Irregular Repeat Accumulate Codes 1


 Merryl Simpson
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1 Irregulr epet Accumulte Codes 1 Hu Jn, Amod Khndekr, nd obert McElece Deprtment of Electrcl Engneerng, Clforn Insttute of Technology Psden, CA 9115 USA Eml: {hu, mod, Abstrct: In ths pper we wll ntroduce n ensemble of codes clled rregulr repetccumulte (IA) codes. IA codes re generlzton of the repetccumlute codes ntroduced n [1], nd s such hve nturl lnertme encodng lgorthm. We shll prove tht on the bnry ersure chnnel, IA codes cn be decoded relbly n lner tme, usng tertve sumproduct decodng, t rtes rbtrrly close to chnnel cpcty. A smlr result ppers to be true on the AWGN chnnel, lthough we hve no proof of ths. We llustrte our results wth numercl nd expermentl exmples. Keywords: repetccumulte codes, turbocodes, lowdensty prtycheck codes, tertve decodng. 1. INTODUCTION Wth the hndsght provded by the pst seven yers of reserch n turbocodes nd lowdensty prtycheck codes, one s tempted to propose the followng problem s the fnl problem for chnnel codng reserchers: For gven chnnel, fnd n ensemble of codes wth (1) lnertme encodng lgorthm, nd () whch cn be decoded relbly n lner tme t rtes rbtrrly close to chnnel cpcty. For turbocodes, both prllel nd serl, (1) holds, but ccordng to the recent work by Dvslr, Dolnr, nd Pollr [7], on the AWGN chnnel there ppers to be gp, lbet usully not lrge one, between chnnel cpcty nd the tertve decodng thresholds for ny turbo ensemble. For DPC codes, the nturl encodng lgorthm s qudrtc n the block length, nd from the work of chrdson nd Urbnke [] we know tht for regulr DPC codes, on the bnry symmetrc nd AWGN chnnels there s gp between cpcty nd the tertve decodng thresholds. On the postve sde, however, uby, Shokrollh et t. [3], [4], [8], hve estblshed the remrkble fct tht on the bnry ersure chnnel rregulr DPC codes stsfy (). ecent work by chrdson, Shokrollh nd Urbnke [5] shows 1 Ths pper s to be presented t the Second Interntonl Conference on Turbo Codes, Brest, Frnce, September. Ths reserch ws supported by NSF grnt no. CC , nd grnts from Sony, Qulcomm, nd Cltech s ee Center for Advnced Networkng. tht on the AWGN chnnel, rregulr DPC codes re mrkedly better thn regulr ones, but whether or not they cn rech cpcty s not yet known. In summry, s yet there s no known nosy chnnel for whch the fnl problem hs been solved, lthough reserchers re very close on the AWGN chnnel nd extremely close on the bnry ersure chnnel. In ths pper, we wll ntroduce promsng clss of codes clled rregulr repetccumulte codes, whch generlzes the repetccumulte codes of [1]. After defnng the codes n Secton, nd observng tht they hve smple lnertme encodng lgorthm, n Secton 3, usng the powerful chrsonurbnke method [], we wll prove rgorously tht IA codes solve the fnl problem for the bnry ersure chnnel. In Secton 4, we wll dscuss, less rgorously, the performnce of IA codes on the AWGN chnnel, nd show tht ther performnce s remrkbly good.. DEFINTION OF IA CODES Fgure 1 shows Tnner grph of n IA code wth prmeters (f 1,...,f J ; ), where f, f = 1 nd s postve nteger. The Tnner grph s bprtte grph wth two knds of nodes: vrble nodes (open crcles) nd check nodes (flled crcles). There re k vrble nodes on the left, clled nformton nodes; there re r =(k f )/ check nodes; nd there re r vrble nodes on the rght, clled prty nodes. Ech nformton node s connected to number of check nodes: the frcton of nformton nodes connected to exctly check nodes s f. Ech check node s conected to exctly nformton nodes. These connectons cn mde n mny wys, s ndcted n Fgure 1 by the rbtrry permutton of the r edges jonng nformton nodes nd check nodes. The check nodes re connected to the prty nodes n the smple zgzg pttern shown n the fgure. If the rbtrry permutton n Fgure 1 s fxed, the Tnner grph represents bnry lner code wth k nformton bts (u 1,...,u k ) nd r prty bts (x 1,...,x r ), s follows. Ech of the nformton bts s ssocted wth one of the nformton nodes; nd ech of the prty bts s ssocted wth one of the
