Sum capacity of multi-source linear finite-field relay networks with fading

Sum Capacity of Multi-source Linear Finite-field Relay Networks W ith F ading Sang-W oon Jeon and Sa e-Y oung Chung School of EECS, K AIST , Daejeo n, Korea Email: swjeon @kaist.ac.kr, sychung @ee.kaist.ac.kr Abstract — W e study a fading linear finite-field re lay network hav ing multiple source-destination pairs. Because of the inter - ference created by diff erent unicast sessions, th e problem of finding its capacity region is in general difficul t. W e observe that, since channels are time-varying, relays can deliver their recei ved si gnals by waiting f or a ppropriate ch annel r ealizations such that the destinations can decode their messages without interference. W e propose a block M arko v encoding and relaying scheme that exploits such channel va riations. By deriving a general cut-set upper bound and an achievable rate re gion, we characterize the sum capacity for some classes of channel distributions and n etwork topologies. For example, when th e channels are uniformly distributed, the sum capacity is giv en by the minimum ave rage ra nk of th e channel matrices constructed by all cuts that separate the entire sources and destinations. W e also describe other ca ses wh ere the capacity is characterized. I . I N T RO D U C T I O N Characterizing the cap acity region of wireless re lay n et- works is o ne of the f undame ntal problem s. Howe ver , if th e network has multiple unicast sessions, the p roblem of findin g its capacity region becomes much mor e ch allenging since the transmission of other sess ions acts as interference and, in general, the cut- set upp er bou nd is not tight. Even fo r th e tw o- user Gaussian interferen ce channel, an approxim ate capacity region was rece ntly cha racterized [1]. For wireless networks, there exist three fundam ental issues, i.e., bro adcast, interferen ce, and fading. In this paper, we consider a mu lti-source fading linear finite-field relay network , which captur es these three key char acteristics of wireless en viron ment. There have been related works dealing with wireless networks assum ing interf erence-fr ee reception s [2], [3] and assuming no br oadcast natur e [4]. The works in [5], [6] ha ve considered deter ministic relay networks a nd th e w ork in [7] has stud ied finite-field erasure network s. Since the channels are time-varying, destinatio ns can de- code their messages without interference by transmitting th em throug h a series o f par ticular channel instances during mul- tihop transmission. As an example, consider th e binary- field two-hop network in Fig. 1. T he symbol in each node den otes the tra nsmit bit of tha t nod e, where s k denotes the informatio n bit of the k -th sour ce. W e no tice the interference-free reception is possible if H 1 H 2 = I , where H 1 and H 2 denote the channe l instances of the first and seco nd hop, respectively . The works in [8], [9], [10] h av e also shown that, by u sing particular channel instances jo intly , one can improve achiev able rates of single-ho p n etworks. Based on this key ob servation, we 1 s 3 s 2 s 1 3 s s  3 s 2 3 s s  Fig. 1. Inte rference mitigatio n for two-hop ne tworks. derive an achiev able rate r egion for general linear