Nested Lattice Codes for Gaussian Relay Networks with Interference
aa r X i v : . [ c s . I T ] F e b Nested Lattice Codes for Gaussian Relay Networkswith Interference
Wooseok Nam,
Student Member, IEEE,
Sae-Young Chung,
Senior Member, IEEE, andYong H. Lee,
Senior Member, IEEE
School of EECS, KAIST,Daejeon, Republic of KoreaE-mail: [email protected], [email protected], [email protected]
Abstract
In this paper, a class of relay networks is considered. We assume that, at a node, outgoing channels to itsneighbors are orthogonal, while incoming signals from neighbors can interfere with each other. We are interested inthe multicast capacity of these networks. As a subclass, we first focus on Gaussian relay networks with interferenceand find an achievable rate using a lattice coding scheme. It is shown that there is a constant gap between ourachievable rate and the information theoretic cut-set bound. This is similar to the recent result by Avestimehr,Diggavi, and Tse, who showed such an approximate characterization of the capacity of general Gaussian relaynetworks. However, our achievability uses a structured code instead of a random one. Using the same idea used inthe Gaussian case, we also consider linear finite-field symmetric networks with interference and characterize thecapacity using a linear coding scheme.
Index Terms
Wireless networks, multicast capacity, lattice codes, structured codes, multiple-access networks, relay networks
I. I
NTRODUCTION
Characterizing the capacity of general relay networks has been of great interest for many years. In thispaper, we confine our interest to the capacity of single source multicast relay networks, which is stillan open problem. For instance, the capacity of single relay channels is still unknown except for somespecial cases [1]. However, if we confine the class of networks further, there are several cases in whichthe capacity is characterized.Recently, in [2], the multicast capacity of wireline networks was characterized. The capacity is given bythe max-flow min-cut bound, and the key ingredient to achieve the bound is a new coding technique callednetwork coding. Starting from this seminal work, many efforts have been made to incorporate wirelesseffects in the network model, such as broadcast, interference, and noise. In [3], the broadcast nature wasincorporated into the network model by requiring each relay node to send the same signal on all outgoingchannels, and the unicast capacity was determined. However, the model assumed that the network isdeterministic (noiseless) and has no interference in reception at each node. In [4], the work was extendedto multicast capacity. In [5], the interference nature was also incorporated, and an achievable multicast ratewas computed. This achievable rate has a cut-set-like representation and meets the information theoreticcut-set bound [27] in some special cases. To incorporate the noise, erasure networks with broadcast orinterference only were considered in [7], [8]. However, the network models in [7], [8] assumed that theside information on the location of all erasures in the network is provided to destination nodes. Noisynetworks without side information at destination nodes were considered in [12] and [13] for finite-fieldadditive noise and erasure cases, respectively.
This work was supported by the IT R&D program of MKE/IITA. [2008-F-004-01, 5G mobile communication systems based on beamdivision multiple access and relays with group cooperation]
Along the same lines of the previous work on wireless networks mentioned above, we consider themulticast problem in a special class of networks called relay networks with interference. More specifically,we assume that all outgoing channels at each node are orthogonal, e.g., using frequency or time divisionmultiplexing, but signals incoming from multiple neighbor nodes to a node can interfere with each other.Since wireless networks are often interference limited, our setup focuses on the more important aspectof them. This model covers those networks considered in [8], [9], [10], [12]. Our interest in the relaynetworks with interference was inspired by [14], in which the capacity of single relay channels withinterference was established. In this paper, we focus on two special subclasses of general networks withinterference; Gaussian relay networks with interference and linear finite-field symmetric networks withinterference.For the Gaussian relay networks with interference, we propose a scheme based on nested lattice codes[19] which are formed from a lattice chain and compute an achievable multicast rate. The basic idea ofusing lattice codes is to exploit the structural gain of computation coding [11], which corresponds to akind of combined channel and network coding. Previously, lattices were used in Gaussian networks in[10], and an achievability was shown. However, our network model differs from the one in [10] in thatwe assume general unequal power constraints for all incoming signals at each node, while an equal powerconstraint was mainly considered in [10]. In addition, our lattice scheme is different from that in [10] inthat we use lattices to produce nested lattice codes, while lattices were used as a source code in [10].We also show that our achievable rate is within a constant number of bits from the information theoreticcut-set bound of the network. This constant depends only on the network topology and not on otherparameters, e.g., transmit powers and noise variances. This is similar to the recent result in [6], whichshowed an approximate capacity characterization for general Gaussian relay networks using a randomcoding scheme. However, our achievability uses a structured code instead of a random one. Thus, ourscheme has a practical interest because structured codes may reduce the complexity of encoding anddecoding.Finally, we introduce a model of linear finite-field symmetric networks with interference, which gen-eralizes those in [12], [13]. In the finite-field case, we use a linear coding scheme, which corresponds tothe finite-field counterpart of the lattice coding scheme. The techniques for deriving an achievable ratefor the finite-field network are basically the same as those for the Gaussian case. However, in this case,the achievable rate always meets the information theoretic cut-set bound, and, thus, the capacity is fullyestablished.This paper is organized as follows. Section II defines notations and parameters used in this paper andintroduces the network model and the problem of interest. In Section III, we analyze Gaussian relaynetworks with interference and give the upper and lower bounds for the multicast capacity. In Section IV,we define a model of linear finite-field symmetric networks with interference and present the multicastcapacity. Section V concludes the paper.II. R
ELAY NETWORKS WITH INTERFERENCE
A. System model and notations
