Cding schemes fr the binary symmetric channel with knwn interference Giusee Caire Eurecm Institute, France Amir Bennatan Tel-Aviv University, Israel hlm hamai Technin, Israel David Burshtein Tel-Aviv University, Israel Abstract We cnsider a binary symmetric channel with additive binary interference knwn t the transmitter but unknwn t the receiver Deending n whether the interference signal is knwn nncausally r causally, this channel falls in the cases studied by Gel fand and insker and by hannn, resectively, fr which cding therems and single-letter caacity exressins are knwn In this wrk we resent effective cde cnstructins fr bth rblems In articular, we shw that the nn-causal interference knwledge case can be turned int an equivalent binary multileaccess channel, fr which standard suersitin cding and successive decding can be used We discuss als tw alternatives fr the causal interference knwledge case, ne based n time-sharing and the ther based n cding ver a ternary alhabet with nn-unifrm rbability assignment 1 Intrductin Memryless channels with inut, utut and state-deendent transitin rbability where the channel state is iid, knwn t the transmitter and unknwn t the receiver, date back t hannn [1, wh cnsidered the case f state sequence knwn causally, and t usnetsv and Tsybakv [2, wh cnsidered the case f state sequence knwn nn-causally Gel fand and insker [3 rved the caacity frmula " " ) (1) fr the nn-causal case, where is an auxiliary randm variable with cnditinal distributin * and is a deterministic functin f and Fr the causal case with iid state sequence, hannn rved the caacity frmula [1 " (2) which can be btained as a secial case f (1) by restricting the suremizatin t indeendent f [ Frm Csta s Writing n dirty-aer famus title [, cding techniques fr the nn-causally knwn state sequence are generally referenced t as Dirty-aer cding By analgy, the case f causally knwn interference is referred t as Dirty-Tae cding Binary Dirty-aer and Dirty-Tae rblems are relevant in data-hiding subject t a maximum distrtin, where the hst signal is a black and white image, r the least-significant bit-layer f a gray-scale image, and where the signal is received thrugh sme memryless transfrmatin mdeled as a BC Dirty-aer alies when the hst signal is available at the encder befre transmissin, while Dirty- Tae is relevant fr n-line data hiding, where the hst signal is revealed causally t the encder 0 The channel utut is given by where additin is ver the binary field 2 3, is the hst signal, assumed iid with unifrm rbability, 0 is the BC nise, : Bernulli-;, and is the channel inut Due t the additive nature f the channel at hand, we shall refer t the hst signal as interference in the fllwing 09
G ` A cding scheme is defined by a sequence f encding functins < =? A A A C A A A ) D 2 G3 H 2 3 frj, such that fr the infrmatin messagem N A A A C ) and the interference realizatin N 2 G3 the crresnding cdewrd is R < M A A A < M If the transmitter has nn-causal knwledge f < U,then M is allwed t deend n the entire interference signal (Dirty-aer) On the cntrary, if the transmitter has causal knwledge f, then < = M is allwed t deend nly n the interference signal u t timej (Dirty-Tae) The maximum distrtin cnstraint is reflected by the inut cnstraint V X R Y [ (3) wherev X ^ ` ^ -vectrs and dentes Hamming distance between tw 11 The binary Dirty-aer caacity When the encder knws nn-causally, it can be shwn (see [6 and references therein) that the maximum achievable rate is given by a b fr g d [ [ b () fr [ [ d i where the functin i, defined fr N k,isgivenby b i a i ; fr ; [ i [ fr [ i () [ ; where ; dentes the binary entry functin and where we let d w y ; and g { ~ d b d 1 The rate is achieved by Dirty-aer cding, and crresnds t the chice f the auxiliary variable,where is a Bernulli randm variable indeendent f, with rbability It is instructive t sketch the rf f achievability, based n randm cding and randm binning We cnstruct a binary randm cdebk, unifrmly distributed, f size w y { ~ ; and we artitin by randm assignment f its cdewrds t bins (subsets)? A A A C C M ) By chsing w y b, with N k;,frlarge the size f each subset is clse t w y { ~ Bth and the artitin ) are revealed t encder and decder Let be the interference sequence and M be the infrmatin message t be sent The encder finds a cdewrd N such that V X [ Then, the encder sends R The decder bserves the standard BC utut R Œ the subset cntaining Ž ince the cdebk is small enugh, fr sufficiently large f errr M Ž M can be made smaller than b Eventually, the rate can be transmitted ince each subset is large enugh, such sequence can be fund with rbability fr sufficiently large,where is the nise realizatin, finds the unique cdewrdž N jintly tyical with Œ V X (ie, such that Ž Œ [ ; fr sme ) 2 and ututs the message MŽ as the index f the rbability 1 When infrmatin rates are measured in bits, lg and ex are base-2, when it is measured in nats, lg and ex are base- 2 Clearly, a better decder such as the minimum distance decder š œ ž š «achieves the same rate 10
º Y Ï ² ³ Á Ï Á ; with errr rbability nt larger than fr sufficiently large The abve achievability argument can be made rigrus The cnverse is rvided, fr examle, in [6 In the range [ [ d, caacity is achieved by time-sharing with duty-cycle d the abve scheme fr Hamming distrtin equal t d and silence (zer rate and zer Hamming distrtin) We ntice the different rles f and its subsets : the cdebk must be a gd channel cde fr the BC with transitin rbability ; while each subset must be a gd Hamming quantizer fr the interference sequence (a Bernulli iid unbiased surce) with Hamming distrtin Unlike the case f AWGN channel with Gaussian interference studied by Csta [, the binary Dirty-aer caacity is strictly less than the caacity if interference was nt resent, which is given by where we define ² ± ³ µ (6) is always larger than () ³ 12 The binary Dirty-Tae caacity ; ± ; (6) ² Clearly, ; ± fr all, hence In the Dirty-Tae case, hannn s caacity frmula (2) is given exlicitly by { ~ ; () The rf f () is a simle examle f the idea f cding ver strategies underlying hannn s result The auxiliary variable in (2) takes n values in the set ¹ f memryless functins (r strategies ) maing the state int the inut When bth and are binary, then ¹ Y º» ¼ ½ ¾ ), ie, the identically *zer*, identically *ne* functins, identity and negatin, resectively The transitin rbability assignment f the assciated channel with inut and utut,givenby * À Á Â Ä À Á Å Å Å is given by the table À 0 1 Y º id nt ; ; ; Due t the cncavity f caacity as a functin f the inut cnstraint symbl has assciated cst zer, we have [ [ U È É Ê * Ë U Í * Ë Î k V X» ¼ where the suremizatin is achieved by either Á r by Á { ~ ½ ¾ and t the fact that the inut ; () In rder t shw that () is indeed the caacity, we find an inut rbability assignment * such that " equals () The inut cnstraint is given by k V X * Y Ñ * º Ñ *» ¼ * ½ ¾ [ (9) We bserve that the timal inut rbability assignment must ut zer rbability mass n the inut,since * º wuld " increase the inut average Hamming weight» ¼ withut increasing ½ ¾ mutual infrmatin ( ) Mrever, by symmetry, it must be * * (ntice that º 11
; R k Ø Ú Œ Ø Œ 0 06 N interference 1 h(01) = 031 bits 0 C (bit/symbl) 0 03 Dirty aer Dirty Tae 02 01 Functin (W) A 0 0 00 01 01 02 02 03 03 0 0 0 Figure 1: Caacity f the inut-cnstrained BC with causally and nn-causally knwn interference, fr ;» ¼ ½ ¾ this Ychice makes the utut distributin unifrm) Hence, we chse * *,and * fr sme N k The resulting mutual infrmatin is given by " { ~ ; (10) By letting, we btain () Fig A 1 shws the binary Dirty-aer and the Dirty-Tae caacities vs the inut cnstraint fr Causal knwledge f the interference sequence incurs a nticeable caacity lss with resect t nn-causal knwledge 2 uersitin cding fr the Dirty-aer rblem A standard arach t imlement the randm binning scheme cnsists f using nested linear cdes [ Ô, and identifying the bins with the csets f in the artitin Let be a linear binary cde, and be a linear subcde f We have the cset decmsitin Ö = Ø = ) (11) Ë where Ø = is the J -th cset f in Ø =, is a cset leader (ie, a cset reresentative with minimum Hamming weight) We define minimum-distance decding fr a cde as Ù Ú Œ fr any Œ N 2 G3, and the mdul eratr as Hence, k Œ k Œ Ú is the cset leader f the cset arg W Û Ü Ý Þ È Ú V X (12) Ù Ú Œ (13) Œ in the artitin Fig 2 shws the blck diagram f the structured binning scheme (see als [) The encder mas the infrmatin message M Ø int the cset leader f the artitin Then, the transmitted sequence is given by Ú (1) 12