finite-field relay networks. By compar ing the achievable r ate region with a cut-set upper bound, we ch aracterize the sum capacity f or some classes o f ch annel d istributions and network top ologies. I I . S Y S T E M M O D E L A. Linear F inite-field Relay Networks W e study a multi-sourc e layered network in which the network consists of M + 1 layers having K m nodes at the m - th layer , w here m ∈ { 1 , · · · , M + 1 } . Let the ( k , m ) -th n ode denote th e k -th node at the m -th layer . The ( k , 1) -th no de and th e ( k , M + 1) -th nod e are the source and the de stination of the k -th sour ce-destination ( S-D) pair , respectively . Thus K = K 1 = K M +1 is the number o f S- D pairs. W e d efine K max = max m { K m } and K min = min m { K m } . Consider the m -th h op transmission. The ( i, m ) -th node and the ( j, m + 1 ) -th node b ecome the i -th transmitter (Tx) a nd the j -th re cei ver (Rx) o f the m -th hop, respec ti vely , wher e i ∈ { 1 , · · · , K m } and j ∈ { 1 , · · · , K m +1 } . Let x i,m [ t ] ∈ F 2 denote the transmit signal of the ( i, m ) -th node at time t and let y j,m [ t ] ∈ F 2 denote the received signal of the ( j, m + 1) -th node at time t 1 . Let h j,i,m [ t ] ∈ F 2 be the channel from the ( i, m ) -th node to the ( j, m + 1) -th n ode at time t . Then the relation b etween the tr ansmit and received signals is g i ven by y j,m [ t ] = K m X i =1 h j,i,m [ t ] x i,m [ t ] , (1) where all operations are perform ed over F 2 . W e assume time- varying channels such that Pr( h j,i,m [ t ] = 1) = p j,i,m and h j,i,m [ t ] are independent from each o ther with d ifferent i , j , m , and t . Let x m [ t ] and y m [ t ] be th e K m × 1 tra nsmit signal vector and K m +1 × 1 recei ved signal vector at the m - th h op, re spectiv ely , where x m [ t ] = [ x 1 ,m [ t ] , · · · , x K m ,m [ t ]] T , 1 W e focus on the binary fie ld F 2 in this paper , b ut some results can be direct ly extended to F q (see Remark 1). y m [ t ] =  y 1 ,m [ t ] , · · · , y K m +1 ,m [ t ]  T . Th us the transmission at the m -th hop can be re presented as y m [ t ] = H m [ t ] x m [ t ] , (2) where H m [ t ] is the K m +1 × K m channel ma trix of th e m -th hop having h j,i,m [ t ] as th e ( j, i ) -th element. W e assume that at time t b oth Txs and Rxs of the m -th ho p know H 1 [ t ] thr ough H m [ t ] . B. Pr o blem Sta tement Consider a set of length n block codes. Let W k be the message of the k -th sourc e un iformly distributed o ver { 1 , 2 , · · · , 2 nR k } , where R k is the ra te of the k -th sou rce. For simplicity , we assume that nR k is an in teger . A  2 nR 1 , · · · , 2 nR K ; n  code co nsists of th e fo llowing encoding, relaying, a nd de coding functio ns. • (E ncoding ) For k ∈ { 1 , · · · , K } , the set of encoding function s o f the k -th sou rce is given b y { f k, 1 ,t } n t =1 : { 1 , · · · , 2 nR k } → F n 2 such that x k, 1 [ t ] = f k, 1 ,t ( W k ) , where t ∈ { 1 , · · · , n } . • (Relay ing) For m ∈ { 2 , · · · , M } and k ∈ { 1 , · · · , K m } , the set of relaying f unctions of the ( k, m ) -th node is gi ven by { f k,m,t } n t =1 : F n 2 → F n 2 such that x k,m [ t ] = f k,m,t ( y k,m − 1 [1] , · · · , y k,m − 1 [ t − 1]) , where t ∈ { 1 , · · · , n } . • (D ecoding) For k ∈ { 1 , · · · , K } , the deco ding function of the k -th d estination is given by g k : F n 2 → { 1 , · · · , 2 nR k } such that ˆ W k = g k ( y k,M [1] , · · · , y k,M [ n ]) . If M = 1 , th e sources tran smit directly to th e intended destinations without relays. The pro bability of error at the k - th destinatio n is given by P ( n ) e,k = Pr( ˆ W k 6 = W k ) . A set of rates ( R 1 , · · · , R K ) is said to be achiev able if there exists a sequence o f (2 nR 1 , · · · , 2 nR K ; n ) code s with P ( n ) e,k → 0 a s n → ∞ fo r all k ∈ { 1 , · · · , K } . The achiev able sum -rate is giv en by R sum = P K k =1 R k and the sum cap acity is the supremum of the ach ie vable sum-rates. C. Notations 1) Notations fo r dir ected graphs: The considered network can be rep resented a s a directed graph G = ( V , E ) consisting of a vertex set V a nd a directed edge set E . Let v k,m denote the ( k , m ) -th node and V m = { v k,m } K m k =1 denote the set of no des in the m -th layer . Then V is given by ∪ m ∈{ 1 , ··· ,M +1 } V m . There exists a dir ected edge ( v i,m , v j,m +1 ) fr om v i,m to v j,m +1 if p j,i,m > 0 . Let H V ′ , V ′′ be the channel matrix fr om the nodes in V ′ ⊆ V to the nod es in V ′′ ⊆ V . 2) Sets of chann el instance s a nd nodes: Suppose ¯ V ′ ⊆ V ′ , ¯ V ′′ ⊆ V ′′ , and G is a | ¯ V ′′ | × | ¯ V ′ | matrix. W e defin e th e following set: H F V ′ , V ′′  G , ¯ V ′ , ¯ V ′′  = { H V ′ , V ′′   H ¯ V ′ , ¯ V ′′ = G , rank( H V ′ , V ′′ ) = rank( G ) , H V ′ , V ′′ ∈ F |V ′′ |×|V ′ | 2 } , (3) i.e., H F V ′ , V ′′  G , ¯ V ′ , ¯ V ′′  is the set of all instances of H V ′ , V ′′ that contain G in H ¯ V ′ , ¯ V ′′ and ha ve th e same rank as G . W e 1 s 1 s 2 s 2 s 1 s 1 s 2 s 2 s 1 s 1 s 1 s 1 s 2 s 2 s 2 s 2 s 1 s 1 s 1 s 1 s 2 s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 1 2 s s  1 2 s s  2 s 1 2 s s  1 s 1 2 s s  2 s Fig. 2. Deterministic pairing of H 1 and H 2 . further d efine the following sets: V ( a, b, V ′ , V ′′ ) =  ( ¯ V ′ , ¯ V ′′ )   | ¯ V ′ | = a, | ¯ V ′′ | = b, ¯ V ′ ⊆ V ′ , ¯ V ′′ ⊆ V ′′  , V ( H V ′ , V ′′ ) =  ( ¯ V ′ , ¯ V ′′ )   | ¯ V ′ | = | ¯ V ′′ | = rank( H V ′ , V ′′ ) , rank( H ¯ V ′ , ¯ V ′′ ) = rank( H V ′ , V ′′ ) , ¯ V ′ ⊆ V ′ , ¯ V ′′ ⊆ V ′′  , (4) where a a nd b are po siti ve integers satisfying a ≤ |V ′ | and b ≤ |V ′′ | . W e define V ( H V ′ , V ′′ ) = φ if ra nk( H V ′ , V ′′ ) = 0 . The set V ( a, b, V ′ , V ′′ ) consists of all ( ¯ V ′ , ¯ V ′′ ) such th at the numb ers of nodes in ¯ V ′ and in ¯ V ′′ are equal to a and b , respectively . The set V ( H V ′ , V ′′ ) consists of all ( ¯ V ′ , ¯ V ′′ ) that H ¯ V ′ , ¯ V ′′ is a full ran k matrix and has th e same rank as H V ′ , V ′′ . I I I . C U T - S E T U P P E R B O U N D In this section, we introduce a sum-rate up per bo und, which is der iv ed from th e g eneral cut-set upper bo und in [10]. Theor em 1: Sup pose a linear finite-field relay n etwork. The achiev able sum-rate is uppe r bound ed by R sum ≤ min m ∈{ 1 , ··· ,M } E (rank( H m )) . (5) Pr oof: W e refer reade rs to [10]. Let us defin e m 0 = arg min m E (rank( H m )) , wh ich is the bottleneck -hop for the entire multiho p transmission 2 . I V . T R A N S M I S S I O N S C H E M E In this section , w e pr opose a transmission scheme for lin ear finite-field relay