We begin with a description of the class of networks that will be dealt with in this paper. The memorylessrelay networks with interference are characterized such that all outgoing channels from a node to itsneighbors are orthogonal to each other. We still assume that incoming signals at a node can interfere witheach other through a memoryless multiple-access channel (MAC). An example of this class of networks isshown in Fig. 1. Some special cases and subclasses of these networks have been studied in many previousworks [8], [9], [10], [13], [14].We will begin by giving a detailed description of the network and some definitions of the parameters.The network is represented by a directed graph G = ( V, E ) , where V = { , . . . , | V |} is a vertex set and E ⊆ V × V is an edge set. Each vertex and edge correspond to a communication node and a channel inthe network, respectively. In this paper, we focus on a multicast network: vertex 1 represents the source Documents\2008_Relay_net㱐(
Non-orthogonal MAC1 s
23 4 5 d OrthogonalBC ab c
Fig. 1. Example of general memoryless relay network with interference. node and is denoted by s , and the set of destination nodes is denoted by D , where s / ∈ D . It will beassumed that the source node has no incoming edge, and the destination nodes have no outgoing edge.All the other nodes, which are neither the source nor the destination, are called the relay nodes. Since allbroadcast channels in the network are orthogonal, we associate a discrete or continuous random variable X ( t ) u,v at time t with edge ( u, v ) ∈ E as a channel input (output of a node). As a channel output (inputof a node), we associate a discrete or continuous random variable Y ( t ) v at time t with node v ∈ V \ { } .From now on, we sometimes drop the superscript ‘ ( t ) ’ when doing so causes no confusion.At node v ∈ V , the set of incoming and outgoing nodes are denoted by ∆( v ) = { u : ( u, v ) ∈ E } , Θ( v ) = { w : ( v, w ) ∈ E } .Set S ⊂ V is called a cut if it contains node s and its complement S c contains at least one destinationnode d ∈ D , i.e., S c ∩ D = ∅ . Let Γ denote the set of all cuts. The boundaries of S and S c are defined as ¯ S = { u : ∃ v s.t. ( u, v ) ∈ E, u ∈ S, v ∈ S c } , ¯ S c = { v : ∃ u s.t. ( u, v ) ∈ E, u ∈ S, v ∈ S c } .For node v ∈ S c , the set of incoming nodes across S is defined as ∆ S ( v ) = ∆( v ) ∩ S = ∆( v ) ∩ ¯ S .For any sets S ⊆ V and S ⊆ V , we define X S ,S = { X u,v : ( u, v ) ∈ E, u ∈ S , v ∈ S } , Y S = { Y v : v ∈ S } ,and X ∆( v ) = { X u,v : u ∈ ∆( v ) } .Using the aforementioned notations, we can formally define the class of networks of interest. Thememoryless relay network with interference is characterized by the channel distribution function p ( y V | x V,V ) = p (cid:0) y | x ∆(2) (cid:1) p (cid:0) y | x ∆(3) (cid:1) · · · p (cid:0) y M | x ∆( M ) (cid:1) over all input and output alphabets. B. Coding for the relay network with interference
The multicast over the relay network consists of encoding functions f ( t ) u,v ( · ) , ( u, v ) ∈ E , t = 1 , . . . , N ,and decoding functions g d ( · ) , d ∈ D . The source node s has a random message W that is uniform over { , . . . , M } and transmits X ( t ) s,w = f ( t ) s,w ( W ) at time t on the outgoing channels ( s, w ) , w ∈ Θ( s ) . The relay node v transmits X ( t ) v,w = f ( t ) v,w ( Y t − v ) at time t on the outgoing channels ( v, w ) , w ∈ Θ( v ) , where Y t − v = (cid:16) Y (1) v , . . . , Y ( t − v (cid:17) . At destinationnode d ∈ D , after time N , an estimate of the source message is computed as ˆ W = g d (cid:0) Y Nd (cid:1) .Then, the probability of error is P e = Pr (cid:26) ∪ d ∈ D (cid:8) g d ( Y Nd ) = W (cid:9)(cid:27) . (1)We say that the multicast rate R is achievable if, for any ǫ > and for all sufficiently large N , encodersand decoders with M ≥ NR exist such that P e ≤ ǫ . The multicast capacity is the supremum of theachievable multicast rates.As stated in Section I, we are interested in characterizing the multicast capacity of the memorylessrelay networks with interference. However, as shown in [13], even for a relatively simple parallel relaychannel, finding the capacity is not easy. Thus, we further restrict our interest to the Gaussian networksin Section III and the linear finite-field symmetric networks in Section IV.III. G AUSSIAN RELAY NETWORKS WITH INTERFERENCE
In this section, we consider Gaussian relay networks with interference. At node v at time t , the receivedsignal is given by Y ( t ) v = X u ∈ ∆( v ) X ( t ) u,v + Z ( t ) v ,where Z ( t ) v is an independent identically distributed (i.i.d.) Gaussian random variable with zero mean andunit variance. For each block of channel input (cid:16) X (1) u,v , . . . , X ( n ) u,v (cid:17) , we have the average power constraintgiven by n n X t =1 (cid:0) X ( t ) u,v (cid:1) ≤ P u,v .In [10], Nazer et al. studied the achievable rate of the Gaussian relay networks with interference for theequal power constraint case, where P u,v = P v for all u ∈ ∆( v ) . In our work, we generalize it such that P u,v ’s can be different. The main result of this section is as follows. Theorem 1:
For a Gaussian relay network with interference, an upper bound for the multicast capacityis given by min S ∈ Γ X v ∈ ¯ S c C X u ∈ ∆ S ( v ) p P u,v , (2) where C ( x ) = log (1 + x ) . For the same network, we can achieve all rates up to min S ∈ Γ X v ∈ ¯ S c "
12 log P u ∈ ∆( v ) P u,v + 1 ! · max u ∈ ∆ S ( v ) P u,v ! + , (3)where [ x ] + , max { x, } . Furthermore, the gap between the upper bound and the achievable rate isbounded by X v ∈ V \{ } log ( | ∆( v ) | ) . (4) Remark 1:
Note that, in the equal power case, i.e., P u,v = P , the achievable multicast rate (3) has termsin the form of log (cid:0) K + P (cid:1) for some integer K ≥ . Similar forms of achievable rate were observed in[10], [15], [16], [25] for some equal power Gaussian networks.The following subsections are devoted to proving Theorem 1. A. Upper bound
The cut-set bound [27] of the network is given by R ≤ max p ( x V,V ) min S ∈ Γ I ( X S,V ; Y S c | X S c ,V ) . (5)Though the cut-set bound is a general and convenient upper bound for the capacity, it is sometimeschallenging to compute the exact cut-set bound in a closed form. This is due to the optimization by thejoint probability density function (pdf) p ( x V,V ) . In some cases, such as the finite-field networks in [5],[8], [12], [13], it is easy to compute the cut-set bound because a product distribution maximizes it. Forthe Gaussian case, however, the optimizing distribution for the cut-set bound is generally not a productdistribution.Thus, we consider another upper bound which is easier to compute than the cut-set bound. This boundis referred to as the relaxed cut-set bound and given by R ≤ min S ∈ Γ max p ( x V,V ) I ( X S,V ; Y S c | X S c ,V ) . (6)Due to the max-min inequality, the relaxed cut-set bound is looser than the original cut-set bound (5).For the relay network with interference, we can further simplify (6) as I ( X S,V ; Y S c | X S c ,V ) = I ( X S,S , X
S,S c ; Y S c | X S c ,V )= I ( X S,S c ; Y S c | X S c ,V )= I ( X ¯ S, ¯ S c ; Y ¯ S c | X S c ,V ) ,where the second and the third equalities follow by the structure of the network, i.e., • X S,S → ( X S,S c , X S c ,V ) → Y S c , • ( X S,S c , Y ¯ S c ) → X S c ,V → Y S c \ ¯ S c , • X S,S c = X ¯ S, ¯ S c .For cut S , the mutual information I ( X ¯ S, ¯ S c ; Y ¯ S c | X S c ,V ) is maximized when there is a perfect coherencebetween all inputs to a Gaussian MAC across the cut. Thus, we have max p ( x V,V ) I ( X ¯ S, ¯ S c ; Y ¯ S c | X S c ,V ) = X v ∈ ¯ S c C X u ∈ ∆ S ( v ) p P u,v . (7)Then by (6) and (7), the upper bound (2) follows. Voronoi region O Fig. 2. Example: two-dimensional lattice constellation.
