THE interference channel problem describes a setup where multiple pairs of transmitters and receivers share a communication

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1 Trade-off betwee Commuicatio ad Cooperatio i the Iterferece Chael Farhad Shirai EECS Departmet Uiversity of Michiga A Arbor,USA fshirai@umich.edu S. Sadeep Pradha EECS Departmet Uiversity of Michiga A Arbor,USA pradhav@umich.edu arxiv: v1 [cs.it] 17 Feb 2016 Abstract We cosider the problem of codig over the multi-user Iterferece Chael (IC). It is well-kow that aligig the iterferig sigals results i improved achievable rates i certai setups ivolvig more tha two users. We argue that i the geeral iterferece problem, seders face a tradeoff betwee commuicatig their message to their correspodig decoder or cooperatig with other users by aligig their sigals. Traditioally, iterferece aligmet is carried out usig structured codes such as liear codes ad group codes. We show through a example that the usual structured codig schemes used for iterferece eutralizatio lack the ecessary flexibility to optimize this tradeoff. Based o this ituitio, we propose a ew class of codes for this problem. We use the example to show that the applicatio of these codes gives strict improvemets i terms of achievable rates. Fially, we derive a ew achievable regio for the three user IC which strictly improves upo the previously kow ier bouds for this problem. I. Itroductio THE iterferece chael problem describes a setup where multiple pairs of trasmitters ad receivers share a commuicatio medium. Each receiver is oly iterested i decodig the message from its correspodig trasmitter. However, sice the chael is shared, sigals from other seders iterfere with the desired sigal at each decoder. The presece of iterferig sigals adds ew dimesios to this problem i terms of strategies that ca be used as compared to poit-to-poit (PtP) commuicatio. For example, the ecoders ca cooperate with each other by choosig their chael iputs i a way that would facilitate their joit commuicatio. It turs out that, ofte, this cooperatio requires a ecoder to employ a strategy which may be sub-optimal from its ow PtP commuicatios perspective. I this paper, we ivestigate this tradeoff ad develop a ew class of codes which allow for more efficiet cooperatio betwee the trasmitters. Characterizig the capacity regio for the geeral IC has bee a challege for decades. Eve i the simplest case of the two user IC, the capacity regio is oly kow i special cases [1] [2]. The best kow achievable regio for the IC was due to Ha ad Kobayashi [3]. However, recetly it was show that the Ha-Kobayashi (HK) rate regio is suboptimal [4] [9]. Particularly, whe there are more tha two trasmitter-receiver pairs, the atural geeralizatio of the HK strategy ca be improved upo by iducig structure i the codebooks used i the scheme [9]. Structured codes such as liear codes ad group codes eable the ecoders to alig their sigals more efficietly. This i tur reduces iterferece at the decoders. Such codebook structures have also prove to give gais i other multi-termial commuicatio problems [7]- [10]. The idea of iterferece aligmet was proposed for maagig iterferece whe there are three or more users. Iitially, the techique was proposed by Maddah-Ali et. al. [5] for the MIMO X chael, ad for the multi-user IC by Jafar ad Cadambe [6]. The iterferece aligmet strategy was developed for cases of additive iterferece ad uiform chael iputs over fiite fields. This was exteded to arbitrary iterferece settigs ad iput distributios i [9]. However, it turs out that aligmet is ot always beeficial to the users i terms of achievable rates. Cosider the example i Figure 1. Ituitively, it would be beeficial to alig the iput from users 1 ad 2 to reduce iterferece at decoder 3. However, if users 1 ad 2 alig their sigals, it becomes harder for decoder 2 to distiguish betwee the two iputs. Oe might suggest that the problem could be alleviated if users 1 ad 2 desiged their codebooks i a way that they would "look" aliged at decoder 3 based o P Y3 X 1,X 2,X 3, but at the same time they would seem differet at decoder 2 based o P Y2 X 1,X 2. I this paper we show that liear codes lack the ecessary flexibility for such a strategy. Based o this ituitio, we propose a ew class of structured codes. Usig these codes we derive a achievable rate regio which improves upo the best kow achievable regio for the three user IC give i [9]. The rest of the paper is orgaized as follows: Sectio II gives the otatio used i the paper as well as the problem statemet. I sectio III, we cosider two differet examples of three user IC. I the first example - where iterferece aligmet is strictly beeficial - we prove that ot oly structured codes are useful for aligmet but that ay arbitrary codig scheme which achieves optimality must possess certai liearity properties. I the secod example, we show the existece of the tradeoff discussed above ad prove that liear codes are suboptimal for that example. Sectio IV, gives the ew codebook costructios ad proves that these ew codes outperform the liear codig scheme i [9]. I Sectio V, we provide a ew geeral achievable rate regio for this problem. Sectio VI, cocludes the paper.