2 (Arbtrry permutton) f f 3 f J Check nodes (ll hve left degree ) x 1 x For exmple, the orgnl A codes re nonsystemtc IA codes wth = 1 nd exctly one f equl to 1, sy f q = 1, nd the rest zero, n whch cse () smplfes to = 1/q. (However, n ths pper we wll be concerned lmost exclusvely wth systemtc IA codes.) In n tertve sumproduct messgepssng decodng lgorthm, ll messges re ssumed to be loglkelhood rtos,.e., of the form m = log(p()/p(1)). The outgong messge from vrble node u to check node v represents nformton bout u, nd messge from check node u to vrble node v represents nformton bout u. Intlly, messges re sent from vrble nodes whch represent trnsmtted symbols. The outgong messge from node u tonodev depends on the ncomng messges from ll neghbors w of u except v. Ifu s vrble messge node, ths outgong messge s Informton nodes (f = frcton of nodes of degree ) x r Prty nodes Fgure 1: Tnner grph for IA code wth prmeters (f 1,...,f J ; ). prty nodes. The vlue of prty bt s determned unquely by the condton tht the mod sum of the vlues of the vrble nodes connected to ech of the check nodes s zero. To see ths, let us conventonlly set x =. Then f the vlues of the bts on the r edges comng out of the permutton box re (v 1,...,v r ), we hve the recursve formul x j = x j 1 + v (j 1)+, (1) for j =1,,...,r. Ths s n effect the encodng lgorthm, nd so f s fxed nd n, the encodng complexty s O(n). There re two versons of the IA code n Fgure 1: the nonsystemtc nd the systemtc versons. The nonsystemtc verson s n (r, k) code, n whch the codeword correspondng to the nformton bts (u 1,...,u k )s(x 1,...,x r ). The systemtc verson s (k + r, k) code, n whch the codeword s (u 1,...,u k ; x 1,...,x r ). The rte of the nonsystemtc code s esly seen to be nsys = f, () wheres for the systemtc code the rte s sys = + f (3) m(u v) = w v m(w u)+m (u), (4) where m (u) s the loglkelhood messge ssocted wth u. (Ifu s not codeword node, ths term s bsent.) If u s check node the correspondng formul s [1] tnh m(u v) = w v tnh m(w u). (5) 3. IA CODES ON THE BINAY EASUE CHANNE The sumproduct lgorthm defned n equtons (4) nd (5) smplfes consderbly on the bnry ersure chnnel (BEC). The BEC s bnry nput chnnel wth three output symbols,, 1 nd ersure. The nput symbol s receved s n ersure wth probblty p nd s receved correctly wth probblty. It s mportnt to note tht no errors re ever mde on ths chnnel. It s not dffcult to see tht the messges defned n (4) nd (5) cn ssume only three vlues on the BEC, vz. +, or, correspondng to vrble vlue, 1, or unknown. No errors cn occur durng the runnng of the lgorthm; f messge s ±, the correspondng vrble s gurnteed to be or 1, respectvely. The opertons t the nodes n the grph gven by eqns (4) nd (5) cn be stted much more smply nd ntutvely n ths cse. At vrble node, the outgong messge s equl to ny nonersure ncomng messge, or n ersure f ll ncomng messges re ersures. At check node, the outgong messge s n ersure f ny ncomng messge s n ersure, nd otherwse s the bnry sum of ll ncomng messges.
