A Partial Decode-Forward Scheme For A Network with N relays

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1 A Partial Decode-Forward Scheme For A etwork with relays Yao Tag ECE Departmet, McGill Uiversity Motreal, QC, Caada yaotag2@mailmcgillca Mai Vu ECE Departmet, Tufts Uiversity Medford, MA, USA maivu@ecetuftsedu Abstract We study a discrete-memoryless relay etwork cosistig of oe source, oe destiatio ad relays, ad desig a scheme based o partial decode-forward relayig The source splits its message ito oe commo ad 1 private parts, oe iteded for each relay It ecodes these message parts usig th-order block Markov codig, i which each private message part is idepedetly superimposed o the commo parts of the curret ad previous blocks Usig simultaeous slidig widow decodig, each relay fully recovers the commo message ad its iteded private message with the same block idex, the forwards them to the followig odes i the ext block This scheme ca be applied to ay etwork topology We derive its achievable rate i a compact form The result reduces to a kow decode-forward lower boud for a -relay etwork ad partial decode-forward lower boud for a two-level relay etwork We the apply the scheme to a Gaussia two-level relay etwork ad obtai its capacity lower boud cosiderig power costraits at the trasmittig odes I ITRODUCTIO The relay chael first itroduced by va der Meule [1] cosists of a source aimig to commuicate with a destiatio with the help of a relay I [2], Cover ad El Gamal itroduce the cut-set boud ad two codig strategies, amely decodeforward ad compress-forward, for the basic three-ode relay chael By allowig the relay to decode oly a part of the trasmitted message, partial decode-forward ca be cosidered as the geeralizatio of decode-forward [2], [3] I the past few years, substatial research activities have bee dedicated to extedig the classical oe-relay chael to a geeral relay etwork cosistig of commuicatig parties I [4], Gastpar ad Vetterli discuss the asymptotic capacity i the limit as the umber of relays teds to ifiity ad the scalig behavior of capacity for a large class of Gaussia relay etworks Recetly, Lim, Kim, El Gamal ad Chug propose a compress-forward based scheme oisy etwork codig) [5] for the geeral multi-source multicast oisy etwork, which icludes etwork codig [6] as a special case However, it is still uclear how to geeralize decodeforward relayig to the multi-source multicast etwork I [7], Xie ad Kumar aalyze a multiple-level relay chael with oe source ad oe destiatio ad give a achievable rate based o full decode-forward This scheme is exteded i [8], i which all relays successively decode oly part of the messages of the previous relay, ad obtais the capacity of semi-determiistic ad orthogoal relay etworks I [9] ad [1], Ghabeli ad Aref geeralize partial decode-forward M source Fig 1 cloud p y, y,, y x, x,, x ) Geeral discrete memoryless relay etwork Mˆ 1 destiatio i a two-level relay etwork, cosiderig all possible partial decodig coditios that ca occur amog differet message parts of at the source ad the relays There are three commo approaches for the decode-forward strategy, amely: a) irregular ecodig/sequetial decodig; b) regular ecodig/simultaeous slidig widow decodig; c) regular ecodig/backward decodig [11] For semidetermiistic relay etworks [12], the secod ad the third approaches ca achieve the same rate, which is greater tha that of the first approach Furthermore, the secod approach creates less delay tha the third oe I this paper, we propose a ovel trasmissio scheme for a sigle-source sigle-destiatio etwork with relays based o regular ecodig ad simultaeous slidig widow decodig The source splits