2 X1 P Y1 X1 Y1 X2 P Y2 X1,X2 Y2 X3 P Y3 X1,X2,X3 Y3 Fig. 1. A setup where iterferece aligmet is beeficial to user 3 but harmful for user 2. II. Problem Statemet ad Notatio I this sectio, we give the otatio used i the paper ad provide the problem statemet. Throughout the paper, we deote radom variables by capital letters such as X, U, their realizatios by small letters x, u, ad their correspodig alphabets (fiite) by sas-serif typeface X, U, respectively. Small letters such as l, k are used to represet umbers. The field of size q is deoted by F q. We represet the field additio by ad the additio o real umbers by +. For m N, We defie the set of umbers [1, m] {1, 2,, m}. Vectors are represeted by the bold type-face such as u, b. For a radom variable X, A ɛ (X) deotes the set of ɛ-typical sequece of legth with respect to the probability distributio P X, where we use the defiitio of frequecy typicality. Let q be a prime umber. For l N, cosider U i, i [1, m] i.i.d radom variables with distributio P U defied o the field F q. U l deotes a radom variable which has the same distributio as i [1,l] U i where the summatio is over F q. We proceed with formally defiig the three user IC problem. A three user IC cosists of three iput alphabets X i, i {1, 2, 3}, three output alphabets Y i, i {1, 2, 3}, ad a trasitio probability matrix P Y X. A code for this setup is defied as follows. Defiitio 1. A three user IC code (, M 1, M 2, M 3, e, d) cosists of (1) Three sets of message idices M i (2) Three ecoder mappigs e i : M i X i, i [1, 3], without loss of geerality, these maps are assumed to be ijective (3) ad three decodig fuctios d i : Y i M i, i [1, 3]. We defie the codebook correspodig to the ecodig map e i as C i = {e i (m i ) m i M i }, i [1, 3]. The rate of user i is defied as r i = 1 log C i. Defiitio 2. A rate-triple (R 1, R 2, R 3 ) is said to be achievable if for every ɛ > 0, there exists a code (, M 1, M 2, M 3, e, d) such that (1) r i R i ɛ, i [1, 3], ad (2) P(d(Y ) = M e(m) = X ) 1 ɛ. We make frequet use of coset codes ad Nested Liear Codes (NLC) which are defied ext. Defiitio 3. A (k, ) coset code C is characterized by a geerator matrix G k ad a dither b defied o the field F q. The code is defied as C {ug b u F k q}. The rate of the code is give by R = k log q. Defiitio 4. For atural umbers k i < k o, k o <, let G ki, G (ko k i ) ad G (k o k i ) be matrices o F q. Defie C i, C o ad C o as the liear codes geerated by G, [G G] ad [G G ], respectively. C o ad C o are called a pair of NLC s with ier code C i. We deote the outer rates as r o = k o ad r o = k o ad the ier rate r i = k i. III. The Iterferece Aligmet tradeoff I this sectio, we ivestigate the iterferece aligmet tradeoff metioed i the itroductio i more detail. We show that i certai three user iterferece setups, o the oe had, aligmet is beeficial to oe of the users, while o the other had, the rates achieved by the aligig users is reduced due to the aligmet. We ivestigate the pheomeo i two examples. The first example ivolves a three user iterferece setup. I this example, the first two ecoders use liear codes to maage the iterferece for the third user. This gives a strictly improved achievable rate regio. It is well-kow that iterferece aligmet ca be iduced efficietly by the applicatio of structured codes. Additioal to this, we show the stroger statemet that the oly esemble of codes which achieve the desired rate-triples i this example, are the oes with specific liearity properties. Next, we build upo the first example to create a setup where aligmet is beeficial to oe of the users ad harmful for the other oe. This secod example provides the motivatio for our ew codebook costructios i the ext sectio. A. Example 1 Cosider the example show i Figure 2. All of the iputs are q-ary ad the additios are defied o the field F q. The three outputs of the chael are Y i = X i N 1 N 3, i {1, 2}, Y 3 = X 1 X 2 X 3 N 3. We are iterested i achievig the followig rates for the first ad secod users: R 1 = R 2 = log q H(N 1 N 3 ). Give these rates, we wat to maximize R 3. The followig lemma shows that liear codes achieve the optimum R 3 for this setup. Furthermore, we show that if a esemble of codes achieves the optimum R 3, the the codes correspodig to the first two users are "almost" the same coset code.

