43] proposed a fixedcomplexity sphere decoder, where the number of candidates to be searched at the ith level of sphere decoding tree is at most n _{ i }, thereby yielding a constant complexity \(\prod _{i} n_{i}\). Another way to reduce complexity is through the various orderings of singular values of R. A comprehensive study in this direction can be found for example in [44]. We shall emphasize that the complexity exponents simulated in [43, 44] are both based on Damens’ definition [41] (cf. Remark 2) because of the aforementioned premises.
Remark 4.
In case of n _{ r }≥K n _{ t }, it has been shown [45, 46] that the DMT optimal MIMOMAC lattice codes can be decoded with subexponential complexity, i.e., having a complexity exponent asymptotically equal to 0, using the LenstraLenstraLovászbased lattice reduction aided regularized lattice decoder. The decoder is a combination of GDFEMMSE, lattice reduction and sphere decoding, and it has a vanishing gap of performance loss to the exact ML decoding as SNR approaches infinity.
3 Objectives for the design of transmission schemes
In Section 2, we have seen that there is a fundamental difference between the channel for DSS repair transmission and the classical MIMOMAC, in the sense that the former includes additional interhelper communication links. Thus, the MIMOMAC DMT (5) and the MIMOMAC codes [26] are no longer optimal in scenarios such as DSS repair transmission. Moreover, due to these additional interhelper channels, it is expected that the DSS repair transmission can have a higher optimal DMT than (5). This then calls for the design of new transmission schemes with good DMT performance for DSS repairing, which is the first design objective considered in this paper.
The second design objective comes from the observation of high decoding complexity of MIMOMAC codes [26] in Fig. 3 when K n _{ t }>n _{ r }. In a DSS, it is often possible that K is large, and n _{ r } is relatively small and fixed. This then calls for the design of new transmission schemes that can yield efficiently spheredecodable spacetime codes avoiding the need to process the ambiguities by bruteforce. Potentially, such an aim could be achieved by reducing the number of “active” helper nodes, i.e., reducing the effective value of K in (1), such that the average number of independent QAM symbols received by the repairing node at each channel use be no larger than n _{ r }, as observed from (6).
In the subsequent sections, we will focus on the case of n _{ t }=1 and K≫n _{ r }, and we will provide three transmission schemes, each for a different configuration of the wireless DSS network and for a different design objective. The first scheme is given in Section 4 for the case of two receive antennas and an arbitrary number of helper nodes, each having one transmit antenna. It is based on a simple time sharing among pairs of helpers and is aimed at having a low spheredecoding complexity at a cost of certain DMT performanceloss due to its neglect of existing interhelper links. The DMT for this scheme falls between the simple time sharing DMT and optimal MIMOMAC DMT (5).
Two more elaborate schemes will be presented in Sections 5 and 6, respectively, where we aim to improve the DMT performance at the possible cost^{8} of higher decoding complexity. These schemes take advantage of interhelper channels and transform the overall DSS network into a series of relay networks, where the conventional halfduplex NAF protocol [37, 38] will be used. In particular, we will see that these schemes can outperform the MIMOMAC DMT (5) at certain multiplexing gains, simply by exploiting interhelper communications in the DSS.
4 Scheme 1: n _{ t }=1, n _{ r }=2, and K helper nodes
We have seen in Section 2.2 that the existing stateoftheart MIMOMAC spacetime codes [26] could incur an extremely high decoding complexity when the repairing node has only a few number of antennas. Thus, our major aim in this section is to provide a new transmission scheme that can yield spacetime codes with reduced decoding complexity. In particular, we would like these potential codes to be efficiently spheredecodable, by which we mean that the H matrix, when writing the channel inputoutput relation in a vector form (cf. (9)), has linearly independent columns with probability one.
Besides the desired property of being efficiently spheredecodable, the complexity of the transmission schemes should also be considered. In other words, if we ignore the existence of interhelper links (2), then the schemes for DSS repair transmission can be made relatively simple. These are the main objectives of Scheme 1.
The following theorem is a straightforward consequence of [34].
Theorem 1.