its message ito oe commo ad 1 private parts ad performs block Markov codig Each relay helps forward the commo part ad the private message part iteded for itself We derive the achievable rate i a compact form ad show that this scheme ca reduce to the etwork decode-forward scheme of [2] ad partial decode-forward for two-level relay etwork i [1] Fially, we aalyze a two-level relay etwork i AWG eviromets ad provide the achievable rate II PRELIMIARIES A Discrete Memoryless etworks Cosider a discrete memoryless relay etwork DM- R) with 2 odes X X 1 X, py 1, y 2,, y x, x 1,, x ), Y 1 Y 2 Y ), where source ode wats to sed a message M to the destiatio ode 1 with the help of relay odes 1,,,

2 U1, Source Ecoder Ecoder Ecoder W1, X1 W2, X2 1 2 U2, U, W U,, U, X, W 1 W1, X1 Ecoder W, X 2 2 W, X Destiatio Decoder Fig 2 The proposed private message scheme for a sigle-source sigledestiatio etwork with relays as show i Figure 1 A 2 R, ) code for this DM-R cosists of: A message set M = [1 : 2 R ] A source ecoder that assigs a codeword x m) to each message m [1 : 2 R ] A set of relay ecoders k [1 : ], which assigs a symbol x ki y i 1 k ) to each received sequece y i 1 k for i [1 : ] A destiatio decoder, which assigs a estimate ˆm to each received sequece y Y, or declares a error message e Defiitios for the average error probability, achievable rate ad capacity follow the stadard oes i [13] B Defiitios ad otatio To make the followig aalysis more cocise ad readable, we itroduce some defiitios ad clarify otatio i this sectio Defie T = {1,, } as the complete set of all relays Defie S to be a subset of T, that is S T ad S c = T S Either S or S c ca be empty ad the largest S is T Deote M j i = {M i, M i1,, M j }, where j i For example, U2 1 meas {U 2, U 3,, U 1 }, where 3 Give oempty set L ad variable M, let M L = {M i } i L = {M a, M b, M c, }, where a, b, c, L ad are differet from each other L sigifies the cardiality of L III A ETWORK PARTIAL DECODE-FORWARD SCHEME Cosider a etwork cosistig of a sigle source, sigle destiatio ad relays as i Figure 2 The source has direct liks to all relays ad to the destiatio, ad coectios amog the relays ad the destiatio are arbitrary We desig a ovel codig scheme for this relay etwork based o partial decode-forward relayig Let all relay odes be ordered i a arbitrary order permutatio π ) I each order, we assume that the kth relay decodes iformatio from all odes below it, ie order {1,, k 1}) ad forwards iformatio to odes above it ie order {k 1,, }) ext, we will describe the scheme for the omial order π = [1, 2,, ] to simplify the otatio, keepig i mid that it ca also be applied to ay other order π The ew idea i this scheme is the way it performs rate splittig At each block trasmissio, the source splits its message ito 2 parts: a commo message ad 1 private messages, oe iteded for each relay ad oe for the destiatio These messages are the ecoded usig th order block Markov codig Each relay fully recovers the commo message ad its iteded private message with same block idex as the commo message, the forwards them together i the ext block Specifically, let the source message i block j be split as m j = m j, m 1j,, m )j ), where m deotes the commo message that is forwarded amog all relays, m k deotes the message iteded to be decoded at relay k, but ot at other relays, ad m deotes the message iteded to be decoded oly at the destiatio The rate is R = i= R i Block Markov superpositio codig is used to geerate the idepedet codewords i each block as follows for simpler otatio we suppress block idex i codewords here, but will iclude it i the detailed proof later) W k, k {} T, carries commo message m,j k of differet