3 X1 N1 q N3 N1 q N3 X2 N3 X3 Fig. 2. A Three User IC Where Aligmet Is Strictly Beeficial Lemma 1. For a give family of codes (, M 1, M 2, M 3, e, d), N satisfyig the rate ad error costraits at decoders 1 ad 2, user 3 ca achieve the rate R 3 = H(N 1 N 3 ) H(N 3 ) iff there exists a dither b such that for every radom variable N defied o F q with positive etropy, the followig holds: P(e 1 (M 1 ) e 2 (M 2 )) C 1 C 2 A ɛ (N) b) 1, as. (1) Equivaletly, the optimal rate is achieved iff there exists aother family of codes (, M 1, M 2, M 3, e, d ), N for which 1) P(e i(m i ) C i A ɛ (N)) 1 as, 2) C 1 = C 2 is a coset code, ad 3) they also achieve the rate triple (R 1, R 2, R 3 ) = (log q H(N 1 N 3 ), log q H(N 1 N 3 ), H(N 1 N 3 ) H(N 3 )). Proof: We provide a outlie of the proof i Appedix A. The lemma proves that eve if we expad our search to arbitrary -legth codebook costructios (as opposed to the usual radom codebook geeratio based o sigle-letter distributios), coset codes are the oly efficiet esemble of codes for the classes of iterferece chaels uder cosideratio up to small perturbatios. This is a stroger assertio tha the well-kow result that liear codes are useful for aligig the iterferig sigals. The lemma ca be used to provide a coverse result provig that schemes ivolvig radom ustructured codes (e.g. the geeralized versio of the sigle-letter HK scheme), ca t achieve the desired rate-triple without directly aalyzig the bouds correspodig to their achievable rate regio as doe i [9]. B. Example 2 Next, we cosider a example where iterferece aligmet results i a tradeoff betwee two of the users. Cosider the setup i Figure 3. Similar to the previous example, all iput alphabets, output alphabets, ad additios are defied o the field F q. The outputs of the chael are Y 1 = X 1 q N 1 q N 2 q N 3, Y 2 = X 1 q X 2 q N 2 N 3, ad Y 3 = 2X 1 q X 2 q X 3 q N 3. Followig our argumets i the previous example, for user 3 to be able to trasmit its messages at rate R 3 = H(X 3 N 3 ) H(N 3 ), the iputs for users 1 ad 2 must alig. However, if these two users alig their iputs, user 2 would ot be able to decode its message which is beig corrupted by its aliged iterferig sigal comig from user 1. Hece, we have a tradeoff. We proceed with evaluatig the rate-triples achievable i this example. The followig lemma proves that we must have R 1 + R 2 log q H(N 2 N 3 ), otherwise the rate-triple (R 1, R 2, R 3 ) is ot achievable. N1 q N2 q N3 X1 2 N2 q N3 X2 N3 X3 Fig. 3. A Three User IC Where Aligmet Results i a Tradeoff Lemma 2. Give that (R 1, R 2, R 3 ) is achievable, we must have R 1 + R 2 logq H(N 2 N 3 ). Proof: Sice the rate triple is achievable, there exists a family of codes (, M 1, M 2, M 3, e, d) for which the codewords set by the secod user, X2, is decoded at decoder 2 with error probability approachig 0. Assumig errorless decodig at decoder 2, the decoder has access to X2, X 1 X 2 N 2. The decoder ca subtract X 2 from X 1 X 2 N 2 to get X 1 N 2. Now, sice by assumptio decoder 1 ca decode X1 from X 1 N 1 N 2 with error goig to 0, decoder 2 ca use X 1 N 2 to recover X 1 with