5 Scheme 2: n _{ t }=1, general n _{ r }, and K helper nodes
The aim of Scheme 1 presented in the previous section was to have a small decoding complexity, at a cost of certain DMT performance loss due to the neglect of interhelper links in DSS repaircommunication. In this section as well as the next, we will shift our focus to designing transmission schemes that take into account these interhelper links and beat the DMT performance \(d^{*}_{n_{t}, n_{r}, K}(r)\).
Consider a DSS repair channel with K helpers, each having n _{ t }=1 transmit antenna, and a repairing node with n _{ r } receive antennas. To make good use of the interhelper links, we interpret in Scheme 2 some of the links as links of a relay channel. More specifically, in this scheme each of the K helper nodes will take turns acting as the source in a cooperative relay network [37], while the remaining K−1 helper nodes play the role of relays helping the source to send information to the repairing node.
With the above, the proposed scheme is a modification of the NAF protocol [37, 38] for a cooperative relay network with K−1 relays. It consists of K phases, and each phase requires at least 2(K−1) channel uses. Thus, the total number of channel uses required by Scheme 2 is at least 2K(K−1).
Then, at channel use t=2i, node u _{ i } joins node k and sends the amplified signal \(a_{u_{i}, k} r_{u_{i},2i1}\) to the repairing node.
5.1 DMT achieved by Scheme 2
Note firstly that by the symmetry among the phases of Scheme 2, it suffices to analyze the DMT achieved within the first phase, i.e., for k=1, where the helper node 1 acts as the source, and the remaining helper nodes are relays. Thus, for notational convenience, we will henceforth drop the subindex k.

the scheme takes K phases to complete, and

each phase requires 2(K−1) channel uses.
5.2 DMT achieved by Scheme 2 when n _{ r }=1
When n _{ r }=1, it can be seen that the DMT achieved by Scheme 2 is exactly the DMT for the NAF protocol derived by Azarian et al. [38] with K−1 relays and multiplexing gain Kr. Hence, the following result is immediate from [38].
Theorem 2.
5.3 Upper and lower bounds on d _{2}(r) with general n _{ r }
Analyzing the outage probability (23) turns out to be very challenging in general when the repairing node has multiple antennas, i.e., n _{ r }≥2. Almost all existing works such as [38, 47] consider only the case n _{ r }=1. In [39] Yang and Belfiore investigated the DMT for the MIMONAF protocol and provided a lower bound for such DMT. Their result can be modified to yield a lower bound for d _{2}(r). We will comment more on that particular lower bound at the end of this subsection.
To provide bounds on the DMT d _{2}(r) for general values of n _{ r }, let U be an (n _{ r }×n _{ r }) unitary matrix such that \( U \, \underline {h}_{1} =\; [\! \begin {array}{cccc} {\underline {h}_{1}}&0&\cdots &0 \end {array} ]^{\top } := \underline {h}. \)
Equations (27) and (29) then yield the following theorem for bounding the DMT d _{2}(r) for Scheme 2.
Theorem 3.
Proof.
Note that the random variables g _{ i }’s are i.i.d., hence there is no need to distinguish them in (27) and (29) when deriving the DMT. The same holds also true for a _{ i }, \(\underline {\ell _{i}}\), and its elements ℓ _{ i,j } for i=2,…,K. Thus, we set \({a_{i}}^{2} \doteq \text {SNR}^{b}\), \({g_{i}}^{2} \doteq \text {SNR}^{g}\), \({\underline {h}}^{2} \doteq \text {SNR}^{\alpha }\), \({\underline {\ell }_{i}}^{2} \doteq \text {SNR}^{\beta }\), and \({\ell _{i,j}}^{2} \doteq \text {SNR}^{\beta _{j}}\) with \(\beta = \min _{j=1,\ldots,n_{r}} \beta _{j}\). Moreover, we note that \({\underline {h}}^{2}\) is a χ ^{2} random variable with 2n _{ r } degrees of freedom, hence it contributes the term n _{ r } α to (27). Each \(\underline {\ell }_{i}\) consists of n _{ r } i.i.d. \(\mathbb {C}{\cal {N}}(0,1)\) complex random variables, and there is no need to distinguish ℓ _{ i,j } for i=2,…,K and for j=2,…,n _{ r } as can be seen from (27) and (29). Hence, we can set \({\ell _{i,j}}^{2} \doteq \text {SNR}^{\beta _{2}}\) for i=2,…,K and for j=2,…,n _{ r }. Similarly, there is no need to distinguish ℓ _{ i,1} for i=2,…,K, hence we set \({\ell _{i,1}}^{2} \doteq \text {SNR}^{\beta _{1}}\) for i=2,…,K. Finally, note that g _{ i }^{2} and ℓ _{ i,j }^{2} are i.i.d. χ ^{2} random variables with two degrees of freedom. Plugging the above into (27) and (29) and applying the Laplace principle as in [27] yield the desired upper and lower bounds (30) and (31).