networks when M ≥ 2 . W e refer to the results in [10] for th e single-h op case. As men tioned befor e, due to the time-varying nature of wireless channels, informa tion b its c an be transmitted throu gh particular instan ces f rom H 1 to H M such th at th e cor respond- ing destination s receiv e inform ation bits without interfer ence. That is, inform ation bits are transmitted using time indices t 1 , · · · , t M such th at H 1 [ t 1 ] = H 1 , · · · , H M [ t M ] = H M . A block Markov encoding and relayin g stru cture makes a series of pa iring from H 1 to H M possible. W e first study 2 - 2 - 2 networks and then extend th e idea to general linear finite-field relay n etworks. A. 2 - 2 - 2 Networks Consider 2 - 2 - 2 networks with p j,i,m = 1 / 2 for all i , j , and m . There are 1 6 possible instances fo r each H 1 [ t ] and H 2 [ t ] . 2 Tie s are broken arbitr arily . For each time t , if in formatio n bits ar e tr ansmitted throug h H 1 [ t ] and H 2 [ t + 1] , there exist 256 possible in stances from H 1 [ t ] to H 2 [ t + 1] a nd R sum = E (rank( H 1 H 2 )) = 177 256 is achiev able in this case. Howe ver , we can get an achievable sum-rate h igher than 177 256 by appro priately pairing H 1 and H 2 . Fig. 2 illustrates the deterministic pairing o f H 1 and H 2 and related en coding an d relaying, where the dashed lines and the solid lines de note the correspo nding ch annels are z eros and ones, respecti vely . The symbols in the figure d enote the transmit bits o f the nod es and the nod es with no symbo l transmit zeros, wh ere s k denotes the informa tion bit o f the k -th source . T his deterministic pairing achieves R sum = E (rank ( H 1 )) = 21 16 , which coincides with the up per bound in (5). Th us, th is simple schem e achieves the sum cap acity . Based on th e determin istic pairing in Fig. 2, we characterize the sum capacity for more ge neral ch annel distributions. Theor em 2: Sup pose a linear finite-field r elay n etwork with M = 2 and K 1 = K 2 = K 3 = 2 . Then the sum capacity is characterized for the f ollowing cases. • Sym metric channel satisfying p 1 , 1 , 1 = p 2 , 2 , 1 = p 1 , 1 , 2 = p 2 , 2 , 2 and p 2 , 1 , 1 = p 1 , 2 , 1 = p 2 , 1 , 2 = p 1 , 2 , 2 . • Z chann el satisfying p 2 , 1 , 1 = p 2 , 1 , 2 = 0 with p 1 , 1 , 1 = p 2 , 2 , 2 , p 2 , 2 , 1 = p 1 , 1 , 2 and p 1 , 2 , 1 = p 1 , 2 , 2 . • Π channel satisfying p 1 , 1 , 1 = p 2 , 2 , 2 = 0 with p 2 , 1 , 1 = p 1 , 2 , 2 , p 1 , 2 , 1 = p 2 , 1 , 2 and p 2 , 2 , 1 = p 1 , 1 , 2 . Pr oof: W e refer r eaders to the full paper [11]. B. General Multihop Networks In this subsection , we p ropose a transmission sch eme for general lin ear finite-field r elay networks whe n M ≥ 2 and p j,i,m = p for all i , j , and m . W e a ssume symme tric rates for all S-D pairs, that is R 1 = · · · = R K , and consider the following class of networks. Definition 1 : A linear finite-field relay network is said to have a minimum- dimensional bo ttleneck-ho p m 0 if K m ≥ K m 0 and K m +1 ≥ K m 0 +1 (or K m ≥ K m 0 +1 and K m +1 ≥ K m 0 ) for all m ∈ { 1 , · · · , M } . Notice that any n etworks having K m = K f or all m or any 2 -h op networks are includ ed in this class of networks regardless of th e value of p . If a series of pairin g from H 1 to H M satisfies th e