3 3.1. Notton In ths secton nd the next, t wll be convenent to use slghtly dfferent representton for n IA code thn the one used n Secton. Frstly, we wll begn wth the ssumpton tht the degrees of both the nformton nodes nd the check nodes re nonconstnt, though we wll soon restrct ttenton to the rghtregulr cse, n whch the check nodes hve constnt degree. Secondly, let λ be the frcton of edges between the nformton nd the check nodes tht re djcent to n nformton node of degree, nd let ρ be the frcton of such edges tht re djcent to check node of degree + (.e. one whch s djcent to nformton nodes). We wll use these edge frctons λ nd ρ to represent the IA code rther thn the correspondng node frctons. We defne λ(x) = λ x 1 nd ρ(x) = ρ x 1 to be the genertng functons of these sequences. The pr (λ, ρ) s clled degree dstrbuton. It s qute esy to convert between the two representtons. We demonstrte the converson wth the nformton node degrees. et the f s be s defned n Secton nd let (x) = f x. Then we hve f = (x) = λ / j λ j/j, (6) x λ(t)dt/ λ(t)dt. (7) The rte of the systemtc IA code (we shll be delng only wth these) gven by ths degree dstrbuton s gven by ( j te = 1+ ρ ) 1 j/j j λ (8) j/j (Ths s n esy exercse. For proof, see [8].) 3.. Fxed pont nlyss of tertve decodng In [], t ws shown tht f for code ensemble, the probblty of the depthl neghborhood of n edge (n the Tnner grph) beng cyclefree goes to 1 s the length of the code goes to nfnty (we wll cll ths condton the cyclefree condton), then densty evoluton gves n ccurte estmte of the bt error rte fter l tertons, gn s the length of the codes goes to nfnty. In densty evoluton, we evolve the probblty densty of the messges beng pssed ccordng to the opertons beng performed on them, ssumng tht ll ncomng messges re ndependent (whch s true f the depthl neghbourhood s treelke). The cyclefree condton does ndeed hold for IA codes. The proof of ths fct s lmost exctly the sme s n the rregulr DPC codes cse, whch ws done n []. Now, n the cse of the ersure chnnel, we hve seen tht the messges re only of three types, so n effect we hve dscrete densty functon, nd the probblty of error s merely the probblty of ersure. Wth ths n mnd, we wll now study the evoluton of the ersure probblty, nd derve condtons whch gurntee tht t goes to zero s the number of tertons goes to nfnty. Under these condtons tertve decodng wll be successful n the sense of [],.e., t wll cheve rbtrrly smll BEs, gven enough tertons nd long enough codes. et p be the chnnel probblty of ersure. We wll terte the probblty of ersure long the edges of the grph durng the course of the lgorthm. et x be the probblty of ersure on n edge from n nformton node to check node, x 1 the probblty of ersure on n edge from check node to prty node, x the probblty of ersure on n edge from prty node to check node, nd x 3 the probblty of ersure on n edge from check node to n nformton node. The ntl probblty of ersure on the messge bts s p. We now ssume tht we re t fxed pont of the decodng lgorthm nd solve for x. We get the followng equtons: x 1 = 1 (1 x )(1 x ), (9) x = px 1, (1) x 3 = 1 (1 x ) ρ(1 x ), (11) x = pλ(x 3 ). (1) where (x) s the polynoml n whch the coeffcent of x denotes the frcton of check nodes of degree. (x) s gven by (cf. eq. (7)) (x) = x ρ(t)dt ρ(t)dt (13) We elmnte x 1 from the frst two of these equtons to get x n terms of x nd then keep substtutng forwrds to get n equton purely n x, henceforth denoted by x. We thereby obtn the followng equton for fxed pont of tertve decodng: ( [ ] pλ 1 ρ(1 x)) = x. (14) (1 x) If ths equton hs no soluton n the ntervl (, 1], then tertve decodng must converge to probblty of ersure zero. Therefore, f we hve