□
where d _{RP}(r) is the DMT for the Rayleigh product channel \(\underline {h}_{i} \cdot g_{i}\), and an exact expression for d _{RP}(r) can be found in ([39], Proposition 1).
5.4 Another upper bound on d _{2}(r) with general n _{ r }
Theorem 4.
Proof.
Solving the above optimization problem gives the desired result. □
5.5 Remarks on the complexity exponents of Scheme 2
Determining the complexity exponents of the second scheme requires much more effort than determining the DMT. At least two major difficulties must be resolved before any identification of complexity exponents is possible. Notice that the notion of complexity exponents resides in an actual construction of spacetime codes for the scheme, and that the complexity exponents can vary from one code to another. Codes with a smaller complexity exponent are more favorable in practice, provided that the codes are optimal in the DMT sense, i.e., achieve the DMT d _{2}(r). Therefore, we have to at least identify a spacetime code for Scheme 2 first. In [39], Yang and Belfiore provided a systematic construction of spacetime codes that is approximately universal [48] for NAFbased cooperative relay communications. It is certainly possible to adapt their construction to the transmission using Scheme 2.
The second issue complicating the investigation of complexity exponents arises from the need of an exact characterization of eigenvalues of the matrices \(H_{i}^{\dag } K_{i}^{1} H_{i}\) for i=2,…,K, appearing in (21). Determining these eigenvalues is particularly difficult. It is in fact the main reason preventing us from obtaining an exact expression for d _{2}(r) in previous subsections, and we are only able to provide bounds on d _{2}(r) in this paper.
Nevertheless, it can be seen from (20) that the equivalent channel matrix H is of size (N n _{ r }×N) and has linearly independent columns with probability 1. This implies that when applying a sphere decoder to decode the codes—for instance, the code constructed by Yang and Belfiore [39]—transmitted using Scheme 2, the QR decomposition of the matrix H would result in an upper triangular matrix R; hence, there is no ambiguity to be resolved prior to processing the root of the sphere decoding tree. Therefore, the code must be efficiently sphere decodable.
6 Scheme 3: n _{ t }=1, n _{ r }≥2, and K helper nodes
In the previous section, we presented a powerful scheme that makes a good use of the interhelper links to improve the DMT performance of DSS repair transmission. The scheme allows one helper node to transmit information in each phase, and the remaining helper nodes are regarded as relays. Furthermore, we have introduced a novel technique that allows us to upperbound the DMT for the NAF protocol in a cooperative relay network with multiple antennas at the repairing node. In this section, we will present our third scheme, which can be seen as an enhancement of Schemes 1 and 2 and can provide a further improvement on the DMT performance.
The third proposed scheme concerns the case n _{ t }=1, n _{ r }≥2 and K helper nodes. It allows L helper nodes, L≤ min{n _{ r },K−1}, to transmit simultaneously and noncooperatively to the repairing node as well as to the remaining (K−L) helper nodes, which will function as relays^{9} in the network. To achieve an average multiplexing gain r, each of the selected L helper nodes must transmit at a higher multiplexing gain of \(\frac {K}{L} r\). In particular, we could later seek to improve the overall DMT performance by optimizing over the choices of L. Therefore, L can actually be a function of the multiplexing gain r.