blocks W k are successively superimposed o each other as i block Markov ecodig {U k }, k T, carries private message m k to be decoded at relay k ad ot decoded at other relays Each U k is superimposed o all W k X k, k T, is the codeword set by relay k which supports the forwardig of the message i U k of the previous block) ad all W l l k) X is the codeword set by the source which carries all messages icludig the remaiig message m to be decoded oly at the destiatio X is superimposed o all { }, {U1 } ad {X1 } At each block j, the source seds X which cotais all { }, {U1 } ad {X1 } Each relay k decodes {,, W k 1 } from all previous odes ad U k from the source The i block j 1, it trasmits X k which carries its private message of block j k m k,j k ) superimposed o all previous-block commo messages The destiatio uses joit decodig simultaeously over all blocks Specifically, it waits util the ed of the last block to decode all messages carried by { }, {U1 } ad {X } simultaeously usig sigals received i the last blocks This codig scheme is illustrated i Figure 2 Theorem 1 For a sigle-source sigle-destiatio etwork with relays X X 1 X, py 1, y 2,, y x, x 1,, x ), Y 1 Y 2 Y ), by usig partial decode-forward, the capacity C is lower bouded by 1), where π ) deotes a permutatio order for the relay odes Proof: We use a block codig scheme i which each user seds b messages over b blocks of symbols Each relay ad the destiatio employ simultaeous decodig A Codebook geeratio Fix the joit distributio i 2) where the meaig of each compoet is as follows: pw k Wk1 ): at relay k, the curret commo message is superimposed o previous commo messages

3 I ) U1, X, ; Y, C sup sup mi mi I X, X S, U S ; Y X S c, U S c, πt ) p π S πt ) π I ) j S U c πj ; Y πj, X πj where T = {1, 2,, }, πt ) is a permutatio order of T, S is a subset of πt ), ad p π = pu 1, W, X ) = px U 1, W, X 1 ) π π k =π 1 pu πk W, X πk )p W 1 ), j, j x m 1, j m1, j, m, j, mk, jk ) ) miπj S W c I, Wπ πj 1 1 ; Y πj X πj, W π π j k[1: ] π px πk Wπ π k π k =π 1 1) )pw πk W π π k1 ) 2) u m m m, 1 1, j, j 1, j 1 j, j, j, u2 m2, j m, j, m2, j2 u m, j m, j, m, j w m m, j 1, j, j x m m, j 1 1 1, j1, j x m m, j 2 2 2, j2, j x m m, j, j Fig 3 w m m, j 2 1, j1, j w m m, j 3 2, j2, j w m, j Ecodig diagram of a sigle-source sigle-destiatio etwork with relays arrows deote superpositio structure) px k Wk ): at relay k, the curret private message is superimposed o the curret ad previous commo messages p W1 ): at the source, the curret commo message is superimposed o the commo messages of all previous blocks pu k, X k ): at the source, the private message for a specific relay is superimposed o all commo messages ad the previous private message for that relay px U1,, X1 ): at the source, the private message for the destiatio is superimposed o all commo messages ad all private messages for all relays Figure 3 illustrates the superpositio codig structure I block j, the source splits its message as m j = [m,j, m,j,, m 1,j, m,j ] For every relay ode k =,, 1 ad every message set {m,j,, m,j k, m k,j k }: Radomly ad idepedetly geerate 2 R sequeces wk m,j k m,j k 1,j ) for all m,j k [1 : 2 R ], each accordig to i=1 p W k Wk1 w ki w,i k1,i ), Radomly ad idepedetly geerate 2 R k sequeces x k m k,j k m,j k,j ) for all m k,j k [1 : 2 R k ], each accordig to i=1 p X k Wk x ki w,i k,i ) For source ode k = ad each message set m,j,j For all sequece wk m,j k m,j k 1,j ) with k T, radomly ad idepedetly geerate 2 R sequeces w m,j m,j 1,j ) for m,j [1 : 2 R ], each accordig to i=1 p W1 w,i w,i 1,i ), For each k T, radomly ad idepedetly geerate 2 R k sequeces u k m k,j m,j,j, m k,j k) for all m k,j k [1 : 2 R