B. Lattices and nested lattice codes
Before proving the achievable part of Theorem 1, let us establish some preliminaries for the latticesand nested lattice codes, which are key ingredients of our achievability proof. For a more comprehensivereview on lattices and nested lattice codes, see [19], [20], [23]. An n -dimensional lattice Λ is defined asa discrete subgroup of Euclidean space R n with ordinary vector addition. This implies that for any latticepoints λ, λ ′ ∈ Λ , we have λ + λ ′ ∈ Λ , λ − λ ′ ∈ Λ , and ∈ Λ . The nearest neighbor lattice quantizerassociated with Λ is defined as Q ( x ) = arg min λ ∈ Λ k x − λ k ,and the mod Λ operation is x mod Λ = x − Q ( x ) .The (fundamental) Voronoi region of Λ , denoted by R , is defined as the set of points in R n closer to theorigin than to any other lattice points, i.e., R = { x : Q ( x ) = } ,where ties are broken arbitrarily. In Fig. 2, an example of a two-dimensional lattice, and its Voronoi regionare depicted.We now define some important parameters that characterize the lattice. The covering radius of thelattice r cov is defined as the radius of a sphere circumscribing around R , i.e., r cov = min { r : R ⊆ r B} ,where B is an n -dimensional unit sphere centered at the origin, and, thus, r B is a sphere of radius r . Inaddition, the effective radius of Λ , denoted by r eff , is the radius of a sphere with the same volume as R ,i.e., r eff = (cid:18) Vol( R )Vol( B ) (cid:19) n , where Vol( · ) denotes the volume of a region. The second moment per dimension of Λ is defined as thesecond moment per dimension associated with R , which is given by σ ( R ) = 1Vol( R ) · n Z R k x k d x .In the rest of this paper, we also use Vol(Λ) and σ (Λ) , which have the same meaning as Vol( R ) and σ ( R ) , respectively. Finally, we define the normalized second moment of Λ as G (Λ) = σ ( R )(Vol( R )) /n .For any Λ , G (Λ) is greater than πe , which is the normalized second moment of a sphere whose dimensiontends to infinity. Goodness of lattices
We consider a sequence of lattices Λ n . The sequence of lattices is said to be Rogers-good if lim n →∞ r cov r eff = 1 ,which implies that Λ n is asymptotically efficient for sphere covering [20]. This also implies the goodnessof Λ n for mean-square error quantization, i.e., lim n →∞ G (Λ n ) = 12 πe .We now define the goodness of lattices related to the channel coding for the additive white Gaussian noise(AWGN) channel. A sequence of lattices is said to be Poltyrev-good if, for ¯ Z ∼ N ( , ¯ σ I ) , Pr { ¯ Z / ∈ R} ≤ e − nE P ( µ ) , (8)where E P ( · ) is the Poltyrev exponent [22] and µ is the volume-to-noise ratio (VNR) defined as µ = (Vol( R )) /n πe ¯ σ .Note that (8) upper bounds the error probability of the nearest lattice point decoding (or equivalently,Euclidean lattice decoding) when we use lattice points as codewords for the AWGN channel. Since E P ( µ ) > for µ > , a necessary condition for reliable decoding is µ > . Nested lattices codes
Now we consider two lattices Λ and Λ C . Assume that Λ is coarse compared to Λ C in the sense that Vol(Λ) ≥ Vol(Λ C ) . We say that the coarse lattice Λ is a sublattice of the fine lattice Λ C if Λ ⊆ Λ C andcall the quotient group (equivalently, the set of cosets of Λ relative to Λ C ) Λ C / Λ a lattice partition . Forthe lattice partition, the set of coset leaders is defined as C = { Λ C mod Λ } , { Λ C ∩ R} ,and the partitioning ratio is ρ = |C| n = (cid:18) Vol(Λ)Vol(Λ C ) (cid:19) n .Formally, a lattice code is defined as an intersection of a lattice (possibly translated) and a bounding(shaping) region, which is sometimes a sphere. A nested lattice code is a special class of lattice codes,whose bounding region is the Voronoi region of a sublattice. That is, the nested lattice code is defined in terms of lattice partition Λ C / Λ , in which Λ C is used as codewords and Λ is used for shaping. The codingrate of the nested lattice code is given by n log |C| = log ρ .Nested lattice codes have been studied in many previous articles [18], [19], [23], [24], and proved to havemany useful properties, such as achieving the capacity of the AWGN channel. In the next subsection, wedeal with the nested lattice codes for the achievability proof of Theorem 1. C. Nested lattice codes for a Gaussian MAC
As an achievable scheme, we use a lattice coding scheme. In [10], lattices were also used to prove anachievable rate of Gaussian relay networks with interference (called Gaussian MAC networks). However,they used the lattice as a source code with a distortion and then related the achievable distortion to theinformation flow through the network. Our approach is different from [10] in that we use lattices to producecoding and shaping lattices, and form nested lattice codes. As a result, our approach can handle unequalpower constraints where incoming links have different power at a MAC. Our scheme is a generalizationof the nested lattice codes used for the Gaussian two-way relay channel in [15], [16].Let us consider a standard model of a Gaussian MAC with K input nodes: Y = K X j =1 X j + Z , (9)where Z denotes the AWGN process with zero mean and unit variance. Each channel input X i is subjectto the average power constraint P i , i.e., n P nt =1 ( X ( t ) i ) ≤ P i . Without loss of generality, we assume that P ≥ P ≥ · · · ≥ P K .The standard MAC in (9) is a representative of MACs in the Gaussian relay network with interference.Now, we introduce encoding and decoding schemes for the standard MAC. Let us first consider thefollowing theorem which is a key for our code construction. Theorem 2:
For any P ≥ P ≥ · · · ≥ P K ≥ and γ ≥ , a sequence of n -dimensional lattice chains Λ n ⊆ Λ n ⊆ · · · ⊆ Λ nK ⊆ Λ nC exists that satisfies the following properties.a) Λ ni , ≤ i ≤ K , are simultaneously Rogers-good and Poltyrev-good while Λ nC is Poltyrev-good.b) For any δ > , P i − δ ≤ σ (Λ ni ) ≤ P i , ≤ i ≤ K , for sufficiently large n .c) The coding rate of the nested lattice code associated with the lattice partition Λ nC / Λ nK can approachany value as n tends to infinity, i.e., R K , n log |C K | = γ + o n (1) ,where C K = { Λ nC mod Λ nK } and o n (1) → as n → ∞ . Furthermore, for ≤ i ≤ K − , the coding rateof the nested lattice code associated with Λ nC / Λ ni is given by R i , n log |C i | = R K + 12 log (cid:18) P i P K (cid:19) + o n (1) ,where C i = { Λ nC mod Λ ni } . Proof:
See Appendix A.A conceptual representation of the lattice chain and the corresponding sets of coset leaders are givenin Fig. 3 for a two-dimensional case.
Encoding
We consider a lattice chain as described in Theorem 2. We assign the i -th input node to the MAC withthe set of coset leaders C i . For each input node, the message set (cid:8) , . . . , nR i (cid:9) is arbitrarily mapped onto O Fig. 3. Example of lattice chain and sets of coset leaders. C i . We also define random dither vectors U i ∼ Unif( R i ) , ≤ i ≤ K , where R i denotes the Voronoiregion of Λ i (we dropped the superscript ‘ n ’ for simplicity). These dither vectors are independent of eachother and also independent of the message of each node and the noise. We assume that each U i is knownto both the i -th input node and the receiver. To transmit a message that is uniform over (cid:8) , . . . , nR i (cid:9) ,node i chooses W i ∈ C i associated with the message and sends X i = ( W i + U i ) mod Λ i .Let us introduce a useful lemma, which is known as the crypto-lemma and frequently used in the restof this paper. The lemma is given in [23], and we repeat it here for completeness. Lemma 1 (Crypto-lemma [23]):