6.1 DMT analysis for Scheme 3
The communication channel deduced from Scheme 3 resembles the multipleaccess relay channel (MARC), which was first introduced by Kramer and van Wijngaarden [49]. The DMTs for the twouser and singlerelay MARC—in terms of our notation this means n _{ t }=1, n _{ r }=1, K=3 and L=2—using various protocols have been studied in the past. For instance, Azarian et al. [50] investigated the DMT for such MARC using the dynamicdecodeandforward (DDF) strategy, and Yuksel and Erkip [51] focused on the compressforward (CF) protocol. Furthermore, a protocol similar to Scheme 3 was proposed in [40] and was termed multipleaccess amplifyandforward (MAF), which is a variation of the NAF protocol. It was found in [40] that the MAF outperforms the DDF in the high multiplexing gain regime and the CF protocol [51] in the low multiplexing gain regime when n _{ t }=1, n _{ r }=1, K=3, and L=2. The MAF thus provides a nice balance between complexity and performance.
Scheme 3 considers a much more complicated scenario than the one in [40], with n _{ t }=1, and general values of n _{ r }, K and L≤ min{n _{ r },K−1}. To the best of our knowledge, the DMT analysis for the MAF protocol has never been taken to such complexity level. On the other hand, our novel bounding technique employed in the proof of Theorem 4 is extremely powerful and enables us to analyze the DMT for general MARC using the MAF protocol.
To this end, for any subset \({{\mathcal {U}}}=\left \{u_{1}, \ldots, u_{k}\right \} \subseteq {{\mathcal {L}}}\) of the selected helper nodes, let \({{\mathcal {E}}}_{{\mathcal {U}}}\) denote the event that helper nodes u _{1},…,u _{ k } are in outage. The probability for \({{\mathcal {E}}}_{{\mathcal {U}}}\) is given by
The technique introduced in Section 5.4 can be applied to yield the following upper bound on d _{3}(L,r).
Theorem 5.
Proof.

\(\lambda _{s} = \text {SNR}^{\alpha _{s}}\) for \(s=1, \ldots, \frac N2\), with each α _{ s } contributing the term (2s−1+(n _{ r }−k))α _{ s } to the overall diversity order.

\({v_{s,j}}^{2} = \text {SNR}^{\beta _{s,j}} = \text {SNR}^{\beta _{j}}\) for \(s=1, \ldots, \frac N2\) and j=1,…,k, since there is no need to distinguish v _{ s,j } in these cases when applying the Laplace principle to (53). Each β _{ j }, j=1,…,k, contributes the term \(\frac N2 \beta _{j}\) to the overall diversity order.

\({v_{s,j}}^{2} = \text {SNR}^{\beta _{s,j}} = \text {SNR}^{\beta _{k+1}}\) for \(s=1, \ldots, \frac N2\) and j=k+1,…,n _{ r }, for the same reason. The factor β _{ k+1} contributes the term \(\frac N2 (n_{r}k) \beta _{k+1}\) to the overall diversity order.
Finally, the upper bound d _{3,U }(L,r) is obtained after minimizing \(d_{{\mathcal {U}}}(r)\) for all possible subsets \({{\mathcal {U}}} \subseteq {{\mathcal {L}}}\) (or equivalently for all k=1,…,L) and after simplifying the constraints in (55). □
By optimizing over all possible L=1,2,…, min{n _{ r },K−1} for d _{3,U }(r), we obtain an upper bound on the DMT performance for the third scheme.
Corollary 6.
In particular, we note that the DMT upper bound for Scheme 3 achieves the maximal possible multiplexing gain of \(\frac {2}{10}=0.2\), which is the same as the TDMA scheme and the MIMOMAC. Such possibility for the optimality of Scheme 3 turns out to be generally true, at least from the viewpoint of the upper bound (51). To see this, note that by (51), we have
i.e., \(r \leq \frac LK\). Now, with L=n _{ r }<K we see that Scheme 3 achieves the maximal possible multiplexing gain of \(\frac {n_{r}}{K}\) for each helper node, same as MIMOMAC [28], where the maximal possible multiplexing gain is given by \(\frac {\min \{Kn_{t}, n_{r}\}}{K} = \frac {n_{r}}{K}\).