ç ù ù ü á ï î ø ç í ñ û where Ù Ú Ø weight nt larger than The cde is chsen such that the leaders f its csets in 2 G3 have Hamming Therefre, R satisfies the inut cnstraint Cset â ã ä å Maer è é ê ë Decder fr ì Cset Demaer Figure 2: Dirty-aer cding scheme with linear binary cdes The received sequence is given by ð ñ ò ó õ ó ö ñ ø ó ù ó ö ñ ù ó ö (1) Ntice that, by cnstructin, ù ñ ø ó ù is a cdewrd f û Hence, the decder sees a standard BC with nise ö and transmitted cdewrd ù ü û and decdes û by sme decding scheme (nt necessary minimum distance) Let ý be the decding utcme (nt necessarily a cdewrd f û since decding might nt be cmlete) If ý û, then it belngs t sme cset û ó þ f the artitin û ÿ û and the index ý is utut Otherwise, an errr is declared By using randm binary linear cdes (ie, whse arity-check matrix is generated with iid unifrmly distributed elements) it can be easily shwn that there exist sequences f nested linear cdes such that, fr sufficiently large, û has exnentially vanishing errr rbability fr BC arameter and û has Hamming distrtin nt larger than fr an iid unifrmly distributed surce, rvided that û and û [9, 10, Althugh the abve scheme is timal, it resents sme rblems fr ractical cde cnstructin In fact, while û can be decded by any suitable methd, nt necessarily minimum-distance, the mdul û eratin requires minimum-distance decding r, at least, that û admits cmlete decding, inducing a artitin f the whle sace int decisin regins Unfrtunately, minimum-distance decding has exnential cmlexity We are temted t use fr bth û and û sme randmlike cde cnstructin (eg, LDC [, 11, r turb cdes [12), which rved t be very effective in araching caacity f varius binary-inut channels under lw-cmlexity iterative decding based n Belief-ragatin [13 Unfrtunately, while B decding is very effective fr channel cding, it fails miserably fr quantizatin The reasn f this failure is intuitively exlained as fllws: in channel cding the received signal ð ñ ù ó ö is tyically clse t the transmitted cdewrd ò (within a Hamming shere f radius ) It turns ut that, if the cde û has an arriate Tanner grah, then B is able t recver a large fractin f the symbls f ù, achieving vanishing bit-errr rate (BER) in the limit f large blck length if is belw a certain iterative decding threshld!, that deends n the ensemble defining û There exist several cnstructins f ensembles fr which! can be made very clse t with cding rate very clse t On the ther hand, in quantizatin the surce signal õ is unifrmly distributed ver and it is tyically far frm the cdewrds f û It turns ut that B alied t quantizatin yields very r erfrmance, since with high rbability it des nt even cnverge t a cdewrd f û Relaxing the requirement n the quantizatin cde û, and searching fr û in sme ensemble f structured cdes fr which minimum-distance decding can be imlemented with mderate cmlexity, is f little hel In fact, since û must be a linear subcde f û, the Tanner grah f û must satisfy a certain structure thus reventing the use f standard randmlike ensembles (such as LDCs) fr û Because f these shrtcmings, we take a different rute and cnstruct Dirty-aer cdes based n suersitin cding and successive decding Cnsider tw cdes û and, referred t as the quantizatin cde and the auxiliary cde in the fllwing The quantizatin cde is randmly generated with unifrm iid elements, while the auxiliary cde is randmly generated accrding t an iid Bernulli-% rbability distributin, with % The suersitin cde is defined as û ó, 13