Let C be a finite or compact group with group operation + . Forindependent random variables a and b over C , let c = a + b . If a is uniform over C , then c is independentof b and uniform over C .By Lemma 1, X i is uniformly distributed over R i and independent of W i . Thus, regardless of W i , theaverage transmit power of node i is equal to σ (Λ i ) , which approaches P i as n tends to infinity. Thus,the power constraint is met. Decoding
Upon receiving Y = P Kj =1 X j + Z , where Z is a vector of i.i.d. Gaussian noise with zero mean andunit variance, the receiver computes ˜ Y = α Y − K X j =1 U j ! mod Λ = " K X j =1 ( W j + U j ) mod Λ j − K X j =1 X j + α K X j =1 X j + α Z − K X j =1 U j mod Λ = (cid:16) T + ˜ Z (cid:17) mod Λ , where T = " K X j =1 ( W j − Q j ( W j + U j )) mod Λ = " W + K X j =2 ( W j − Q j ( W j + U j )) mod Λ , (10) ˜ Z = − (1 − α ) K X j =1 X j + α Z , ≤ α ≤ is a scaling factor, and Q j ( · ) denotes the nearest neighbor lattice quantizer associated with Λ j . We choose α as the minimum mean-square error (MMSE) coefficient to minimize the variance of theeffective noise ˜ Z . Thus, α = P Kj =1 P j P Kj =1 P j + 1 ,and the resulting noise variance satisfies n E (cid:26)(cid:13)(cid:13)(cid:13) ˜ Z (cid:13)(cid:13)(cid:13) (cid:27) ≤ P Kj =1 P j P Kj =1 P j + 1 . (11)Note that, though the relation in (11) is given by an inequality, it becomes tight as n → ∞ by Theorem2. By the chain relation of the lattices in Theorem 2, it is easy to show that T ∈ C . Regarding T , wehave the following lemma. Lemma 2: T is uniform over C and independent of ˜ Z . Proof:
Define ˜ W , P Kj =2 ( W j − Q j ( W j + U j )) mod Λ , and, thus, T = (cid:16) W + ˜ W (cid:17) mod Λ .Note that ˜ W is correlated with X i , ≤ i ≤ K , and ˜ Z . Since W is uniform over C and independentof ˜ W , T is independent of ˜ W and uniformly distributed over C (crypto-lemma). Hence, if T and ˜ Z arecorrelated, it is only through W . However, W and ˜ Z are independent of each other, and, consequently, T is also independent of ˜ Z .The receiver tries to retrieve T from ˜ Y instead of recovering W i , ≤ i ≤ K , separately. For thedecoding method, we consider Euclidean lattice decoding [19]-[23], which finds the closest point to ˜ Y in Λ C . From the symmetry of the lattice structure and the independence between T and ˜ Z (Lemma 2),the probability of decoding error is given by p e = Pr n T = Q C (cid:16) ˜ Y (cid:17)o = Pr n ˜ Z mod Λ / ∈ R C o , (12)where Q C ( · ) denotes the nearest neighbor lattice quantizer associated with Λ C and R C denotes the Voronoiregion of Λ C . Then, we have the following theorem. Theorem 3:
Let R ∗ = "
12 log P P Kj =1 P j + P ! + .For any ¯ R < R ∗ and a lattice chain as described in Theorem 2 with R approaching ¯ R , i.e., R =¯ R + o n (1) , the error probability under Euclidean lattice decoding (12) is bounded by p e ≤ e − n “ E P “ R ∗ − ¯ R ” − o n (1) ” . Proof:
See Appendix B.According to Theorem 3, the error probability vanishes as n → ∞ if ¯ R < R ∗ since E p ( x ) > for x > . This implies that the nested lattice code can achieve any rate below R ∗ . Thus, by c) of Theorem2 and Theorem 3, the coding rate R i , ≤ i ≤ K , can approach R ∗ i arbitrarily closely while keeping p e arbitrarily small for sufficiently large n , where R ∗ i = "
12 log P i P Kj =1 P j + P i ! + . (13) Remark 2:
In theorem 3, we showed the error exponent of lattice decoding and the achievability of R directly followed. However, if we are only interested in finding the achievability of R , not in the errorexponent, we can use the argument on the bounding behavior of lattice decoding in [21], which gives thesame result in a much simpler way. Remark 3:
Since P ≥ · · · ≥ P K , we have R ∗ ≥ · · · ≥ R ∗ K . Now, consider the case that, for some ˆ i < K , the rates R ∗ i , ˆ i + 1 ≤ i ≤ K , are zero while R ∗ i , ≤ i ≤ ˆ i , are nonzero. In this situation, nodes ˆ i + 1 , . . . , K cannot transmit any useful information to the receiver, and, thus, we can turn them off so asnot to hinder the transmissions of nodes , . . . , ˆ i . Then, the variance of ˜ Z decreases and we have extendedrates given by R ∗ i = "
12 log P i P ˆ ij =1 P j + P i ! + , ≤ i ≤ ˆ i .However, for the ease of exposition, we do not consider the transmitter turning-off technique and assumethat nodes ˆ i + 1 , . . . , K just transmit X i = U i when their coding rates are zero. D. Achievable multicast rate
We consider B blocks of transmissions from the source to destinations. Each block consists of n channeluses. In block k ∈ { , . . . , B } , an independent and uniform message W [ k ] ∈ { , . . . , nR } is sent from thesource node s . It takes at most L , B + | V | − blocks for all the B messages to be received by destinationnodes. After receiving L blocks, destination nodes decode the source message W , ( W [1] , . . . , W [ B ]) .Thus, the overall rate is BL R , which can be arbitrarily close to R by choosing B sufficiently large. Time-expanded network
For ease of analysis, we consider the B blocks of transmissions over the time-expanded network [2],[5], G TE , obtained by unfolding the original network G over L + 1 time stages. In G TE , node v ∈ V atblock k appears as v [ k ] , and v [ k ] and v [ k ′ ] are treated as different nodes if k = k ′ . There are a virtualsource and destination nodes, denoted by s TE and d TE , respectively. We assume that s TE and s [ k ] ’s areconnected through virtual error-free infinite-capacity links, and, similarly, d TE and d [ k ] ’s are. For instance,the network in Fig. 1 is expanded to the network in Fig. 4. Dealing with the time-expanded network doesnot impose any constraints on the network. Any scheme for the original network can be interpreted toa scheme for the time-expanded network and vice-versa. In our case, the transmissions of B messages W [ k ] , k = 1 , . . . , B , from s to d ∈ D over G correspond to the transmission of a single message W from s TE to d TE ∈ D TE over G TE , where D TE denotes the set of virtual destination nodes.A main characteristic of the time-expanded network is that it is always a layered network [5] whichhas equal length paths from the source to each destination . We define the set of nodes at length k fromthe virtual source node as V TE [ k ] = { v [ k ] : v ∈ V } Another characteristic is that the time-expanded network is always acyclic [2]. …… Fig. 4. Time-expansion of the network in Fig. 1. and call it the k -th layer. We use the subscript ‘ TE ’ to differentiate parameters of G and G TE . The set ofnodes and edges of G TE are defined as V TE = { s TE } ∪ D TE ∪ (cid:18) L +1 ∪ k =1 V TE [ k ] (cid:19) , E TE = { ( u [ k ] , v [ k + 1]) : ( u, v ) ∈ E, k = 1 , . . . , L }∪ { ( s [ k − , s [ k ]) : k = 1 , . . . , L }∪ { ( d [ k ] , d [ k + 1]) : k = 1 , . . . , L } ,where we define s [0] = s TE and d [ L + 2] = d TE . Note that, since G TE is layered, edges only appearbetween adjacent layers. From V TE and E TE , the other parameters, e.g., ∆ TE ( · ) , Θ TE ( · ) , S TE , ¯ S TE , Γ T E ,and ∆ TE ,S ( · ) , are similarly defined as ∆( · ) , Θ( · ) , S , ¯ S , Γ , and ∆ S ( · ) , respectively. Encoding