k ], each accordig to i=1 p U k,x u k ki w,i,i, x k,i), Radomly ad idepedetly geerate 2 R sequeces x m,j m,j 1,j, m,j,j, {m k,j k} k T ) for all m,j [1 : 2 R ], each accordig to i=1 p X U1, x i u,i,x 1 1,i, w,i,i, x,i 1,i ) The codebook is idepedetly geerated i each block as above ad the is revealed to all the parties B Ecodig To sed {m,j,, m j } i block j, the source trasmits x m,j m,j 1,j, m,j,j, {m k,j k} k T ) from codebook C j At the ed of block j, each relay k T has a estimate m k,j k1 of message m k,j k1 ad m,j k1 of message m,j k1 I the block j 1, each relay k T trasmits x k m k,j k1, m,j k1,j ) from codebook C j1 C Decodig 1) Simultaeous decodig at the first relay: At the ed of block j, by kowig m,j 1,j ad m 2,j 1, the first relay k = 1 decodes m 1,j ad m,j such that: u 1 m 1,j m,j,j, m 1,j 1), w m,j m,j 1,j ), w 1 m,j 1 m,j 2,j ), x 1 m 1,j 1 m,j 1,j ), w 2, w 3,, w, y 1 j)) T ) ɛ 3)

4 The decodig error probability goes to as, if R 1 < IU 1 ; Y 2 W, X 1 ), 4) R 1 R < IU 1, ; Y 2 X 1, W 1 ) 5) 2) Simultaeous slidig widow decodig at other relays k [2 : ]: At the ed of block j, by kowig m,j k,j k1 ad m k,j k, the relay ode k will decode m k,j k1 ad m,j k1 such that the followig coditios hold simultaeously: wk 1m,j k1 m,j k,j ), w k m,j k m,j k 1,j ), x km k,j k m,j k,j ), w k1, w k2,, w, y k j) ) T ) ɛ w1 m,j k1 m,j k,j k 2 ), w 2,, wk 1, wk, x k, w k1,, w, y k j k 2) ) T ) ɛ u km k,j k1 m,j k1,j k1, m k,j 2k1), w m,j k1 m,j k u,j k1 ), w 1, w 2,, w k 1, x k, w k, w k1,, w, y k j k 1) ) T ) ɛ 6) Therefore, there are k decodig rules to be satisfied simultaeously at relay k The decodig error probability goes to, as, if R k < IU k ; Y k W, X k ), 7) R k R < IU k, W k 1 ; Y k X k, W k ) 8) Detailed error aalysis at relay k is i Appedix A 3) Simultaeous slidig widow decodig at destiatio ode 1: At the ed of block j, the destiatio ode 1 will decode m k,j for all k T { 1} ad m,j such that the followig coditios hold simultaeously: x m,j m,j ), w m,j ), yj) ) T ɛ ) x i1m i1,j m,j,j i ), w i1m,j m,j,j i ), x i1, w i1,, x, w, yj i 1) ) T ɛ ) x 1 m 1,j m,j,j 2 ), w 1 m,j m,j 1,j 2 ), Ad, x 2, w 2,, x, w, y j 1) ) T ) ɛ {u km k,j m,j,j 2, m k,j k)} k T, x m,j m,j 1,j, m,j,j 2, {m k,j k } k T ), w m,j m,j 1,j 2 ), x 1, w 1, x 2, w 2,, x, w, y j ) ) T ) ɛ 9) We have 1 decodig rules to be satisfied simultaeously The decodig error probability goes to, as, if i= R i < IU 1, X, W ; Y ), 1) R i R < IX, X S, U S ; Y X S c, U S c, ), 11) i S R < IX ; Y U 1, X 1, W ), 12) i=1 R i < IU 1, X ; Y W ), 13) where S is a subset of T that cotais wrogly decoded messages at the relay Detailed error aalysis at the destiatio is i Appedix B D Combiatio Process From 1), we ca directly get that: R < IU 1, X, W ; Y ) 14) From 11), 7) ad 8), we get R = R i S ) R i mi R j R ) j S c i S c,i j < IX, X S, U S ; Y X S c, U S c, W ) 15) j 1 mi IW j Sc ; Y j X j, Wj ) IU i ; Y i, X i ), i S c for all S T From 7), 8), 12) ad 13), we get 2R < i=1 mi i,j T R i R R i R ) R j R ) l T,l i,j < IU 1, X ; Y W ) IX ; Y U 1, X 1, W ) j 1 2 mi I ; Y j X j, Wj ) j T 2 i T R l R i IU i ; Y i W, X i ), 16) However, if we let S = i 15) ad double the right-hadside RHS) expressio, the we ca get a smaller expressio tha the RHS of 16) Thus, 16) is redudat After this combiatio process, we get the rate i 1) E Special etworks If = 2, we have the partial decode-ad-forward lower boud for a two-level relay etwork as show i Figure 4, which coicides with the result i [1] For a geeral, if we set private parts U1 = ad m =, we ca get Xie ad Kumar s [7] etwork decodeforward lower boud as show i Figure 5 Furthermore, if = 1, it reduces to the decode-forward lower boud [2] for the discrete-memoryless relay chael DM-RC)