( R Œ + + [ [ [ " 30 ie, The encder chses a cdewrd where Ù Ú k? N N ', and sends the sequence Ú Ù Ú The decder receives the signal R N ' ) (16) (1) The errr rbability f this scheme is given by the rbability f the unin f the fllwing events, V X R ) and ( Ž ) Tw bservatins are in rder: 1 ince is unifrmly distributed ver 2 G3 and indeendent f,thenals is unifrmly distributed ver 2 3 By chsing in the ensemble f randm linear cdes with rate { ~ ( we can find a sequence f cdes fr which vanishes exnentially as H ) 2 Again because f the fact that is unifrmly distributed ver 2 G3 and it is indeendent f,the cdewrd is indeendent f Hence, (1) cincides with a multile-access channel (MAC) where tw virtual users with cdebks and ' send indeendent messages by sending the cdewrds and, resectively + + + + ' ' ' - ' ' 0 2 3 0 + - N Given the frmal analgy with the MAC, we cnclude that is als vanishing if the rate air (where is the rate f and is the rate f ) is inside the caacity regin f the MAC This cnditin is sufficient, ie, it yields an achievability result In fact, the decder is nly interested in reliable decding f, while in the MAC bth and must be reliably decded Hwever, as we shall see in the fllwing, fr an arriate chice f the arameter gverning the ensemble f, the additinal cnditin f decding reliably als incurs n lss f timality Mrever, decding f the suersitin cde can be accmlished by (lw cmlexity) successive decding f and We have: Lemma 1 Cnsider the MAC (1) where all variables are in,where is Bernulli-;, and where user 1 has the Hamming weight inut cnstraint The caacity regin is given by all airs satisfying + + + [ ; ± - { ~ ( ; ; (19) rf The rf fllws immediately frm the general MAC caacity We give the details fr the sake f cmleteness Fr any inut rduct distributin 2 2, define the regin 2 given by + + [ " " The caacity regin f the MAC (1) is given by the clsure f the cnvex hull f the unin f all regins 2,frall 2 satisfying the inut cnstraint, ie, fr which 2 satisfies V X k [ - We bserve that " [ ; ± - ; (20) 1
" 0 since the RH in (20) is the caacity f the inut-cnstrained BC (under cnditining with resect t, the cntributin f user 0 can be remved frm the received signal) We bserve als that { ~ [ ; (21) 0 since the RH in (21) is the caacity f the BC Finally, we bserve that " 6 { ~ ; [ ; (22) By letting 2 be Bernulli-- and be Bernulli-, the uer bunds (20), (21) and (22) are simultaneusly achieved ince the resulting regin is clsed and cnvex, n clsure and cnvex hull eratins are needed Nw, we chse - such that - ± ; Frany; [ [ there exist - N k fr which this cnditin hlds Hence, the rate air + { ~ 3, + ; crresnds t a vertex f the MAC caacity regin (see Fig 3) We cnclude that airs f quantizatin and auxiliary cdes and ' can be fund that achieve rates arbitrarily clse t ; under successive decding Namely, we first decde by treating as nise Then, we subtract the decded cdewrd Ž f the first stage frm the received signal and decde ' based n Œ Ž E F 9 : ; =? B C D 9 : ; =? B C I D B C I D? B C D E G Figure 3: Caacity regin f the MAC riginated by the binary dirty-aer channel with suersitin cding Numerical examle We exerimented with the cnstraint W = 03 ver a dirty aer channel with nise = 01 and unifrmly distributed knwn interference The caacity f this channel is 01 Fr the quantizatin cde, we selected a cnvlutinal cde, enabling efficient quantizatin using the Viterbi algrithm The selectin f the cde arameters was guided by the maximum cdewrd weight requirement, ie, the ability t quantize a unifrmly distributed surce ( A O A R ) with distrtin less than The rate-distrtin functin at is and hence we sught lw-rate cnvlutinal cdes at arximately this rate We selected a cde at rate and cnstraint length b U (ie, states) rvided A O in [1 imulatin results indicate that the cde is caable f quantizatin t a distrtin f The channel cding abilities f the cnvlutinal cde are less favrable than its quantizing abilities A W imulatin results A X W indicate that the abve cde is able t crrect a bit errr rate f arximately, instead f as wuld be exected f a randm cde f the same rate Mrever, it rduces a bit errr f arximately 000 This has tw imlicatins fr the design f the auxiliary cde ' First, 1