4 small error. So decoder 2 has access to M 1 ad M 2. By the coverse of the poit-to-poit chael codig theorem, we must have 1 log M 1 M 2 log q H(N 2 ), which completes the proof. We wat to achieve the rate R 1 = log q H(N 1 N 2 N 3 ). I other words, ecoder 1 is to operate at PtP optimality. The goal is to optimize the liear combiatio R 2 + R 3. We argue that the liear codig scheme preseted i [9] ca t achieve the triple (R 1, R 2, R 3 ) for R 2 + R 3 > H(N 1 N 2 N 3 ) H(N 3 ). Lemma 3. Give R 1 = log q H(N 1 ), the scheme i [9] ca t achieve R 2 + R 3 > H(N 1 N 2 N 3 ) H(N 3 ). Proof: We provide the ituitio behid the proof here. Let us use two NCL s C 1 ad C 2 as defied i Defiitio 4 to trasmit the messages at ecoders 1 ad 2. Let the rate of C j, j {1, 2} be r j ad let the ier code have rate r i. If we assume that the codig scheme exists which achieves the rate-triple, the by the proof of Lemma 2, oe should be able to recover X 1, X 2 from X 1 X 2 N 2 N 3 with small error probability. Also, at decoder 3, the decoder ca recostruct X 3 with low error probability ad by subtractio it ca have 2X 1 X 2 N 3. Note that i the liear codig scheme, both 2X 1 X 2 ad X 1 X 2 come from radomly ad uiformly geerated liear codes of rate r 1 + r 2 r i. So, give that X 1 ad X 2 ca be recovered from X 1 X 2 N 2 N 3, decoder 3 must be able to recover X 1 ad X 2 from 2X 1 X 2 N 3. The similar to the proof of Lemma 1, by the poit-to-poit chael codig coverse, we must have R 1 + R 2 + R 3 < log q H(N 3 ). The argumets i the proof of the previous lemma suggest that NLC s lack the ecessary flexibility whe it comes to determiig the size of differet liear combiatios of such codes. We explai this i more detail. Cosider two NLC s, C ad C, with rates r o ad r o, respectively, ad with ier code rate r i. The rate of ay liear combiatio of the two, αc βc, α, β F q \{0}, is equal to r o + r o r i. Whereas i settigs such as the oe at had, it is desirable to have differet rates for differet values of α ad β. I this setup, decoder 2 requires C 1 C 2 to be large (sice by Lemma 2 i order to icrease R 2 it eeds to icrease the rate of this liear combiatio) ad decoder 3 wats the size of the iterferig codebook 2C 1 C 2 to be small, so that it ca decode the iterferece. I the ext sectio, we provide a ew class of codes. The ew costructio allows for differet rates for differet liear combiatios of such codes. This i tur results i higher achievable sum-rates. IV. A New Class of Code Costructios I this sectio, we preset our ew codig costructios. These ew codes are called Quasi Liear Codes (QLC). They are ot liearly closed but maitai a degree of liearity. I order to costruct a QLC, we first costruct a liear code. The, we take a subset of that codebook to trasmit the messages. More precisely, QLC s are defied as follows: Defiitio 5. A (k, ) QLC o the field F q, is characterized by a geerator matrix G k, a dither b ad a set U. The codebook is defied as C {ug b u U}. For ijective G o U, the rate of the code is give by R = 1 log C = 1 log U. I this paper we oly cosider the cases whe U is a cartesia product of typical sets: C { u i G i b u i A k i ɛ (U i )}. i [1,m] A pair of Nested Quasi Liear Codes (NQLC) is defied