co ndition H 1 H 2 · · · H M = I , each destinatio n can receive info rmation bits witho ut interferenc e. But if som e instances are rank- deficient, we can not construct such p airs by u sing rank- deficient instances. Furthermor e, th e n umber of possible p airs increases exponen tially as the nu mber of nodes in a layer or the number o f lay ers incr eases. In stead, w e ran domize a series of pairing such that H m is paired at rand om with one in stance in a subset of H m +1 ’ s. 1) Block Mark ov encod ing a nd r elaying: The proposed scheme d i vides a block in to B + M − 1 su b-block s ha ving length n B for e ach sub- block, where n B = n B + M − 1 . Sin ce block Markov encoding and relaying ar e applied over M h ops, the numb er of e ffecti ve sub-b locks is equal to B . Thus, the overall rate is given by B B + M − 1 R , w here R is the symm etric rate of each sub-blo ck. As n → ∞ , th e fractio nal rate loss 1 − B B + M − 1 will be negligible bec ause we can make both n B and B arbitrarily large. F or simplicity , we omit the block index in describ ing the proposed scheme. 2) Ba lancing th e average rank o f each h op: R ecall that the m 0 -th hop b ecomes a bottleneck f or th e e ntire multihop transmission, which can be seen from the sum-rate upper bound in (5). As an example, consider 3 - 2 - 2 - 3 networks in which the second h op becomes a b ottleneck. If each sour ce in V 1 transmits at a rate of 1 K E (rank( H 1 )) , then it will cause an error at the second hop. T o p revent this error event, the rate of each source should b e decr eased to 1 K E (rank( H m 0 )) . For this reason , we select V m, tx [ t ] ⊆ V m and V m, rx [ t ] ⊆ V m +1 random ly su ch that ( V m, tx [ t ] , V m, rx [ t ]) ∈ V ( K m 0 , K m 0 +1 , V m , V m +1 ) (6) with equ al pro babilities (or in V ( K m 0 +1 , K m 0 , V m , V m +1 ) ). For each time t , only the nodes in V m, tx [ t ] and V m, rx [ t ] will become activ e at the m -th ho p. Notice that si nce the considered network has a minim um-dimen sional b ottleneck- hop, it is possible to construct such V m, tx [ t ] and V m, rx [ t ] . In the case of 3 - 2 - 2 - 3 networks, o nly the nod es in V 1 , tx [ t ] a nd V 1 , rx [ t ] satisfying ( |V 1 , tx [ t ] | , |V 1 , rx [ t ] | ) = (2 , 2 ) be come active at the first ho p. The same is true for the last ho p. Wh ereas the whole nodes in V 2 and V 3 become acti ve at the second ho p, that is V 2 , tx [ t ] = V 2 and V 2 , rx [ t ] = V 3 . The following lemma sho ws the probability distribution of H V m, tx [ t ] , V m, rx [ t ] [ t ] , which will be used to deriv e the achie v- able rate region of general multihop n etworks. Lemma 1: Sup pose a linear finite- filed relay with p j,i,m = p for all i , j , and m . If the network has a minim um- dimensiona l bottleneck-h op, th en Pr( H V m, tx [ t ] , V m, rx [ t ] [ t ] = H ) = p u (1 − p ) K m 0 +1 K m 0 − u , (7) where u is the number o f zero s in H ∈ F K m 0 +1 × K m 0 2 or in H ∈ F K m 0 × K m 0 +1 2 . Pr oof: W e refer reade rs to the full paper [11]. The probability distribution o f H V m, tx [ t ] , V m, rx [ t ] [ t ] is the same as th at of H m 0 [ t ] , which is the chan nel matrix of the bottleneck -hop. T hus if only the nod es in V m, tx [ t ] and V m, rx [ t ] are activ ated at the m -th hop, each ho p can deliv er inform ation bits th at are sustainable at the bottlene ck-hop . 