7 Conclusions
The communications within a wireless storage network can be modeled as a multipleaccess channel with additional intersource communication links. Motivated by this observation, we have proposed three physical layer transmission schemes based on different timesharing and relaying strategies that are suitable for the given channel model. In contrast to the stateoftheart MAC DMT optimal algebraic spacetime codes, our schemes are efficiently spheredecodable with only one or two antennas. Their DMT performance reaches between the timesharing DMT and the optimal MAC DMT—the one for conventional MIMOMAC having no intersource links—in the highmultiplexing gain regime. When the desired multiplexing gain is low, the schemes even outperform the optimal MAC DMT. Naturally, the schemes are also applicable to DSS file reconstruction, as well as to any MAC communications with intersource links.
In the future, even small devices with very limited power may be equipped with several antennas thanks to massive MIMO at 60 Hz. However, implementation of a practical massive MIMO system still calls for a considerable amount of research efforts regarding pilot design, channel estimation, and code design. Before all that is realized, we believe that the proposed schemes provide a good and efficient alternative.
8 Endnotes
^{1} Devicetodevice (D2D) communication networks provide one such example, see e.g. [53, 54].
^{2} Preliminary results related to this work were reported in the Global Wireless Summit 2014 GWS’14 [18] (invited abstract which is considered a preprint), 21st International Symposium on Mathematical Theory of Networks and Systems MTNS’14 [19] (short invited abstract, Scheme 1), and 2014 International Symposium on Information Theory and Its Applications (ISITA) [20] (Schemes 1–3, now combined to Scheme 2). We point out that the numbering of the schemes has been changed so that the schemes previously called 2 and 3 [20] have been combined to Scheme 2, and the new scheme is hence now called Scheme 3 and has not appeared anywhere before. This paper extends the results by additional proofs for the bounds related to Scheme 2, and with a completely new scheme, Scheme 3, that improves upon the other schemes.
^{3} By efficiently spheredecodable spacetime code we mean that the code can be spheredecoded without the need of performing an exhaustive search for part of the symbols before starting processing the root of a spheredecoding tree. See discussions in Section 2.2.
^{4} By halfduplex we mean each node can choose to either transmit or receive, but not both at the same time.
^{5} Such a comparison might not seem fair to some readers as (5) assumes no interhelper links. However, the DMT (5) is the best DMT result that can be found in the related literature.
^{6} This latter condition might seem unrealistic in certain (logical) distributed storage codes. However, it would be extremely difficult to determine the mutual information between the helpers and the repairing node if one takes into account the shared information among helper nodes.
^{7} A lattice is a discrete abelian subgroup of a real or complex vector space, and its rank is given by its rank as a module over \({\mathbb Z}\). By an algebraic lattice we refer to one constructed from a number field extension or a division algebra, see e.g. [29].
^{8} It is unfortunate that measuring the exact complexity exponents for these schemes is extremely complicated, and we are unable to complete the task in this paper. Nevertheless, it can still be seen that these schemes can yield efficiently spheredecodable spacetime codes without the need of resolving ambiguities when processing the spheredecoding tree.
^{9} Here we have implicitly assumed K−L≥1 such that at least one helper node will function as a relay.
^{10} Cf. the corresponding upper bound \(\phantom {\dot {i}\!}d_{2,U'}(r)=d_{2}(r)\) for Scheme 3 that turned out to be tight.
Declarations
Acknowledgements
C. Hollanti is supported by the Academy of Finland grants #276031, #282938, and #283262, and by Magnus Ehrnrooth Foundation, Finland. D. Karpuk is supported by the Academy of Finland grant #268364. The support from the European Science Foundation under the ESF COST Action IC1104 is also gratefully acknowledged. The research of H. F. Lu was funded in part by Taiwan Ministry of Science and Technology under Grants MOST 1012923E009001MY3 and MOST 1032221E009043MY3.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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