below: Defiitio 6. For atural umbers k 1, k 2,, k m, let G ki, i [1, m] be matrices, ad b, b dithers all defied o F q. Also, let (U 1, U 2,, U m ) ad (U 1, U 2,, U m) be a pair of radom vectors o F q. The pair of QLC s characterized by the matrices G ki, i [1, m] ad each of the two dithers ad vectors of radom variables are called a pair of NQLC s. As explaied i the previous sectio, our motivatio for defiig NQLC s, is to costruct codes such that differet liear combiatios of those codes have differet rates. The ext lemma shows that NQLC s have this property. Lemma 4. Let C ad C be two QLC s as defied i Defiitio 6, whose geerator matrices ad dithers are take radomly ad uiformly from F q. The, αc 1 βc 2 has rate close to i [1,m] k i H(αU i βu i ) for large with high probability. Proof: The proof follows from the ijectiveess of the G i s ad the usual typicality argumets ad is omitted. Havig defied QLC s, we retur to our iterferece chael setup i Example 2. We claim that NQLC s ca achieve a sum-rate R 2 + R 3 which is higher tha H(N 1 N 2 N 3 ) H(N 3 ). Lemma 5. There exists achievable rate-triples (R 1, R 2, R 3 ) = (log q H(N 1 N 2 N 3 ), r 2, r 3 ) such that r 2 + r 3 > H(N 1 N 2 N 3 ) H(N 3 ). Proof: Refer to Appedix B. So far we have proved that NQLC s outperform NLC s i this specific example. It is straightforward to show that NQLC s are a geeralizatio of NLC s. To see this, cosider a arbitrary pair of NLC s with the parameters as i defiitio 4. These

5 two codes are a pair of NQLC s with parameters m = 3, U 1, U 2 ad U 1, U 3 uiform, U 3 ad U 2 costats ad k 1 = k 1 = k i ad k 2 = k o k i, k 3 = k o k i. So, ay rate regio achievable by NLC s is also achievable usig NQLC s. V. New Achievable Rate Regio I this sectio, we provide a geeral achievable rate regio for the three user IC. The scheme is similar to the oe preseted i [9] (Theorem 2). The mai differece is that here istead of NLC s we use NQLC s. The radom variables ivolved i the codig scheme are depicted i Figure 4. Note that i cotrast with the scheme i [9], decoder 2 recostructs a liear combiatio of U 1 ad U 2. By settig α 2 = 0, β 2 = 1, we recover the radom variables i [9]. The ext theorem provides the achievable rate regio. X1, U1 X1, U1 X2, U2 P (Y1, Y2, Y3 X1, X2, X3) X2, α2u1 + β2u2 X3 X3, α3u1 + β3u2 Fig. 4. The LHS radom variables are the oes set by each ecoder, the RHS radom variables are the oes decoded at each decoder. Defiitio 7. For a give three user IC problem with q-ary iputs ad outputs, defie the set R 3-IC as the set of rate triples (R 1, R 2, R 3 ) such that there exist 1) a joit probability distributio P U1,X 1 P U2,X 3 P X3, 2) A vector of positive reals (K 1, K 2, L 1, L 2, T 1, T 2 ), ad 3) a vector of parameters (m,, k 1, k 2,, k m ) ad pair of vectors of radom variables (V i j ) j [1,m], i {1, 2}, such that the followig iequalities are satisfied:,where r α,β i [1,m] k i H(αV 1 βv 2 ), α, β F q. R 1 = L 1 + T 1, R 2 = L 2 + I(U 2; α 2 U 1 α 2 U 2 ) T 2 H(U 2 ) (2) r 1,0 T 1 log q H(U 1 ), r 0,1 T 2 log q H(U 2 ), (3) K 1 + r 1,0 T 1 log q + H(X 1 ) H(X 1, U 1 ) (4) K 2 + r 0,1 T 2 log q + H(X 2 ) H(X 2, U 2 ) (5) r 0,1 log q H(U 1 X 1, Y 1 ) (6) r 0,1 + L 1 + K 1 log q + H(X 1 ) H(U 1, X 1 Y 1 ) (7) L 1 + K 1 I(X 1 ; U 1 Y 1 ) (8) r α2,β 2 log q H(α 2 U 1 β 2 X 2, Y 2 ) (9) r α2,β 2 + L 2 + K 2 log q+h(x 2 ) H(α 2 U 1 β 2 U 2, X 2 Y 2 ) (10) L 2 + K 2 I(X 2 ; U 2 Y 2 ) (11) r α3,β 3 + L 3 + K 3 log q+h(x 3 ) H(α 3 U 1 β 3 U 2, X 3 Y 3 ) (12) R 3 I(X 3 ; Y 3, α 3 U 1 β 3 U 2 ) (13) Theorem 1. A rate triple (R 1, R 2, R 3 ) is achievable if it belogs to cl(r 3-IC ). Proof: We provide a outlie of the proof. The codig scheme is similar to the oe i [9]. Except that 1) decoder 2 also decodes a liear combiatio α 2 U 1 + β 2 U 2, 2) The uderlyig codes for U 1 ad U 2 are QNLC s istead of ested coset codes, ad 2) There is a outer code o U 2 which allows decoder 2 to decode U 2 from α 2 U 1 + β 2 U 2. As a result the rate regio is similar to the oe i [9] except for a few chages. Bouds (3)-(5) esure the existece of joitly typical codewords at each ecoder. These bouds are the same with the oes i [9]. Bouds (6)-(8) esure errorless decodig at decoder 1, they also remai the same. Iequalities (9)-(11) correspod to the error evets at decoder 2, these bouds are altered to esure recostructio of α 2 U 1 + β 2 U 2, also the rate R 2 is chaged ad the liear codig rate T 2 is multiplied by I(U 2;α 2 U 1 α 2 U 2 ) H(U 2 ), which is due to the outer code. Lastly, (12)-(13) are for the error evets at decoder 3, which is also similar to the oes i [9]. Remark 1. For ease of otatio, we have dropped the time-sharig radom variable Q. The scheme ca be ehaced by addig the variable i the stadard way. Remark 2. By takig α 2 = 0 ad β 2 = 1 ad choosig the NQLC parameters so that the codes become a pair of NLC s we recover the boud i [9] as expected.

6 Remark 3. Followig the geeralizatios i [9], this codig scheme ca be ehaced by addig additioal layers cotaiig the public message codebooks correspodig to the HK strategy. VI. Coclusio The problem of three user IC was cosidered. We showed that there is a iheret tradeoff i the geeral IC. The users ca choose to commuicate their messages by usig optimal PtP strategies or cooperate with other users to facilitate their commuicatio. It was show that the previously used codig structures are uable to optimize this tradeoff. New codig structures were proposed. It was show through a example that these ew structures give strict improvemets. Usig these ew codebooks, a achievable regio for the three user IC was derived which improves upo the previous kow ier bouds for this problem. Appedix A Proof of Lemma 1 Assume the family (, M 1, M 2, M 3, e, d), N achieves the rate-triple. Let M i, i {1, 2, 3} be uiform radom variables defied o sets M i. I the first step, we argue that the size of the set C 1 C 2 C 3 A ɛ (N 3 ) is close to C 1 C 2 A ɛ (N 3 ) C 3. More precisely, we prove the followig claim: Claim 1. For every ɛ R +, there exists a sequece of umbers α,ɛ R +, N such that the followig iequality holds: ad α,ɛ goes to 0 as ad ɛ 0. 