G ± W Ñ e l l g the cde s weight cnstraint - must satisfy - ± A W; the cnvlutinal decder des nt exceed A A A Z U A W A R W O Z, in rder that the level f nise at the inut t This yields - ecnd, the nise level at the inut t the auxiliary cde must accunt fr the nise rduced by the cnvlutinal cde, ie; [ The auxiliary cde was thus designed fr a BC channel with errr; [ and the cdewrd weight cnstraint - We selected a GQC-LDC (2 \ quantized-cset LDC) cde as suggested in [1 GQC-LDC cdes are btained by taking LDC cdewrds defined ver an enlarged, nnbinary alhabet (2 ^ in ur case) and maing them t the channel alhabet It was shwn [1 that under maximum-likelihd decding, GQC-LDC cdes are caable f achieving the caacity f any discrete-memryless channel Iterative decding f GQC-LDC cdes is discussed in [16 alhabet t the binary digit 1 and the rest t 0 ince the2 ^ symbls are arximately unifrmly distributed in each 2 ^ LDC cdewrd (as shwn in [1), we btain that the nrmalized weight f eacho resulting Z GQC- LDC cdewrd (rduced by the maing t the binary alhabet) A O is arximately, as desired The randm cding caacity f the abve BC (-,; [ ) channel is Using the methds f [1 and [16, we btained a GQC-LDC cde at rate 02 A W a Finally, c simulatin results fr d the abve dirty aer scheme f rate indicate a bit errr rate f at a blck length f (100 simulatins) In the cntext f ur dirty aer cnstructin, we have maed 3 elements f the 2 ^ 3 Cde cnstructins fr the Dirty-Tae rblem ince the caacity-cst functin () is linear in the inut cst, any int can be achieved by time-sharing between { ~ and ; This imlies that cding fr the binary Dirty- Tae rblem reduces t cnventinal cding fr the BC with arameter ; and multilexing with a fixed C zer symbl Namely, fr given and increasing blck length we chse a sequence G ) f { e G -cdes such that their errr rbability ver the standard BC with arameter; satisfies ÜÛ G f g G [ { and their rate satisfies ÜÛ ÜÝ i G e { ~ C { ~ f G j ;,frsme, and { ~ ince the BC caacity is equal t ;,such sequence f cdes exists fr arbitrary e Fr examle, such cdes can be fund in the ensemble f binary linear randm cdes In rder t transmit a cdewrd N, the encder multilexes its symbls with G e dummy additinal symbls, indicated by l, accrding t a redetermined time-sharing atter knwn t the receiver The resulting vectr is, f length ver the ternary alhabet l ) The transmitted signal R is btained by symbl-by-symbl maing f the interference signal accrding t the sequence f functins determined by,as m = a Ü i Á = Å = Á = Ü i Á = (23) Frm the law f large numbers it is immediate t shw that V X R Y e can be made smaller than fr sufficiently large Hence, this scheme achieves rate { ~ ; An alternative arach cnsists f cnstructing cdes ver the ternary alhabet l ) with rbability Hwever, fr finite length and randmlike cde cnstructins this arach des nt yield better results, as shwn by the fllwing randm Ï cding errr exnent analysis The Gallager randm cding errr exnent is given by + È r q Ï t t + ) fr + [ [ Fr the BC with arameter; and unifrm inut } rbability we have } Ï v w y t t { ~ t { ~ z 0 ; z 0 ; (2) The Ï exnent fr the Dirty-Tae channel under the reviusly described time-sharing arach is given by w + Ï v w y + Ï,where v w y + dentes the BC exnent 16
Ï Ï " 012 Errr exnent fr the binary Dirty Tae channel, =001, W=01 01 Ternary cding 00 Time sharing with binary cding E r (R) 006 00 002 0 0 002 00 006 00 01 012 01 R (nat) A A Figure : Randm cding exnents fr the binary Dirty-Tae channel with ; and Fr the randm cding errr exnent achieved by ternary cding ver the alhabet l ) with inut rbability we have t { ~ t { ~ ƒ z 0 } z 0 } z 0 } ; ; (2) It can be shwn that Ï w + + fr all j + { ~ N k ; In ther wrds, standard cding fr the BC (with a unifrm inut rbability) and multilexing the cde wrd with dummy l symbls accrding t a redetermined time-sharing atters uterfrms A direct ternary A cde cnstructin Fig shws the randm cding exnents fr the case ; and This result can be interreted as fllws: under n Hamming weight inut cnstraint, the symbl l shuld nt be used since its cnditinal mutual infrmatin is l We are frced t use it in rder t satisfy the inut cnstraint, since this is the symbl with zer cst Hwever, it is better t use this symbl in redetermined sitins (time-sharing) rather than cnstructing cdes ver the inut alhabet including this symbl Cnclusins We have shwn that the binary Dirty-aer caacity can be arached by suersitin cding and successive decding This makes the cde design simler, since we can searately cnstruct a structured quantizatin cde allwing cmlete minimum-distance decding, and an auxiliary Hammingweight cnstrained randmlike cde ' that can be decded using