We apply the nested lattice codes in Section III-C over the all Gaussian MACs in the network. Thus,node v [ k ] is assigned with sets of coset leaders C v [ k ] ,w [ k + ] , w [ k + ] ∈ Θ TE ( v [ k ]) , where k + , k + 1 . We donot change the lattice scheme over blocks, and, thus, C v [ k ] ,w [ k + ] = C v,w At node s [ k ] , the indices (cid:8) , . . . , nR (cid:9) are uniformly randomly mapped onto vectors in C s,w , w ∈ Θ( s ) .We define the random mapping as f s [ k ] ,w [ k + ] ( · ) . Then, node s [ k ] receives W = ( W [1] , . . . , W [ B ]) from s [ k − ] through the error-free link, where k − , k − , and transmits W s [ k ] ,w [ k + ] = f s [ k ] ,w [ k + ] ( W [ k ]) on channel ( s [ k ] , w [ k + ]) using a random dither vector U s [ k ] ,w [ k + ] . At node v [ k ] that is not s [ k ] or d [ k ] ,the received signal is given by ˜ Y v [ k ] = (cid:16) T v [ k ] + ˜ Z v [ k ] (cid:17) mod Λ v , (14) where T v [ k ] = X u [ k − ] ∈ ∆TE( v [ k ]) (cid:0) W u [ k − ] ,v [ k ] − Q u,v (cid:0) W u [ k − ] ,v [ k ] + U u [ k − ] ,v [ k ] (cid:1)(cid:1) mod Λ v , (15)and ˜ Z v [ k ] is an effective noise vector. In (14), Λ v denotes the lattice associated with the incoming channelto node v with the largest power. Then, T v [ k ] is decoded using Euclidean lattice decoding, which yieldsan estimate ˆ T v [ k ] . Next, ˆ T v [ k ] is uniformly and randomly mapped onto vectors in C v,w , w ∈ Θ( v ) . Thismapping is denoted by f v [ k ] ,w [ k + ] ( · ) , and node v [ k ] transmits W v [ k ] ,w [ k + ] = f v [ k ] ,w [ k + ] (cid:16) ˆ T v [ k ] (cid:17) on channel ( v [ k ] , w [ k + ]) using a random dither vector U v [ k ] ,w [ k + ] . Node d [ k ] , d ∈ D , receives ˜ Y d [ k ] andcomputes ˆ T d [ k ] . It also receives (cid:16) ˆ T d [1] , . . . , ˆ T d [ k − ] (cid:17) from d [ k − ] through the virtual error-free infinite-capacity link and passes (cid:16) ˆ T d [1] , . . . , ˆ T d [ k ] (cid:17) to node d [ k + ] .We assume that all the random mappings f u [ k ] ,v [ k + ] , ( u [ k ] , v [ k + ]) ∈ E TE are done independently. Decoding
While decoding, a virtual destination node d TE ∈ D TE assumes that there is no error in decoding T v [ k ] ’s in the network and that the network is deterministic. Therefore, with knowledge of all deterministicrelations (15) in the network, node d TE decodes W by simulating all nBR messages and finding onethat yields the received signal ˆ T d TE , (cid:16) ˆ T d [1] , . . . , ˆ T d [ L +1] (cid:17) . Calculation of the probability of error
In the above decoding rule, we will declare an error if at least one of the following events occurs. • E : there is an error in decoding T v [ k ] at at least one node in the network. • E : a message W ′ = W exists that yields the same received signal ˆ T d TE , which is obtained under W , at at least one virtual destination node d TE ∈ D TE .Thus, the error probability is given by P e = Pr {E ∪ E }≤ Pr {E } + Pr {E |E c } . (16)Let us consider the first term in (16). Using the union bound, we have Pr {E } ≤ L +1 X k =2 X v [ k ] ∈ V [ k ] \{ s [ k ] } p e,v [ k ] ,where p e,v [ k ] , Pr n ˆ T v [ k ] = T v [ k ] o .Note that the summation is from k = 2 since nodes in the first layer do not have any received signalexcept for node s [1] . By Theorem 3, at node v ∈ V \ { } for any ǫ > , p e,v [ k ] is less than ǫ L | V | forsufficiently large n if R u,v = 1 n log |C u,v | =
12 log P u ′∈ ∆( v ) P u ′ ,v + 1 · P u,v − ǫ + (17) It is assumed that the all random dither vectors are known to destination nodes. Thus, (15) is deterministic. for all u ∈ ∆( v ) . Therefore, in this case Pr {E } ≤ ǫ .Now, we consider the second term in (16). Under the condition E c , we have ˆ T v [ k ] = T v [ k ] , and, thus,the network is deterministic. Let us use the notation W u [ k − ] ,v [ k ] ( W ) and T v [ k ] ( W ) to explicitly denote thesignals under message W . We say that node v [ k ] can distinguish W and W ′ if T v [ k ] ( W ) = T v [ k ] ( W ′ ) .Thus, from the argument of a deterministic network in [5], the error probability is bounded by Pr {E |E c } ≤ nBR · Pr ( ∪ d TE ∈ D TE { T d TE ( W ) = T d TE ( W ′ ) } ) = 2 nBR · X S TE ∈ ΓTE Pr { Nodes in S TE can distinguish W , W ′ , and nodes in S c TE cannot } . (18)We briefly denote the probabilities in the summation in (18) as Pr (cid:8) D = S TE , ¯ D = S c TE (cid:9) .Here, we redefine the cut in the time-expanded network G TE for convenience sake. From the encodingscheme, since the source message propagates through nodes s [ k ] , k = 1 , . . . , L + 1 , they can clearlydistinguish W and W ′ . Similarly, if a virtual destination node d TE cannot distinguish W and W ′ , nodes d [ k ] , k = 1 , . . . , L + 1 cannot either. Thus, when we analyze the error probability (18), we can alwaysassume that s [ k ] ∈ S TE and d [ k ] ∈ S c TE , k = 1 , . . . , L + 1 , without loss of generality.From the fact that G TE is layered, we have Pr (cid:8) D = S TE , ¯ D = S c TE (cid:9) = Pr (cid:8) D = S TE , ¯ D = S c TE [1] (cid:9) · L +1 Y k =2 Pr (cid:8) ¯ D = S c TE [ k ] |D = S TE [ k − ] , ¯ D = S c TE [ k − ] (cid:9) ≤ L +1 Y k =2 Pr (cid:8) ¯ D = S c TE [ k ] |D = S TE [ k − ] , ¯ D = S c TE [ k − ] (cid:9) , (19)where S TE [ k ] and S c TE [ k ] denote the sets of nodes in S TE and S c TE in the k -th layer, i.e., S TE [ k ] , S TE ∩ V TE [ k ] , S c TE [ k ] , S c TE ∩ V TE [ k ] .Also, from the fact that the random mapping for each channel is independent, we have Pr (cid:8) ¯ D = S c TE [ k ] |D = S TE [ k − ] , ¯ D = S c TE [ k − ] (cid:9) = Y v [ k ] ∈ Sc TE[ k ] Pr (cid:8) ¯ D = { v [ k ] }|D = S TE [ k − ] , ¯ D = S c TE [ k − ] (cid:9) . (20)Then, we have the following lemma. Lemma 3:
Consider the time-expanded network G TE with independent uniform random mapping ateach node. For any cut S TE in G TE , we have Pr (cid:8) ¯ D = { v [ k ] }|D = S TE [ k − ] , ¯ D = S c TE [ k − ] (cid:9) ≤ − n max u [ k − ] ∈ ∆TE ,S ( v [ k ]) R u,v From the definition, s [ k ] ∈ S TE and d [ k ] ∈ S c TE , k = 1 , . . . , L + 1 . for node v [ k ] ∈ ¯ S c TE [ k ] , where ¯ S c TE [ k ] , ¯ S c TE ∩ V TE [ k ] . For node v [ k ] ∈ S c TE [ k ] \ ¯ S c TE [ k ] , we have Pr (cid:8) ¯ D = { v [ k ] }|D = S TE [ k − ] , ¯ D = S c TE [ k − ] (cid:9) = 1 . Proof:
See Appendix C.Thus, by (18)-(20) and Lemma 3, it follows that Pr {E |E c } ≤ nBR · | Γ TE | · − n min S TE ∈ ΓTE L +1 P k =2 P v [ k ] ∈ ¯ Sc TE[ k ] max u [ k − ] ∈ ∆TE ,S ( v [ k ]) R u,v . (21)We now consider the following lemma. Lemma 4:
In the time-expanded G TE with L + 1 layers, the term in the exponent of (21) min S TE ∈ Γ TE L +1 X k =2 X v [ k ] ∈ ¯ Sc TE[ k ] max u [ k − ] ∈ ∆TE ,S ( v [ k ]) R u,v is upper bounded by L · min S ∈ Γ X v ∈ ¯ S c (cid:18) max u ∈ ∆ S ( v ) R u,v (cid:19) ,and lower bounded by ( L − | Γ | + 2) · min S ∈ Γ X v ∈ ¯ S c (cid:18) max u ∈ ∆ S ( v ) R u,v (cid:19) . Proof:
See Appendix D.Therefore, by (17), (21), and Lemma 4, Pr {E |E c } is less than ǫ for sufficiently large n if R < L − | Γ | + 2 B · min S ∈ Γ X v ∈ ¯ S c "
12 log P u ∈ ∆( v ) P u,v + 1 ! · max u ∈ ∆ S ( v ) P u,v ! − ǫ + . (22)Thus, the total probability of error (16) is less than ǫ , and the achievability follows from (22). E. Gap between the upper and lower bounds
To compute the gap between the upper bound (2) and the achievable rate (3), we can rely on thefollowing lemmas.