4 ( [ ] pλ 1 ρ(1 x)) <x, x. (1 x) (15) then n the sense of [], tertve decodng s successful Cpctychevng sequences of degree dstrbutons We wll now derve sequences of degree dstrbutons tht cn be shown to cheve chnnel cpcty. Frst, we restrct ttenton to the cse ρ(x) =x 1 for some 1, snce t turns out tht we cn cheve cpcty even wth ths restrcton. In ths cse, (x) =x, nd the condton for convergence to zero BE now becomes ( [ ] ) pλ 1 (1 x) (1 x) 1 <x, We now mke the followng new defntons x (16) [ ] f p (x) = 1 (1 x) (1 x) 1 (17) [ ] h p (x) = 1 (1 x) (1 x) (18) g p (x) = h 1 p (x) (19) Notce tht f p (x), h p (x) nd g p (x) re ll monotonc functons n [, 1] nd ttn the vlues t nd 1 t 1. In ddton, h p (x) cn be nverted by hnd (by mkng the substtuton (1 x) = y) nd t cn be shown tht g p (x) hs power seres expnson round wth nonnegtve coeffcents. et ths expnson be g p (x) = g p,x. Now, the condton (16) cn now be rewrtten s whch cn be rewrtten s pλ(f p (x)) <x, x () λ(x) < f p 1 (x) p We mke the followng choce of λ(x): λ(x) = 1 p ( N 1 g p, x + ɛx N ) (1) () where <ɛ<g p,n nd N 1 g p, + ɛ = p. Such choce of N nd ɛ exsts nd s unque snce the g p, s re nonnegtve nd g p, = g p (1) = 1. For ths choce of λ(x), we hve pλ(x) <g p (x) =h 1 p (x) <fp 1 (x) x (3) where the lst nequlty follows becuse f p (x) < h p (x) x. Thus, the condton (1) for BE gong to zero s stsfed nd the degree dstrbutons we hve thus defned yeld codes wth thresholds tht re greter thn or equl to p. We now wsh to compute the rte of these codes n the lmt s to show tht they cheve chnnel cpcty. The rte of the code s gven by eq. (8) whch smplfes to (1 + ( λ /) 1 ) 1 n the rghtregulr cse. Now, lm We lso hve λ = lm lm g p, lm =N N ( N 1 =N g p, + ɛ N ) (4) g p, lm N = (5) where the lst equlty s property of the functon g p (x) nd s lso proved by mnul nverson of h p (x). We therefore hve lm λ = lm g p, = lm = ( 1 g p (x)dx ) h p (x)dx ( ) = x x dx. The ntegrnd on the rght cn be expnded n power seres wth nonnegtve coeffcents, wth the frst nonzero coeffcent beng tht of x. Keepng n mnd tht we re ntegrtng ths power seres, t s esy to see tht +1 < 1 ( ) x x 1 dx h p (x)dx (6) ( ) < x x 1 dx. Both bounds n the bove equton cn be computed esly nd both tend to (1 p)/p n the lmt of lrge. Pluggng ths result nto the formul for the rte, we fnlly get tht the rte tends to 1 p n the lmt of lrge, whch s ndeed the cpcty of the BEC. Thus the sequence of degree dstrbutons gven n eq. () does ndeed cheve chnnel cpcty.
5 3.4. Some numercl results We hve seen tht the condton for BE gong to zero t chnnel ersure probblty of p s pλ(x) <fp 1 (x) x. We lter enforced stronger condton, nmely pλ(x) <h 1 p (x) =g p (x) x nd derved cpctychevng degree sequences stsfyng ths condton. The reson we needed to enforce the stronger condton ws tht h 1 p (x) =g p (x) hs nonnegtve powerseres coeffcents, whle the sme cnnot be sd for fp 1 (x). However, from (6) we see tht enforcng ths stronger condton costs us fctor of 1 /(+1) = 1/(+1) n the rte whch s very lrge for vlues of tht re of nterest, nd therefore the resultng codes re not very good. If, however, fp 1 (x) were to hve nonnegtve power seres coeffcents, then we could use t to defne degree dstrbuton nd we would no longer lose ths fctor of 1/( + 1). We hve found through drect numercl computton n ll cses tht we tred, tht enough terms n the begnnng of ths power seres re nonnegtve to enble us to defne λ(x) by n equton nlogous to eq. (), replcng g p (x) by fp 1 (x). Of course, the resultng code s not theoretclly gurnteed to hve threshold p, but numercl computton shows tht the threshold s ether equl to or very mrgnlly less thn p. Ths desgn turns out to yeld very powerful codes, n prtculr codes whose performnce s n every wy comprble to the rregulr DPC codes lsted n [8] s fr s decodng performnce s concerned. The performnce of some of these dstrbutons s lsted n Tble 1. The threshold vlues p re the sme s those n [8] for correspondng vlues of (IA codes wth rght degree + should be compred to rregulr DPC codes wth rght degree, so tht the decodng complexty s bout the sme), so s to mke comprson esy. The codes lsted n [8] were shown to hve certn optmlty propertes wth respect