1 log C 1 C 2 C 3 A ɛ (N 3 ) 1 log ( C 1 C 2 A ɛ (N 3 ) C 3 ) α,ɛ, Proof: Ituitively, if the size of C 1 C 2 C 3 A ɛ (N 3 ) is much smaller tha C 1 C 2 A ɛ (N 3 ) C 3, that meas there exists a large umber of sets of vectors c 1, c 2, c 3, 3, with differet c 3 s for which the sum is equal. This causes a large error probability i decoder 3 sice the decoder is uable to distiguish betwee these sets of vectors. More precisely, let t be a type o vectors i A ɛ (N 3 ), ad let c 3 C 3, defie B c3, t as follows, B c3, t = {c 1 c 2 3 c 1, c 2, c 3, 3 C 1 C 2 C 3 P t, such that c 1 c 2 c 3 3 = c 1 c 2 c 3 3, c 3 c 3}, where P t is the set of all vectors 3 A ɛ (N 3 ) with type t. That is B c3, t is the set of (c 1, c 2, 3 ) s for which the decoder has o-zero error probability for decodig c 3 or aother codeword c 3. From set theory, we have the followig: C 1 C 2 C 3 A ɛ (N 3 ) C 1 C 2 C 3 P t (14) = C 1 C 2 c 3 P t (15) c 3 C 1 C 2 c 3 P t C 1 C 2 c 3 P t (16) c 3 c 3 c 3 = C 1 C 2 c 3 P t C 1 C 2 c 3 P t (17) c 3 c 3 c 3 = ( C 1 C 2 P t B c3, t ) (18) c 3 C 3 = C 1 C 2 P t C 3 B c3, t (19) c 3 C 3 B = C 1 C 2 P t C 3 C 3 c3,t (20) C c 3 C 3 3 = ( C 1 C 2 P t E( B c3, t ) ) C 3 (21) ( ) E( B c3, = C 1 C 2 P t (1 t ) C 1 C 2 P t ) C 3. (22) O the other had, as, the error probability at decoder 3 goes to 0. This meas that P (d 3 (c 1 c 2 e(m 3 ) 3 ) M 3 ) goes to 0. Cosequetly, there exists a family of types type t such that P (d 3 (c 1 c 2 e(m 3 ) 3 ) M 3 t ) goes to 0. There

7 exists a sequece δ which approaches 0 at the limit such that: δ P (d 3 (c 1 c 2 e(m 3 ) 3 ) M 3 t ) 1 2 P ( ) c 1 c 2 3 B e(m3 ), t t 1 B c3,t 2 C c 3 C 3 C 1 C 2 P t 3 = 1 E( B e3 (M 3 ), t ) 2 C 1 C 2 P t Isertig this last iequality i (22) we have, C 1 C 2 P t C 3 B c3 c 3 C 3 ( C 1 C 2 P t (1 2δ )) C 3 Observe that P t ad A ɛ (N 3 ) have the same expoetial rate. Note that C 1 C 2 A ɛ (N 3 ) C 1 A ɛ (N 3 ) ad sice decoder 1 ca decode X 1 with probability of error approachig 0, we ca use the same argumet to show the followig: 1 log C 1 A ɛ (N 3 ) 1 log C 1 A ɛ (N 3 ) log q H(N 1 ) H(N 3 ) 1 log C 1 C 2 C 3 A ɛ (N 3 ) (23) ( log q H(N1 ) + H(N 3 ) ) + (H(N 1 ) H(N 3 )) = log q. 1 But we kow that log C 1 C 2 C 3 A ɛ (N 3 ) log q. So, we should have equality at all of the iequalities. Hece, 1 log q C 1 C 2 A ɛ (N 3 ) 1 log q C 1 A ɛ (N 3 ). I order to have C 1 C 2 close to C 1, we must have the properties stated i the lemma. Appedix B Proof of Lemma 5 We provide a codig scheme based o NQLC s which achieves the rate vector. Cosider two terary radom variables V 1 ad V 2 such that H(V 1 V 2 ) > H(2V 1 V 2 ). We will show the achievability of the followig rate-triple: R 1 = log q H(N 1 N 2 N 3 ) R 2 = ( H(V 1 V 2 ) 1)(log q H(N 1 N 2 N 3 )) H(V 1 ) R 3 = log q H(N 3 ) H(2V 1 V 2 ) ( log q H(N1 N 2 N 3 ) ). H(V 1 ) Note that i this case R 2 + R 3 = H(N 1 N 2 N 3 ) H(N 3 ) + (H(V 1 V 2 ) H(2V 1 V 2 )) ( H(V 1 ) log q H(N1 N 2 N 3 ) ). Choose radom variable V 3 such that R 3 = H(V 3 N 3 ) H(N 3 ). Codebook Geeratio: Costruct a family of pairs NQLC s with legth ad parameters m = 1, k 1 = (log q H(N 1 N 2 N 3 )) H(V 1 ), U 1 = V 1, ad U 1 = V 2 by choosig the dither b ad geerator matrix G 1 radomly ad uiformly o F q. For a fixed N, Let C 1 ad C 2 be the correspodig pair of NQLC s. Let Φ i = 2 R i for i {1, 2}. Choose Φ i of the codewords i C