5 Fig 4 U1, Source Ecoder Ecoder Ecoder W 1 Ecoder 1 2 U2, X1, W1 U1, U2, U3, X2, W2 3 Destiatio Decoder Two-level relay etwork with partial-decode-forward Ecoder W W 2 W 1 W 2 Ecoder W where I 1 = log g 2 1φ 2 ) 1 g1 2 β2 2 φ2 2 φ2 3 ) 1, I 4 = 1 2 log 1 g2 2α φ 2 ) 2 g 2 α 1 g 12 α 11 ) 2 ) g 2 β 1 g 12 β 11 ) 2 g2 2 φ2 1 φ2 3 ) 1, g2φ 2 2 ) 2, I 3 = log g 2 β 1 g 12 β 11 ) 2 g2 2 φ2 1 φ2 3 ) 1 I 2 = log g 2 1α 2 φ 2 ) 1) g1 2 β2 2 φ2 2 φ2 3 ) 1, I 5 = 1 2 log 1 g 2 3φ 2 3), W 1 I 6 = 1 2 log 1 g 3 β 1 g 13 β 11 ) 2 g 2 3φ 2 1 φ 2 3) ), Source Ecoder Fig 5 Decode-forward relay etwork Destiatio Decoder I 7 = 1 2 log 1 g 3 β 2 g 23 β 22 ) 2 g 2 3φ 2 2 φ 2 3) ), IV GAUSSIA RELAY ETWORKS The Gaussia relay etwork ca be modeled as k 1 Y k = g ik X i Z k, 17) i= where k T { 1}, g ik is the coefficiet of the lik from ode i to ode j, ad Z k is oise at the decoder with Gaussia distributio, 1) There is a power costrait at each trasmittig ode as P k As show i Figure 4, the Gaussia two-level relay etwork ca be modeled as Y 3 = g 3 X g 13 X 1 g 23 X 2 Z 3, Y 2 = g 2 X g 12 X 1 Z 2, Y 1 = g 1 X Z 1, 18) where Z 3, Z 2 ad Z 1 are idepedet AWG oise accordig to the ormal distributio, 1) The sigalig at each ode ca be writte as x 2 = α 22 W 2 w,j 2 ) β 22 V 2 w 2,j 2 ), x 1 = α 11 W 1 w,j 1 ) α 12 W 2 w,j 2 ) β 11 V 1 w 1,j 1 ), x = α w,j ) α 1 W 1 w,j 1 ) α 2 W 2 w,j 2 ) β 1 V 1 w 1,j 1 ) β 2 V 2 w 2,j 2 ) φ 1 U 1 w 1,j ) φ 2 U 2 w 2,j ) φ 3 U 3 w 3,j ), 19) where W 2, V 2, W 1, V 1,, U 1, U 2, U 3 are idepedet, ormalized Gaussia radom variables, 1); {α, β, φ } are power allocatios satisfyig the followig costraits: α 2 22 β 2 22 = P 2, α 2 11 α 2 12 β 2 11 = P 1, α 2 α 2 1 α 2 2 β 2 1 β 2 2 φ 2 1 φ 2 2 φ 2 3 = P, 2) where P, P 1 ad P 2 are power costraits at the correspodig ode, which ca be set equal to each other without loss of geerality Corollary 1 The capacity for a Gaussia two-level relay etwork i 18) is lower bouded by: C mi {I 1 I 4 I 5, I 2 I 3 I 5, I 2 I 7, I 4 I 6, I 8 }, 21) I 8 = 1 2 log 1 g 2 3P g 2 13P 1 g 2 23P 2 2g 3 g 13 α 1 α 11 α 2 α 12 β 1 β 11 ) 2g 3 g 23 α 2 α 22 β 2 β 22 ) 2g 13 g 23 α 12 α 22 ), ad α ij, β ij, φ ij i {, 1, 2},j {, 1, 2, 3}) are power allocatios satisfyig 2) Proof: Applyig the sigalig i 19) to the rate regio i Theorem 1, we obtai 21) V COCLUSIO I this paper, we cosider partial decode-forward relayig i a sigle-source sigle-destiatio etwork with relays We desig a scheme i which each relay forwards the commo message part ad a specific private part to the followig odes The proof is based o block Markov ecodig ad simultaeous slidig widow decodig The key poit is that each relay decodes ad forwards its private part oly whe the last commo part with the same block idex arrives We the obtai the achievable rate for this scheme, which ca be expressed i a compact form over all cutsets ad permutatios of relays We show that this scheme cotais existig results for a -relay etwork with decode-forward ad a two-level relay etwork with partial decode-forward cosiderig all message splittig cases We the study the Gaussia two-level relay etwork ad derive the achievable rate by the proposed scheme APPEDIX A ERROR AALYSIS AT RELAY k Assume without loss of geerality that m k,j k1, m,j k1 ) = 1, 1) is set i block j We first defie the followig evets: E i m k,j k1, m,j k1 ), for i [1 : k], is whe oly the ith decodig rule is satisfied We simplify the otatio as E i i the followig aalysis Em k,j k1, m,j k1 ) as the evet that all decodig rules are satisfied simultaeously