3) Con struction o f transmit a nd receive node sets: Becau se the maxim um number of bits tra nsmitted at the m - th hop is determined by r ank( H V m, tx [ t ] , V m, rx [ t ] [ t ]) , we further select ¯ V m, tx [ t ] ⊆ V m, tx [ t ] an d ¯ V m, rx [ t ] ⊆ V m, rx [ t ] r andomly su ch that ( ¯ V m, tx [ t ] , ¯ V m, rx [ t ]) ∈ V ( H V m, tx [ t ] , V m, rx [ t ] [ t ]) (8) with e qual p robab ilities. For each time t , the nod es in ¯ V m, tx [ t ] transmit and the n odes in ¯ V m, rx [ t ] receiv e th rough th e channel H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] at the m -th hop. Th en, as we will show later , inform ation b its can b e transmitted usin g particular time indices t 1 , · · · , t M such that H ¯ V 1 , tx [ t 1 ] , ¯ V 1 , rx [ t 1 ] [ t 1 ] · · · H ¯ V M, tx [ t M ] , ¯ V M, rx [ t M ] [ t M ] = I , (9 ) which g uarantees interferenc e-free rece ption at the destina- tions. One of the simp lest way is to set H ¯ V 1 , tx [ t ] , ¯ V 1 , rx [ t ] [ t ] = · · · = H ¯ V M − 1 , tx [ t ] , ¯ V M − 1 , rx [ t ] [ t ] = G and H ¯ V M, tx [ t ] , ¯ V M, rx [ t ] [ t ] = ( G M − 1 ) − 1 . It is p ossible to constru ct those pa irs be cause the r esulting H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] is always in vertible 3 . Becau se rank( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ]) = rank( H V m, tx [ t ] , V m, rx [ t ] [ t ]) , there is no rate loss by using ( ¯ V m, tx [ t ] , ¯ V m, rx [ t ]) instead of using ( V m, tx [ t ] , V m, rx [ t ]) . The following lemma shows the probability distribution of H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] , which will b e used to derive the achiev- able rate region of general multihop n etworks. Lemma 2: Sup pose a linear finite-field relay network with p j,i,m = p for all i , j , an d m . If the network has a m inimum- dimensiona l bottlene ck-hop , th en for rank( G ) = r 6 = 0 , we obtain Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G ) = X ( V ′ , V ′′ ) ∈ V ( r,r, V m 0 , V m 0 +1 ) X H ∈H F V m 0 , V m 0 +1 ( G , V ′ , V ′′ ) Pr( H ) |V ( H ) | , (10) where P r( H ) is giv en by (7). If p = 1 / 2 , w e o btain Pr( G ) = 2 − K m 0 +1 K m 0 N K m 0 +1 ,K m 0 ( r ) N r,r ( r ) , (11) where N a,b ( i ) is th e number of instances in F a × b 2 having rank i . Pr oof: W e refer r eaders to the full paper [11]. 4) Encodin g, r elaying, and deco ding function s: Define T m ( G , V ′ m , V ′ m +1 ) as the set of time indice s o f the sub-block at the m -th hop satisfying ¯ V tx ,m [ t ] = V ′ m , ¯ V rx ,m [ t ] = V ′ m +1 , and H V ′ m , V ′ m +1 [ t ] = G , where m ∈ { 1 , · · · , M } . W e further define n ( G ) = n B R min  Pr( G ) , Pr  ( G M − 1 ) − 1  c − 1 , (12) where c = 1 K K min X i =1 i X G ∈ F i × i 2 , rank( G )= i min { Pr( G ) , Pr(( G M − 1 ) − 1 ) } . (1 3) Each sou rce will transmit 1 K P G rank( G ) n ( G ) b its dur ing n B channel uses. Fro m (12) and (13), we can check that R is equal to 1 K n B P G rank( G ) n ( G ) . For all full-rank m atrices G ∈ ∪ K min i =1 F i × i 2 , the detailed encodin g and relaying are as follows, where r = ra nk( G ) . • (E ncoding ) For all ( V ′ 1 , V ′ 2 ) ∈ V ( r, r , V 1 , V 