i radomly ad uiformly, ad idex these sequeces usig the idices [1, Φ i ]. Also, geerate a ustructured codebook C 3 radomly ad uiformly with rate R 3 based o the sigle-letter distributio P V3. Idex C 3 by [1, 2 R 3 ]. Ecodig: Upo receivig message idex M i ecoder i seds the sequece i C 1 which is idexed M i for i {1, 2}. Let the codewords set by ecoder i, i {1, 2} be deoted by v i G 1 b i. Ecoder 3 seds the codeword i C 3 idexed by M 3. Let the codeword set by the third decoder be deoted by c 3. Decodig: Decoder 1 receives X1 N 1 N 2 N 3. Usig typicality decodig, the decoder ca decode the message as log as k 1 H(V 1 ) log q H(N 1 N 2 N 3 ). Decoder 2 receives X1 X 2 N 2 N 3 = (v 1 v 2 )G 1 b 1 b 2 N2 N 3. It ca decode v 1, v 2 joitly as log as 1) k 1 H(V 1 V 2 ) < log q H(N 1 N 3 ), ad 2) R 1 + R 2 k 1 H(V 1 V 2 ). The first coditio esures that v 1 v 2 ca be recovered with probability of error goig to 0 as. After recoverig v 1 v 2, the decoder eeds to joitly decode v 1, v 2 (for reasos explaied i Lemma 2). This is a oiseless additive MAC problem ad coditio 2 esures errorless decodig. Note that i coditio 2, the coefficiet k 1 is preset sice v 1 is of legth k 1. Also, The term H(V 1 V 2 ) is the capacity of the MAC chael. Decoder 3 receives X1 X 2 X 3 N 3 = (2v 1 v 2 )G 1 b 1 b 2 c 3 N 3. The decoder

8 ca recover 2v 1 v 2 as log as k 1 H(2V 1 V 2 ) < log q H(X 3 N 3 ). The, the decoder subtracts 2X 1 X 2 to get X 3 N 3. It ca decode X 3 as log as R 3 H(V 3 N 3 ) H(N 3 ). It is straightforward to check the rate give at the begiig satisfy all of these bouds. Refereces [1] H. Sato, "The capacity of the Gaussia iterferece chael uder strog iterferece," IEEE Tras. If. Theory, vol. 27, o. 6, pp , Nov [2] M. H. M. Costa, ad A. El Gamal, "The capacity regio of the discrete memoryless iterferece chael with strog iterferece," IEEE Tras. If. Theory, vol. 33, o. 5, pp , [3] T. S. Ha, ad K. Kobayashi, "A ew achievable rate regio for the iterferece chael," IEEE Tras. If. Theory, vol. 27, o. 1, pp , [4] C. Nair, K. Xia, ad M. Yazdapaah, "Sub-optimality of the Ha Kobayashi Achievable Regio for Iterferece Chaels", arxiv preprit arxiv: , 2015 [5] M. Maddah-Ali, A. Motahari, ad A. Khadai, "Commuicatio over MIMO X chaels: Iterferece aligmet, decompositio, ad performace aalysis," Iformatio Theory, IEEE Trasactios o, vol. 54, o. 8, pp , Aug [6] V. Cadambe, ad S. Jafar, "Iterferece aligmet ad degrees of freedom of the k -user iterferece chael," IEEE Tras. o Ifo. Th., vol. 54, o. 8, pp , [7] J. Korer, K. Marto, How to ecode the modulo-two sum of biary sources, IEEE Trasactios o Iformatio Theory, vol. 25, o. 2, pp , [8] A. Padakadla, S.S. Pradha, Achievable rate regio for three user discrete broadcast chael based o coset codes, i IEEE Iteratioal Symp. o Iformatio Theory. IEEE, pp.1277,1281, 7-12 July 2013 [9] A. Padakadla, A.G. Sahebi, S.S. Pradha, A ew achievable rate regio for the 3-user discrete memoryless iterferece chael, Proceedigs of IEEE Iteratioal Symposium o Iformatio Theory, July, 2012 [10] F. Shirai, S.S. Pradha, "A Achievable Rate-Distortio Regio for Multiple Descriptios Source Codig Based o Coset Codes," arxiv: , 2016