to the trdeoff between 1 δ/(1 ) (dstnce from cpcty) nd (decodng complexty), so t s very hertenng to note tht the codes we hve desgned re comprble to these. We end ths secton wth bref dscusson of the cse = 1. In ths cse, t turns out tht fp 1 (x) does ndeed hve nonnegtve powerseres coeffcents. The resultng degree sequences yeld codes tht re better thn conventonl A codes t smll rtes. An entrely smlr exercse cn be crred out for the cse of nonsystemtc A codes wth =1 nd the codes resultng n ths cse re sgnfcntly better thn conventonl A codes for most rtes. However, nonsystemtc A codes turn out to be useless for hgher vlues of, s cn be seen by mnully followng the decodng lgorthm for one terton, whch shows tht decodng does not proceed t ll. For ths reson ll the precedng nlyss ws Tble 1: Performnce of some codes desgned usng the procedure descrbed n Secton 3.4. t rtes close to /3 nd 1/. δ s the code threshold (mxmum llowble vlue of p), N the number of terms n λ(x), nd the rte of the code. δ N 1 δ/(1 ) performed for systemtc A codes. 4. IA CODES ON THE AWGN CHANNE In ths secton, we wll consder the behvor of IA codes on the AWGN chnnel. Here there re only two possble nputs, nd 1, but the output lphbet s the set of rel numbers: f the x s the nput, then the output s y =( 1) x + z, where z s men zero, vrnce σ Gussn rndom vrble. For gven nose vrnce σ, our objectve wll be to fnd left degree sequence λ(x) such tht the ensemble messge error probblty pproches zero, whle the rte s s lrge s possble. Unlke the BEC, where we del only wth probbltes, n the cse of the AWGN we must del wth probblty denstes. Ths complctes the nlyss, nd forces us to resort to pproxmte desgn methods Gussn Approxmton Wberg [9] hs shown tht the messges pssed n tertve decodng on the AWGN chnnel cn be well pproxmted by Gussn rndom vrbles, provded the messges re n loglkelhood rto form. In [6], ths pproxmton ws used to desgn good DPC codes for the AWGN chnnel. In ths subsecton, we use ths Gussn pproxmton to desgn good IA codes for the AWGN chnnel. Specfclly, we pproxmte the messges from check nodes to vrble nodes (both nformton nd prty) s Gussn t every terton. For vrble node, f ll the ncomng messges re Gussn, then ll the outgong messges re lso Gussn becuse of (4). A Gussn dstrbuton f(x) s clled consstent [5] f f(x) =f( x)e x for x. The consstency condton mples tht the men nd vrnce stsfy σ =µ. For the sumproduct lgorthm, t hs been shown [] tht consstency s preserved t messge updtes of both the vrble nd
6 check nodes. Thus f we ssume Gussn messges, nd requre consstency, we only need to keep trck of the mens. To ths end, we defne consstent Gussn densty wth men µ to be G µ (z) = 1 e (z µ) /4µ. (7) 4πµ The expected vlue of tnh z for consstent Gussn dstrbuted rndom vrble z wth men µ s then E[tnh z + ]= G µ (z) tnh z dz = φ(µ). (8) It s esy to see tht φ(u) s monotonc ncresng functon of u; we denote ts nverse functon by φ ( 1) (y). et µ (l) nd µ(l) be the mens of the messge from check nodes to vrble nodes on the left (.e., nformton nodes) nd on the rght (.e., prty nodes) t the lth terton. We wnt to obtn expressons for µ (l+1) nd µ (l+1) n terms of µ (l) nd µ (l). A messge from degree nformton node to checknodetthelth terton, s Gussn wth men ( 1)µ (l) + µ o, where µ o s the men of messge m o n (4). Hence f v denotes the messge on rndomly selected edge from n nformton node to check node, the densty of v s From (9) nd (8) we obtn: λ G (l) ( 1)µ +µo(z). (9) E[tnh v ]= λ φ(( 1)µ (l) + µ o). (3) Smlrly, f v denotes the messge on rndomly selected edge from prty node to check node, E[tnh v ]=φ(µ(l) + µ o). (31) Becuse of (5) we hve E[tnh m(u v) ]= E[tnh w v m(w u) ]. (3) Denote messge from check node to n nformton node, resp. prty node, by u, resp, u. eplcng E[tnh m(w u) ] wth the rght sde of (3) or (31) dependng upon whether the messge comes from the left or rght, (3) mples: E[tnh u ]=E[tnh v ] 1 E[tnh v ] = ( λ φ(( 1)µ (l) + µ o)) 1 (φ(µ (l) + µ o)), E[tnh u ]=E[tnh v ] E[tnh v ] = ( λ φ(( 1)µ (l) + µ o)) φ(µ (l) + µ o). Usng the defnton of φ(µ) n (8), we thus hve the followng recurson for µ (l) nd µ(l) : φ(µ (l+1) ) = ( λ φ(( 1)µ (l) + µ o)) 1 (φ(µ (l) + µ o)), (33) φ(µ (l+1) ) = ( λ φ(( 1)µ (l) + µ o)) φ(µ (l) + µ o). (34) In order to hve rbtrry smll bt error probblty, the mens µ (l) nd µ(l) should pproch nfnty s l pproches nfnty. In the next subsecton, we derve suffcent condton for ths. 4.. Fxed pont nlyss We now ssume tht tertve dedodng hs reched fxed pont of (33) nd (34),.e., µ (l+1) = µ (l) = µ nd µ (l+1) = µ (l) = µ. Denote J λ φ(( 1)µ + µ o )byx. From (3) we cn see tht <x<1 nd x 1 f nd only f µ. From (34) t s esy to show tht µ s functon of x, denoted by f,.e., µ = f(x). Then, dvdng (33) by the squre of (34) gves us: φ(µ )=φ (µ )/x +1 = φ (f(x))/x +1. (35) Now replcng µ wth φ ( 1) (φ (f(x))/x +1 ) nto the defnton of x, we obtn the followng equton for the fxed pont x: x = λ φ(µ o +( 1)φ ( 1) ( φ (f(x)) x +1 )). (36) If ths equton doesn t hve soluton n the ntervl [, 1], then the decodng bt error probblty converges to zero. Therefore, f we hve F (x) = λ φ(µ o +( 1)φ ( 1) ( φ (f(x)) x +1 )) >x, (37) for ny x [x, 1), where x s the vlue of x t the frst terton, then (the Gussn pproxmton to) tertve decodng s successful. Snce the rte of the code s gven by (cf. (8)): λ / 1/ + λ /, (38)
7 to mxmze the rte, we should mxmze λ /. Thus, under the Gussn pproxmton, the problem of fndng good degree sequence for IA codes s converted to the followng lner progrmmng problem: ner Progrmmng Problem. Mxmze under the condton λ /, (39) F (x) >x, x [x, 1]. (4) We hve desgned some degree sequences for IA codes usng ths lner progrmmng methodology. The results re presented n Tbles (code rte 1/3) nd 3 (code rte 1/). After usng the heurstc Gussn pproxmton method to desgn the degree sequences, we used exct densty evoluton to determne the ctul nose threshold. (In every cse, the true tertve decodng threshold ws better thn the one predcted by the Gussn pproxmton.) 3 4 λ λ λ λ λ λ λ λ λ 14 λ 16 λ λ rte σ GA σ ( E b N ) (db) S.. (db) Tble : Good degree sequences yeldng codes of rte pproxmtely 1/3 for the AWGN chnnel nd wth =, 3, 4. For ech sequence the Gussn pproxmton nose threshold, the ctul sumproduct decodng threshold, nd the correspondng ( E b N ) n db re gven. Also lsted s the Shnnon lmt (S..) For exmple, consder the = 3 column n Tble. We djust Gussn pproxmton nose threshold σ GA to be to hve the returned optml sequence hvng rte Then pplyng the exct densty evoluton progrm on ths code, we obtn the ctul sumproduct decodng threshold σ =1.67, whch corresponds to E b /N =.5 db. Ths should be compred to the Shnnon lmt for the ensemble of ll lner codes of the sme rte, whch s.4958 db. As we ncrese the prmeter, the ensemble mproves. For = 4, the best code we hve found hs tertve decodng threshold E b /N =.371 db, whch s only.1 db bove the Shnnon lmt. The bove nlyss s for bt error probblty. In order to hve zero word error probblty, t s necessry to hve λ =. (Ths cn be proved by the followng rgument: f λ >, then n the ensemble, s n, the verge number of weght codewords s bounded wy from zero. Hence even mxmumlkelhood decoder would hve nonzero decodng error probblty.) In Tble 3, we compre the nose thresholds of codes wth nd wthout λ =. 8 8 λ λ λ λ λ λ λ λ.7559 λ λ λ λ rte σ ( E b N ) (db) Shnnon lmt Tble 3: Two degree sequences yeldng codes of rte 1/ wth = 8. For ech sequence, the ctul sumproduct decodng threshold, nd the correspondng ( E b N ) n db re gven. Also lsted s the Shnnon lmt. We chose rte onehlf becuse we wnted to compre our results wth the best rregulr DPC codes obtned n [5]. Our best IA code hs threshold.66 db, whle the best rte onehlf rregulr DPC code found n [5] hs threshold.5 db. These two codes hve roughly the sme decodng complexty, but unlke DPC codes, IA codes hve smple lner encodng lgorthm.