lw-cmlexity belief ragatin iterative decding The suersitin cding arach uts less cnstraints n the cde cnstructin than the cnventinal nested linear cding arach since it des nt require that ' be the set f cset reresentatives f the artitin fr sme linear suercde f Desite these advantages, ur first attemts t exlicit cnstructing cdes fr the binary Dirty-aer channel were nt fully satisfactry We believe that the reasn fr these results is twfld On ne hand, careful timizatin f the GQC-LDC cde ensemble is called fr On the ther hand, successive decding shuld be relaced by iterative B decding f the whle suersitin cde, as currently rsed in Turb multiuser decding f cded CDMA (see fr examle [1, 1 and references therein) Iterative B decding relaxes the cnstraint n the BER erfrmance f the cnvlutinal Hamming quantizer as a channel cde, which clearly aears t be the main limiting factr reventing ractically gd erfrmances in the cde design examle rerted abve Further results including iterative decding and careful timizatin f the GQC-LDC cmnent cde are rerted in [19 1
Fr the binary Dirty-Tae case we have shwn that cnventinal binary linear cding with timesharing achieves caacity and a generally better errr exnent than the direct (ternary) cding arach This result rvides a theretical justificatin f sme heuristically rsed watermarking/data hiding schemes, where the infrmatin is first encded and then suerimsed t the hst signal accrding t sme redetermined attern (time-sharing) ince in several cases the Dirty-aer and the Dirty-Tae caacities are nt t far aart, such heuristic araches can achieve a large fractin f the maximum achievable data-hiding rate References [1 C hannn, Channels with side infrmatin at the transmitter, IBM J Res Dev, 29 293, 19 [2 A usnetsv and B Tsybakv, Cding in a memry with defective cells, rbl ered Infrm, vl 10, n 2, 2 60, 19 [3 Gelfand and M insker, Cding fr channel with randm arameters, rblems f Cntrl and Infrmatin Thery, vl 9, n 1, 19 31, January 190 [ R Zamir, hamai, and U Erez, Nested linear/lattice cdes fr structured multiterminal binning, IEEE Trans n Infrm Thery, vl, n 6, 120 126, June 2002 [ M Csta, Writing n dirty aer, IEEE Trans n Infrm Thery, vl 29, n 3, 39 1, May 193 [6 radhan, J Chu and Ramchandran, Duality between surce cding and channel cding with side infrmatin, UCB/ERL Technical Memrandum N M01/3, UC Berkeley, Dec 2001 [ Verd, On Channel Caacity er Unit Cst, IEEE Trans n Infrm Thery, Vl 36, N, 1019 1030, etember 1990 [ R Gallager, Lw-density arity check cdes, MIT ress, Cambridge, MA, 1963 [9 V Blinwskii, A lwer bund n the number f wrds f a linear cde in an arbitrary shere with given radius in 2 rbl ered Infrm (rblems f Infrm Trans), Vl 23, N 2, 0 3, 19 [10 R Dbrushin, Asymttic timality f gru and systematic cdes fr sme channels, Ther rbab Al, Vl, 2-66, 1963 [11 T Richardsn and R Urbanke, The caacity f lw-density arity check cdes under message assing decding, IEEE Trans n Infrm Thery, vl, n 2, 99 61, Feb 2002 [12 C Berru, A Glavieux and Thitimajshima, Near hannn limit errr-crrecting cding and decding: Turb-cdes, IEEE Intern Cnf n Cmmun ICC 93, 106-100, Geneva, witzerland, May 1993 [13 ecial issue n iterative decding, IEEE Trans n Infrm Thery, vl, n 2, Feb 2002 [1 Frenger, Orten, T Ottssn, and A venssn, Multirate cnvlutinal cdes, Tech Re 21, Det f ignals and ystems, Cmmunicatin ystems Gru, Chalmers University f Technlgy, Gtebrg, weden, Ar 199 [1 A Bennatan and D Burshtein, On the Alicatin f LDC Cdes t Arbitrary Discrete-Memryless Channels, submitted fr ublicatin IEEE Trans n Infrm Thery Als resented at the Int ym Inf Thery, Ykhama, Jaan, 2003 [16 A Bennatan and D Burshtein, Iterative Decding f LDC Cdes ver Arbitrary Discrete-Memryless Channels, The 1st Annual Allertn Cnference n Cmmun, Cntrl and Cmuting, Mnticell, IL, Oct 1-3, 2003 [1 J Butrs and G Caire Iterative Multiuser Jint Decding: United Framewrk and Asymttic Analysis, IEEE Trans n Infrm Thery, Vl, N, July 2002 [1 G Caire, Guemghar, A Rumy and Verdu, Maximizing the sectral efficiency f cded CDMA, t aear n IEEE Trans n Infrm Thery, 200 [19 G Caire, A Bennatan, D Burshtein and hamai, Cding schemes fr channels with knwn interference, in rearatin, 200 1