2 ) , declare an error if |T 1 ( G , V ′ 1 , V ′ 2 ) | < n ( G ) /  K 1 r  K 2 r  , o therwise ea ch source in V ′ 1 transmits n ( G ) /  K 1 r  K 2 r  informa tion bits using the time indices in T 1 ( G , V ′ 1 , V ′ 2 ) to the n odes in V ′ 2 . • (Relay ing for m ∈ { 2 , · · · , M − 1 } ) For all ( V ′ m , V ′ m +1 ) ∈ V ( r , r, V m , V m +1 ) , dec lare an error if |T m ( G , V ′ m , V ′ m +1 ) | < n ( G ) /  K m r  K m +1 r  , 3 W e ignore the instances havi ng all zeros, which gi ve zero rate. otherwise ea ch n ode in V ′ m relays n ( G ) /  K m r  K m +1 r  bits using the time ind ices in T m ( G , V ′ m , V ′ m +1 ) to the nodes in V ′ m +1 . • (Relay ing for m = M ) For all ( V ′ M , V ′ M +1 ) ∈ V ( r, r , V M , V M +1 ) , de- clare an error if |T M (( G M − 1 ) − 1 , V ′ M , V ′ M +1 ) | < n ( G ) /  K M r  K M +1 r  , otherwise each nod e in V ′ M re- lays n ( G ) /  K M r  K M +1 r  bits using the time ind ices in T M (( G M − 1 ) − 1 , V ′ M , V ′ M +1 ) to the destinations in V ′ M +1 . For simplicity , we assum e that n ( G ) /  K m r  K m +1 r  is an integer . Notice that the deco ding er ror does not occur if there is n o encodin g and rela ying erro r since eac h destinatio n can receive 2 n B R informa tion bits w ithout interfere nce for this case. 5) Rela ying of the received b its: Let us n ow con sider h ow each relay distributes its recei ved bits to the nodes in the next lay er . For given G and V ′ m , each nod e in V ′ m receives n ( G ) /  K m r  bits and then transmits n ( G ) /  K m r  K m +1 r  received bits to the no des in V ′ m +1 , wh ere r = rank( G ) . Since there exist  K m +1 r  possible V ′ m +1 ’ s, the total nu mber of received bits is the same as the to tal num ber of tran smit bits at each n ode in V ′ m . For m ∈ { 2 , · · · , M − 1 } , we distrib ute the received bits that arrive from different paths evenly to form n ( G ) /  K m r  K m +1 r  bits. T hen, among n ( G ) /  K m r  received b its, n ( G ) /  K 1 r  K m r  received bits originate f rom the sources in V ′ 1 . Therefo re, at the last hop, the no des in V ′ M +1 , which are the corresp onding destinations of the sources in V ′ 1 , can collect all received bits that origin ate from the sources in V ′ 1 . This is becau se, for each n ode in V ′ M , th e number of the rec ei ved bits or iginated from V ′ 1 is the same as the numb er of bits able to transmit to V ′ M +1 , wh ich are given b y n ( G ) /  K 1 r  K M r  and n ( G ) /  K M r  K M +1 r  respectively , wh ere we use the fact that K = K 1 = K M +1 . V . A C H I E V A B L E R AT E R E G I O N In this section, we d eriv e an achiev able rate region by applying the proposed blo ck Markov enco ding and relaying. Let E m denote the encod ing e rror at the m -th hop. Then E m occurs if |T m ( G , V ′ m , V ′ m +1 ) | < n ( G )  K m rank( G )  K m +1 rank( G )  (14) for any V ′ m , V ′ m +1 , a nd G , wh ere m ∈ { 1 , · · · , M − 1 } . Similarly , E M occurs if |T M ( G , V ′ M , V ′ M +1 ) | < n  ( G M − 1 ) − 1   K M rank( G )  K M +1 rank( G )  (15) for any V ′ M , V ′ M +1 , and G . Since dec oding error does n ot occur if th ere is no en coding and relayin g error, from the union b ound, we obtain P ( n B ) e,k ≤ P M m =1 Pr( E m ) . Theor em 3: Sup pose a linear finite-field r