Lemma 5:
Assume that P ≥ · · · ≥ P K ≥ . For any nonempty set A ⊆ { , . . . , K } and l = min A ,we have
12 log X j ∈ A p P j ! − "
12 log P Kj =1 P j + 1 ! P l ! + ≤ log K . Lemma 6: min { a , . . . , a k } − min { b , . . . , b k }≤ max { ( a − b ) , . . . , ( a k − b k ) } .The proof of Lemma 5 is given in Appendix E, and the proof of Lemma 6 is omitted since it isstraightforward. Using Lemmas 5 and 6, the gap in (4) directly follows. Fig. 5. Linear finite-field symmetric MAC.
IV. L
INEAR FINITE - FIELD SYMMETRIC NETWORKS WITH INTERFERENCE
Let us consider a particular class of discrete memoryless relay networks with interference. The linearfinite-field symmetric networks with interference are characterized by a special structure of MACs inthem, which is shown in Fig. 5. In more detail, the linear finite-field symmetric network with interferenceis described as follows: • Every input alphabet to a MAC at node v is the finite field, F q . • The received symbol at node v , Y ( t ) v , is determined to be the output of a symmetric discrete memorylesschannel (DMC) ( F q , p ( y v | x v ) , Y v ) with input X ( t ) v = X u ∈ ∆( v ) β u,v X ( t ) u,v ,where β u,v ∈ F q \ { } denotes the channel coefficient. For the definition of the symmetric DMC, see[28, Sec. 4.5]. • The input field size q and channel transition function p ( y v | x v ) associated with node v need not beidentical.A major characteristic of the symmetric DMC is that linear codes can achieve the capacity [28, Sec.6.2]. Using this, Nazer and Gastpar [11] showed that the computation capacity for any linear function ofsources can be achieved in the linear finite-field symmetric MAC in Fig. 5. Also, in [12], [13], it wasshown that linear codes achieve the multicast capacity of linear finite-field additive noise and erasurenetworks with interference, which are special cases of the class of networks stated above. Extending thisline, we characterize the multicast capacity of the linear finite-field symmetric network with interference. Theorem 4:
The multicast capacity of a linear finite-field symmetric network with interference is givenby min S ∈ Γ X v ∈ ¯ S c C v ,where C v is the capacity of the channel ( F q , p ( y v | x v ) , Y v ) .The proof of Theorem 4 is very similar to the proof of Theorem 1. The difference is that we use linearcodes instead of the nested lattice codes. We show the outline of the proof in the next subsections. Remark 4:
The capacity proof for linear finite-field additive noise networks in [12] can also be extendedto the linear finite-field symmetric networks in Theorem 4. However, the proof in [12] relies on algebraicnetwork coding, and, thus, it has a restriction on the field size, i.e., q > | D | . In our proof, we do not usethe algebraic network coding, and the field size is not restricted. A. Upper bound
As in the Gaussian case in Section III-A, the upper bound follows from the relaxed cut-set bound (6).In particular, for the linear finite-field symmetric network with interference, we have the Markov chainrelation ( X ¯ S, ¯ S c , X S c ,V ) → X ¯ S c → Y ¯ S c , where X ¯ S c = { X v : v ∈ ¯ S c } . Using the data processing inequality,we have I ( X ¯ S, ¯ S c ; Y ¯ S c | X S c ,V ) ≤ I ( X ¯ S c ; Y ¯ S c | X S c ,V ) ≤ I ( X ¯ S c ; Y ¯ S c ) .Thus the upper bound is given by R ≤ min S ∈ Γ max p ( x V,V ) I ( X ¯ S, ¯ S c ; Y ¯ S c | X S c ,V ) ≤ min S ∈ Γ max p ( x V,V ) I ( X ¯ S c ; Y ¯ S c )= min S ∈ Γ X v ∈ ¯ S c C v . B. Achievability
Let us denote the vectors of channel input and output of the symmetric DMC ( F q , p ( y v | x v ) , Y v ) as X v = h X (1) v , . . . , X ( n ) v i T and Y v = h Y (1) v , . . . , Y ( n ) v i T , respectively. Without loss of generality, we assumethat the encoder input is given by a uniform random vector W v ∈ F ⌊ nR ′ v ⌋ q for some R ′ v ≤ . Then wehave the following lemma related to linear coding for the DMC. Lemma 7 (Lemma 3 of [11]):
For the symmetric DMC ( F q , p ( y v | x v ) , Y v ) , a sequence of matrices F v ∈ F n ×⌊ nR ′ v ⌋ q and associated decoding function g v ( · ) exist such that when X v = F v W v , Pr { g ( Y v ) = W v } ≤ ǫ for any ǫ > and n large enough if R v , R ′ v log q < C v .We now consider linear encoding for nodes in the network. We let X u,v = β − u,v F v W u,v , and thus, X v = X u ∈ ∆( v ) β u,v X u,v = F v T v ,where T v , X u ∈ ∆( v ) W u,v . (23)By Lemma 7, a linear code with sufficiently large dimension exists such that node v can recover T v withan arbitrarily small error probability if R v < C v . Now, we can do the same as in Section III-D with (23)replacing (15), and the achievability part follows.V. C ONCLUSION
In this paper, we considered the multicast problem for relay networks with interference and examinedroles of some structured codes for the networks. Initially, we showed that nested lattice codes can achievethe multicast capacity of Gaussian relay networks with interference within a constant gap determined bythe network topology. We also showed that linear codes achieve the multicast capacity of linear finite-field symmetric networks with interference. Finally, we should note that this work is an intermediate steptoward more general networks. As an extension to multiple source networks, we showed that the samelattice coding scheme considered in this work can achieve the capacity of the Gaussian two-way relaychannel within bit [15], [17]. As another direction of extension, we can consider applying structuredcodes to networks with non-orthogonal broadcast channels. There is a recent work on the interferencechannel [26] which is related to this issue. A PPENDIX
A. Proof of Theorem 2