6 P e P c E c {1}, 1)) P c E{m i,j } i T, m,j, m,j )) m,j 1,{m i,j } i T,m,j P c E{m i,j } i S, {1} Sc, m,j )) 22) m,j =1,{m i,j } i S 1,{m i,j 2 } i S c =1,m,j The, by the uio bouds, the probability of error is bouded as P e P c E c 1, 1)) m,j k1 =1,m k,j k1 1 P c Em k,j k1, 1)) m,j k1 1,m k,j k1 P c Em k,j k1, m,j k1 )), where P c is the coditioal probability give that 1, 1) was set By the law of large umber, P c E c 1, 1)) as By the joit typicality lemma, we have P c Em k,j k1, 1)) m,j k1 =1,m k,j k1 1 2 R k 2 IU k;y k W,X k) δɛ)), which goes to as, if R k < IU k ; Y k W, X k ) δɛ) Accordig to the idepedece of the codebooks ad the joit typicality lemma, m,j k1 1,m k,j k1 P c Em k,j k1, m,j k1 )) = P c mk,j k1 m,j k1 1 E 1 E 2 E k )) P E 1 ) P E 2 ) P E k ) m,j k1 1 m k,j k1 2 R k 2 R 2 IW k 1;Y k X k,w k ) δɛ)) 2 IW k 2;Y k X k,w k 1 ) δɛ)) 2 IW2;Y k X k,w 3 ) δɛ)) 2 IU k,w 1;Y k X k,w 2 ) δɛ)), which teds to as if R k R < IU k, W k 1 1 ; Y k X k, W k ) kδɛ) APPEDIX B ERROR AALYSIS AT DESTIATIO 1 Assume without loss of geerality that {m k,j } k T, m,j, m,j ) = 1, 1,, 1) was set i block j We first defie the followig evets: E i {m k,j } k T, m,j, m,j ), i [1 : 1], as the evet that oly the ith decodig rule is satisfied We simplify the otatio as E i i the followig aalysis E{m k,j } k T, m,j, m,j ) as the evet that all 1 decodig rules are satisfied simultaeously We defie set S to be the set of wrogly decoded private messages ad S c as the set of correctly decoded private messages at the destiatio The, by the uio of evets boud, the probability of error is bouded as i 22), where P c is the coditioal probability give that {m k,j } k T, m,j, m,j ) = 1, 1,, 1) was set By the law of large umber, P c E c {1}, 1)) as It is impossible to correctly decode m,j if ay of {m k,j } k T is t decoded correctly Accordig to the idepedece of codebooks ad the joit typicality lemma, the third term i 22) teds to as if R i R <IX, {X i, U i } i S ; Y {X j, U j } j S c, ) i S S 1)δɛ) Similarly, accordig to the idepedece of codebooks ad the joit typicality lemma, the secod term i 22) teds to as if i=1 R i R < IU 1, X, W ; Y ) 1)δɛ) REFERECES [1] E va der Meule, Three-termial commuicatio chaels, Advaces i Applied Probability, vol 3, o 1, pp , 1971 [2] T Cover ad A El Gamal, Capacity theorems for the relay chael, IEEE Tras o Ifo Theory, vol 25, o 5, pp , 1979 [3] A E Gamal ad M R Aref, The capacity of the semidetermiistic relay chael, IEEE Tras o Ifo Theory, vol 28, o 3, p 536, 1982 [4] M Gastpar ad M Vetterli, O the capacity of large Gaussia relay etworks, IEEE Tras o Ifo Theory, vol 51, o 3, pp , March 25 [5] S H Lim, Y-H Kim, A El Gamal, ad S-Y Chug, oisy etwork codig, IEEE Tras o Ifo Theory, vol 57, o 5, pp , May 211 [6] S-Y R L R Ahlswede, Cai ad R W Yeug, etwork iformatio flow, IEEE Tras o Ifo Theory, vol 46, o 4, pp , July 2 [7] L-L Xie ad P R Kumar, A achievable rate for the multiple level relay chael, IEEE Tras o Ifo Theory, vol 51, o 4, pp , April 25 [8] L Ghabeli ad M R Aref, A ew achievable rate ad the capacity of a class of semidetermiistic relay etworks, i IEEE It l Symp o Ifo Theory ISIT), Jue 27 [9], Simultaeous partial ad backward decodig approach for twolevel relay etworks, i IEEE It l Symp o Ifo Theory ISIT), Jue 29 [1], O achievable rate for relay etworks based o partial decodead-forward, i IEEE It l Symp o Ifo Theory ISIT), Jue 21 [11] G Kramer, M Gastpar, ad P Gupta, Cooperative strategies ad capacity theorems for relay etworks, IEEE Tras o Ifo Theory, vol 51, o 9, pp , Sept 25 [12] L Ghabeli ad M R Aref, Symmertric semidetermiistic relay etworks with o iterferece at the relays, IEEE Tras o Ifo Theory, vol 57, o 9, pp , September 211 [13] A E Gamal ad Y-H Kim, etwork iformatio theory Cambridge Uiversity Press, 211