elay n etwork with M ≥ 2 and p j,i,m = p for all i , j , and m . If the network has a m inimum-d imensional bo ttleneck-ho p, then for any δ > 0 , R k = 1 K K min X i =1 i X G ∈ F i × i 2 , rank( G )= i min { Pr( G ) , Pr(( G M − 1 ) − 1 ) } − δ (1 6) is ach iev able for all k ∈ { 1 , · · · , K } , where Pr( G ) is given by ( 10). Pr oof: Le t us first consider |T m ( G , V ′ m , V ′ m +1 ) | , where r = rank( G ) . By the w eak law of large numbers [12], th ere exists a sequence ǫ n B → 0 as n B → ∞ such that th e probab ility |T m ( G , V ′ m , V ′ m +1 ) | ≥ n B (Pr( G , V ′ m , V ′ m +1 ) + δ n B ) (17) for all G , V ′ m , and V ′ m +1 is gr eater than or equa l to 1 − ǫ n B , where δ n B → 0 as n B → ∞ . T his indicates that Pr( E m ) ≤ ǫ n B if n ( G ) ≤ n B  Pr( G ) −  K m r  K m +1 r  δ n B  (18) for all G , where m ∈ { 1 , · · · , M − 1 } . Note th at we u se the fact that Pr( G , V ′ m , V ′ m +1 ) = Pr( G ) /  K m r  K m +1 r  . Similarly , P r ( E M ) ≤ ǫ n B if n ( G ) ≤ n B  Pr  ( G M − 1 ) − 1  −  K M r  K M +1 r  δ n B  (19) for all G . Then, P ( n B ) e,k ≤ M ǫ n if (18) and ( 19) hold fo r all G . T his co ndition is satisfied if R ≤ c − c ( K max !) 2 δ n B min { Pr( G ) , Pr(( G M − 1 ) − 1 ) } (20) for all G , wh ere we u se th e definition of n ( G ) in (1 2) an d the fact that  K m r  ≤ K max ! . Thus we set R = c − δ ∗ n B , where δ ∗ n B = c ( K max !) 2 δ n B min G { Pr( G ) } , which tends to zer o as n B → ∞ . In conclusion , we obtain R = 1 K K min X i =1 i X G ∈ F i × i 2 , rank( G )= i min { Pr( G ) , Pr(( G M − 1 ) − 1 ) } − δ ∗ n B (21) is achie vable. Since R is the symmetr ic rate and δ ∗ n B → 0 as n B → ∞ , this p roves the assertion. Now let us consider the capacity ach ieving case. If Pr( G ) = Pr(( G M − 1 ) − 1 ) for all p ossible G , then the achiev able sum- rate in Theor em 3 will co incide with the upper bo und in (5). When the ch annel instan ces are u niformly d istributed, the above condition ho lds and , as a result, the sum capacity is character ized. The following corollary sho ws that th e sum capacity is given by the average ra nk of the channel matrix of the bottleneck-hop when p = 1 / 2 . Cor ollary 1: Supp ose a line ar fin ite-field relay network with M ≥ 2 and p j,i,m = 1 / 2 for all i , j , and m . If the network ha s a minimum -dimension al bottlen eck-ho p, th e sum capacity is giv en by C sum = 2 − K m 0 +1 K m 0 X H ∈ F K m 0 +1 × K m 0 2 rank( H ) . (22) Pr oof: Con sider the case p j,i,m = 1 / 2 . Then, f rom (11), Pr( G ) is a fun ction of rank( G ) . Since G and ( G M − 1 ) − 1 are full-rank matrices, Pr( G ) = Pr(( G M − 1 ) − 1 ) for all p ossible G . He nce (2 1) is gi ven by R = 1 K K min X i =1 i X G ∈ F i × i 2 , rank( G )= i Pr( G ) − δ ∗ n B = 1 K 2 − K m 0 +1 K m 0 K min X i =1 iN K m 0 +1 ,K m 0 ( i ) − δ ∗ n B = 1 K 2 − K m 0 +1 K m 0 X H ∈ F K m 0 +1 × K m 0 2 rank( H ) − δ ∗ n B , (23 ) where the second equality holds f rom (11) and the fact that P G ∈ F i × i 2 , rank( G )= i 1 = N i,i ( i ) an d the third equ ality ho lds since K min = min { K m 0 , K m 0 +1 } . Since δ ∗ n B → 0 as n B → ∞ the ach iev able sum -rate K R asym ptotically coincides with the up per bou nd in (5), which completes the p roof. 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