Consider a lattice (more precisely, a sequence of lattices) Λ n with σ (Λ n ) = P , which is simultaneouslyRogers-good and Poltyrev-good (simultaneously good shortly). In [20], it was shown that such a latticealways exists. Then, by the argument in [24], we can find a fine lattice Λ n such that Λ n ⊆ Λ n and Λ n isalso simultaneously good. We let the partitioning ratio be (cid:18) Vol(Λ n )Vol(Λ n ) (cid:19) n = (cid:18) P P − δ ′ (cid:19) (cid:18) πeG (Λ n ) (cid:19) (24)for some δ ′ > . Since the partitioning ratio can approach an arbitrary value as n tends to infinity, forany δ > , n ′ exists such that we can choose δ ′ ≤ δ when n ≥ n ′ . We now have σ (Λ n ) = G (Λ n ) · Vol(Λ n ) n = G (Λ n ) · πe ( P − δ ′ ) ,where the second equality follows from (24). Since Λ n is Rogers-good, n ′′ exists such that ≤ πeG (Λ n ) ≤ P P − δ ′ , for n ≥ n ′′ . Thus, for n ≥ max { n ′ , n ′′ } , we have P − δ ≤ σ (Λ n ) ≤ P .By repeating the same procedure, we obtain a lattice chain Λ n ⊆ Λ n ⊆ · · · ⊆ Λ nK , where Λ ni , ≤ i ≤ K ,are simultaneously good and P i − δ ≤ σ (Λ ni ) ≤ P i for sufficiently large n .Moreover, by Theorem 5 of [19], if Λ nK is simultaneously good, a Poltyrev-good lattice Λ nC exists suchthat Λ nK ⊆ Λ nC and the coding rate R K can be arbitrary as n → ∞ , i.e., R K = 1 n log (cid:18) Vol(Λ nK )Vol(Λ nC ) (cid:19) = γ + o n (1) .Given R K , the coding rates R i , ≤ i ≤ K − , are given by R i = 1 n log (cid:18) Vol(Λ ni )Vol(Λ nC ) (cid:19) = 1 n log (cid:18) Vol(Λ ni )Vol(Λ nK ) (cid:19) + R K = 12 log (cid:18) σ (Λ ni ) σ (Λ nK ) (cid:19) + R K + o n (1)= 12 log (cid:18) P i P K (cid:19) + R K + o n (1) ,where the third equality follows by the fact that Λ ni and Λ nK are both Rogers-good, and the fourth followsby the fact that σ (Λ ni ) = P i − o n (1) . (cid:3) B. Proof of Theorem 3
Let r cov i and r eff i denote the covering and effective radii of Λ i , respectively. Then the second momentper dimension of r cov i B is given by σ i , σ ( r cov i B ) = ( r cov i ) n + 2 .Next, we define independent Gaussian random variables Z i ∼ N ( , σ i I ) , i = 1 , . . . , K , and Z ∗ = (1 − α ) K X j =1 Z j + α Z .Then, we have the following lemmas. Lemma 8:
The variance of Z ∗ , each element of Z ∗ , is denoted by Var( Z ∗ ) and satisfies Var( Z ∗ ) = (1 − α ) K X j =1 σ j + α ≤ max j r cov j r eff j ! · P Kj =1 P j P Kj =1 P j + 1 . Lemma 9:
The pdf of ˜ Z , denoted by p ˜ Z ( x ) satisfies p ˜ Z ( x ) ≤ e n P Kj =1 ǫ j · p Z ∗ ( x ) ,where ǫ j = log r cov j r eff j ! + 12 log 2 πeG ( B ) + 1 n . The above two lemmas are slight modifications of Lemmas 6 and 11 in [19]. The proofs also followimmediately from [19].Now, we bound the error probability by p e = Pr n ˜ Z mod Λ / ∈ R C o ≤ Pr n ˜ Z / ∈ R C o ≤ e n P Kj =1 ǫ j · Pr { Z ∗ / ∈ R C } , (25)where (25) follows from Lemma 9. Note that Z ∗ is a vector of i.i.d. zero-mean Gaussian random variables,and the VNR of Λ C relative to Z ∗ is given by µ = (Vol(Λ C )) /n πe Var( Z ∗ ) ≥ (Vol(Λ )) /n / R πe · P Kj =1 P j P Kj =1 P j +1 − o n (1) (26) = 12 R · πeG (Λ ) · P P Kj =1 P j + P ! − o n (1) (27) = 12 R · P P Kj =1 P j + P ! − o n (1) , (28)where (26) follows from Lemma 8 and the fact that Λ i , ≤ i ≤ K , are Rogers-good, (27) from thedefinition of G (Λ ) , and (28) from the fact that Λ is Rogers-good and R = ¯ R + o n (1) . When weconsider the Poltyrev exponent, we are only interested in the case that µ > . Thus, from the definitionof R ∗ and (28), we can write µ = 2 R ∗ − ¯ R ) − o n (1) , for ¯ R < R ∗ . Finally, from (25) and by the fact that Λ C is Poltyrev-good, we have p e ≤ e n P Kj =1 ǫ j · e − nE P ( µ ) = e − n “ E P “ R ∗ − ¯ R ” − o n (1) ” . (cid:3) C. Proof of Lemma 3
For notational simplicity, we prove this lemma in the standard MAC in Section III-C. We assume thatthe uniform random mapping is done at each input node of the standard MAC, as was done in the network.Let A and A c be nonempty partitions of { , . . . , K } , i.e., A ∪ A c = { , . . . , K } , and A ∩ A c = ∅ . Weassume that A implies the set of nodes that can distinguish W and W ′ , and A c implies the set of nodesthat cannot. For node i ∈ A , W i ( W ) and W i ( W ′ ) are uniform over C i and independent of each otherdue to the uniform random mapping. However, for node i ∈ A c , we always have W i ( W ) = W i ( W ′ ) .Thus, if A = ∅ , T ( W ) = T ( W ′ ) always holds, i.e., Pr (cid:8) T ( W ) = T ( W ′ ) |D = A, ¯ D = A c (cid:9) = 1 .If A = ∅ , given D = A and ¯ D = A c , the event T ( W ) = T ( W ′ ) is equivalent to ˜ T ( W ) = ˜ T ( W ′ ) , where ˜ T ( W ) = "X j ∈ A ( W j ( W ) − Q j ( W j ( W ) + U j )) mod Λ ,and ˜ T ( W ′ ) is given accordingly. Now, let l , min A , then T ′ ( W ) , ˜ T ( W ) mod Λ l = W l ( W ) + X j ∈ A \{ l } ( W j ( W ) − Q j ( W j ( W ) + U j )) mod Λ l ,which follows from the fact that Λ ⊆ Λ l , and ,thus, ( x mod Λ ) mod Λ l = x mod Λ l . Note that, dueto the crypto-lemma and the uniform random mapping, T ′ ( W ) and T ′ ( W ′ ) are uniform over C l andindependent of each other. Therefore, Pr (cid:8) T ( W ) = T ( W ′ ) |D = A, ¯ D = A c (cid:9) = Pr n ˜ T ( W ) = ˜ T ( W ′ ) |D = A o ≤ Pr { T ′ ( W ) = T ′ ( W ′ ) |D = A } = 1 |C l | = 2 − nR l .Thus, by changing notations properly to those of the network, we complete the proof. (cid:3) D. Proof of Lemma 4
In the time-expanded network, there are two types of cuts, steady cuts and wiggling cuts [5]. The steadycut separates the nodes in different layers identically. That is, for a steady cut S TE , v [ k ] ∈ S TE for some k if and only if v [1] , . . . , v [ L + 1] ∈ S TE . Let us denote the set of all steady cuts as ˜Γ TE . Then, since ˜Γ TE ⊆ Γ TE , min S TE ∈ Γ TE L +1 X k =2 X v [ k ] ∈ ¯ Sc TE[ k ] max u [ k − ] ∈ ∆TE ,S ( v [ k ]) R u,v ≤ min S TE ∈ ˜Γ TE L +1 X k =2 X v [ k ] ∈ ¯ Sc TE[ k ] max u [ k − ] ∈ ∆TE ,S ( v [ k ]) R u,v = L · min S ∈ Γ X v ∈ ¯ S c (cid:18) max u ∈ ∆ S ( v ) R u,v (cid:19) . We now prove the lower bound. For any two cuts S and S in G , i.e., S , S ∈ Γ , define that ξ ( S , S ) = X v ∈ S c (cid:18) max u ∈ S R u,v (cid:19) ,where R u,v = 0 if ( u, v ) / ∈ E . Then, we have the following lemma Lemma 10:
Consider a sequence of non-identical cuts S , . . . , S L ′ ∈ Γ and define S L ′ +1 = S . For thesequence, we have L ′ X k =1 ξ ( S k , S k +1 ) ≥ L ′ X k =1 ξ ( S ′ k , S ′ k ) ,where for k = 1 , . . . , L ′ , S ′ k = ∪ { i ,...,ik }⊆{ ,...,L ′} ( S i ∩ · · · ∩ S i k ) .The proof of Lemma 10 is tedious but straightforward. Similar lemmas were presented and proved in[5, Lemma 6.4], [8, Lemma 2], and the proof of Lemma 10 also follows similarly.Now, since S ′ k ∈ Γ , it follows that ξ ( S ′ k , S ′ k ) ≥ min S ∈ Γ X v ∈ S c (cid:18) max u ∈ S R u,v (cid:19) = min S ∈ Γ X v ∈ ¯ S c max u ∈ ∆ S ( v ) R u,v ! . (29)Also, since S TE [ k ] ’s correspond to cuts in V , we can rewrite min S TE ∈ Γ TE L +1 X k =2 X v [ k ] ∈ ¯ Sc TE[ k ] max u [ k − ] ∈ ∆TE ,S ( v [ k ]) R u,v = min S TE ∈ Γ TE L +1 X k =2 ξ (cid:0) S TE [ k − ] , S TE [ k ] (cid:1) .Since there are | Γ | = 2 | V |− different cuts, at least the first L −| Γ | +2 of the sequence S TE [1] , . . . , S TE [ L +1] form loops, and, thus, by Lemma 10 and (29), we have min S TE ∈ Γ TE L +1 X k =2 ξ (cid:0) S TE [ k − ] , S TE [ k ] (cid:1) ≥ ( L − | Γ | + 2) · min S ∈ Γ X v ∈ ¯ S c max u ∈ ∆ S ( v ) R u,v ! . (cid:3) E. Proof of Lemma 5
We first consider the case that ∈ A , and the case that / ∈ A afterward.a) ∈ A In this case, l = 1 , and the gap is
12 log X j ∈ A p P j ! − "
12 log P Kj =1 P j + 1 ! P ! + ≤
12 log K X j =1 p P j ! −
12 log P Kj =1 P j + 1 ! P ! ≤
12 log (cid:0) K P (cid:1) −
12 log (cid:18) K + P (cid:19) ≤ log K . b) / ∈ A Since / ∈ A , | A | ≤ K − . Now, the gap is given by
12 log X j ∈ A p P j ! − "
12 log P Kj =1 P j + 1 ! P l ! + ≤
12 log (cid:0) K − P l (cid:1) − (cid:20)
12 log P l (cid:21) + ≤
12 log(1 + ( K − ) ≤ log K . (cid:3) R EFERENCES [1] T. M. Cover and A. A. El Gamal, “Capacity theorems for the relay channels,”
IEEE Trans. Inform. Theory , vol. 51, no. 5, pp. 572–584,Sep. 1979.[2] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network information flow,”
IEEE Trans. Inform. Theory , vol. 46, no. 4, pp.1204–1216, Oct. 2000.[3] M. R. Aref, “Information flow in relay networks,” Ph.D. dissertation Stanford Univ., Stanford, CA, 1980.[4] N. Ratnakar and G. Kramer, “The multicast capacity of deterministic relay networks with no interference,”
IEEE Trans. Inform. Theory ,vol. 52, no. 6, pp. 2425–2432, June 2006.[5] A. S. Avestimehr, S. N. Diggavi, and D. N. C. Tse, “Wireless network information flow,” in
Proc. 45th Annual Allerton Conference ,Sept. 2007.[6] ——, “Approximate capacity of Gaussian relay networks,” in
Proc. IEEE International Symp. Inform. Theory , Toronto, Canada, July2008.[7] A. Dana, R. Gowaikar, R. Palanki, B. Hassibi, and M. Effros, “Capacity of wireless erasure networks,” in
IEEE Trans. Inform. Theory ,vol. 52, no. 3, pp. 789–804, Mar. 2006.[8] B. Smith and S. Vishwanath, “Unicast transmission over multiple access erasure networks: Capacity and duality,” in
IEEE InformationTheory Workshop , Tahoe city, California, Sept. 2007.[9] B. Nazer and M. Gastpar, “Computing over multiple-access channels with connections to wireless network coding,” in
Proc. IEEEInternational Symp. Inform. Theory , Seattle, USA, July 2006.[10] ——, “Lattice coding increases multicast rates for Gaussian multiple-access networks,” in
Proc. 45th Annual Allerton Conference ,Sept. 2007.[11] ——, “Computation over multiple-access channels,”
IEEE Trans. Inform. Theory , vol. 53, no. 10, pp. 3498–3516, Oct. 2007.[12] ——, “The case for structured random codes in network capacity theorems,”
European Trans. Telecomm.: Special Issue on NewDirections in Inform. Theory , no. 4, vol. 19, pp. 455–474, June 2008.[13] W. Nam and S.-Y. Chung, “Relay networks with orthogonal components,” in
Proc. 46th Annual Allerton Conference , Sept. 2008.[14] A. El Gamal and S. Zahedi, “Capacity of a class of relay channels with orthogonal components,”
IEEE Trans. Inform. Theory , vol.51, no. 5, pp. 1815–1817, May 2005.[15] W. Nam, S.-Y. Chung, and Y. H. Lee, “Capacity bounds for two-way relay channels,”
Proc. Int. Zurich Seminar on Comm. , Mar. 2008.[16] K. Narayanan, M. P. Wilson, and A. Sprintson, “Joint physical layer coding and network coding for bi-directional relaying,” in
Proc.45th Annual Allerton Conference , Sept. 2007.[17] W. Nam, S.-Y. Chung, and Y. H. Lee, “Capacity of the Gaussian Two-way Relay Channel to within Bit,” submitted to
IEEE Trans.Inform. Theory , available at http://arxiv.org/PS cache/arxiv/pdf/0902/0902.2438v1.pdf.[18] R. Zamir, S. Shamai, and U. Erez, “Nested linear/lattice codes for structured multiterminal binning,”
IEEE Trans. Inform. Theory , vol.48, no. 6, pp. 1250–1276, June 2002.[19] U. Erez and R. Zamir, “Achieving log(1 + SNR ) on the AWGN channel with lattice encoding and decoding,” IEEE Trans. Inform.Theory , vol. 50, no. 10, pp. 2293–2314, Oct. 2004.[20] U. Erez, S. Litsyn, and R. Zamir, “Lattices which are good for (almost) everything,”
IEEE Trans. Inform. Theory , vol. 51, no. 10, pp.3401–3416, Oct. 2005.[21] H. A. Loeliger, “Averaging bounds for lattices and linear codes,”
IEEE Trans. Inform. Theory , vol. 43, no. 6, pp. 1767–1773, Nov.1997.[22] G. Poltyrev, “On coding without restrictions for the AWGN channel,”
IEEE Trans. Inform. Theory , vol. 40, no. 2, pp. 409–417, Mar.1994.[23] G. D. Forney Jr., “On the role of MMSE estimation in approaching the information theoretic limits of linear Gaussian channels:Shannon meets Wiener,” in
Proc. 41st Annual Allerton Conference [25] T. Philosof, A. Khisti, U. Erez, and R. Zamir, “Lattice strategies for the dirty multiple access channel,” in Proc. IEEE InternationalSymp. Inform. Theory , Nice, France, June-July 2007.[26] S. Sridharan, A. Jafarian, S. Vishwanath, S. A. Jafar, and S. Shamai, “A layered lattice coding scheme for a class of three user Gaussianinterference channels,” available at http://arxiv.org/PS cache/arxiv/pdf/0809/0809.4316v1.pdf.[27] T. Cover and J. Thomas,
Elements of Information Theory , Wiley, New York, 1991.[28] R. Gallager,