8 4.3. Smulton esults We smulted the rte onehlf code wth λ = n Tble 3. Fgure shows the performnce of tht prtculr code, wth nformton block lengths 1 3,1 4, nd 1 5. For comprson, we lso show the performnce of the best known rte 1/ turbo code for the sme block length. BE n=1 Asymptotc Threshold.344 db n=1 n=1 IA code Turbo code SN (db) Fgure : Comprson between turbo codes (dshed curves) nd IA codes (sold curves) of lengths n = 1 3, 1 4, 1 5. All codes re of rte onehlf. 5. CONCUSIONS We hve ntroduced clss of codes, the IA codes, tht combnes mny of the fvorble ttrbutes of turbo codes nd DPC codes. ke turbo codes (nd unlke DPC codes), they cn be encoded n lner tme. ke DPC codes (nd unlke turbo codes), they re menble to n exct chrdson Urbnke style nlyss. In smulted performnce they pper to be slghtly superor to turbo codes of comprble complexty, nd just s good s the best known rregulr DPC codes. In our opnon, the mportnt open problem s to prove (or dsprove) tht IA codes cn be decoded relbly n lner tme t rtes rbtrrly close to chnnel cpcty. We know ths to be true for the bnry ersure chnnel, but for no other chnnel model. If ths should turn out ot be true, we would rgue tht IA codes defntvely solve the problem posed mplctly by Shnnon n If t s not true, then reserchers should serch for n even better clss of code ensembles. EFEENCES [1] D. Dvslr, H. Jn, nd. J. McElece, Codng theorems for turbolke codes, pp. 11 n Proc. 36th Allerton Conf. on Communcton, Control, nd Computng. (Allerton, Illnos, Sept. 1998). [] T. J. chrdson nd. Urbnke, The cpcty of lowdensty prtycheck codes under messge pssng decodng, submtted to IEEE Trns. Inform. Theory. [3] M. uby, M. Mtzenmcher, A. Shokrollh, D. Spelmn, nd V. Stemnn, Prctcl lossreslent codes, Proc. 9th ACM Symp. on the Theory of Computng (1997), pp [4] M. uby, M. Mtzenmcher, A. Shokrollh, nd D. Spelmn, Anlyss of lowdensty codes nd mproved desgns usng rregulr grphs, Proc. 3th ACM Symp. on the Theory of Computng (1998), pp [5] T. J. chrdson, A. Shokrollh,, nd. Urbnke, Desgn of provbly good lowdensty prtycheck codes, submtted to IEEE Trns. Inform. Theory. [6] S.Y. Chung,. Urbnke,, nd T. J. chrdson, Anlyss of sumproduct decodng of lowdensty prtycheck codes usng Gussn pproxmton, submtted to IEEE Trns. Inform. Theory. [7] D. Dvslr, S. Dolnr, nd F. Pollr, Itertve turbo decoder nlyss bsed on Gussn densty evoluton, submtted to IEEE J. Selected Ares n Comm. [8] M. A. Shokrollh, New sequences of lner tme ersure codes pprochng chnnel cpcty, Proc ISITA (Honolulu, Hw, November 1999) pp [9] N. Wberg, Codes nd decodng on generl grphs, dssertton no. 44, nköpng Studes n Scence nd Technology, nköpng, Sweden, [1] J. Hgenuer, E. Offer, nd. Ppke, Itertve decodng of bnry block nd convolutonl codes, IEEE Trns. Inform. Theory, vol. IT 4, no. (Mrch 1996). pp
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