Iterative Decoding of Trellis-Constrained Codes inspired by Amplitude Amplification (Preliminary Version)

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1 Iterative Decoding of Trelli-ontrained ode inired by Amlitude Amlification (Preliminary Verion hritian Franck arxiv: v1 [cit] 13 Ar 2019 Abtract We invetigate a novel aroach for the iterative decoding of Trelli-ontrained ode (a uer-cla of errorcorrection code that comrie Turbo-code and LDP code which i inired by the amlitude amlification rincile ued in quantum comuting The idea i to increae the robability of the correct word in every iteration, o that it will eventually tand out from all other word We rooe an aroach that i rovably converging, but the amlification factor can become o mall that the decoding roce ractically tall We how that by heuritically retriggering the decoding roce one can achieve an error-correcting erformance that i imilar to that of looy belief roagation I INTRODUTION The urriing dicovery of Turbo-code [2] in the early 90 wa a major breakthrough in the field of digital communication During the receding decade many exert believed that no ractical error-correction cheme could ever reach Shannon channel caacity [12], but thee code howed that thi i oible by uing a code contruction with two contituent code and an interleaver in conjunction with an iterative decoder baed on belief-roagation [11], [10] Thi lead to the redicovery of LDP code [7] and to the invetigation of more general contruction like Trelli-ontrained ode (T [6], [5] It turned out that only ome clae of T can be can be decoded in a quai-otimal way uing belief roagation, but the otimal decoding of T in general i an unolved roblem In thi aer we rooe a novel aroach in which we iteratively udate the inut vector of the decoder uch that the robability of the correct codeword increae in every te After a ufficient number of iteration, thi robability hould be large enough to ditinguih the correct word uing an ML decoder The idea i imilar to amlitude amlification [3], [4] which i ued in quantum algorithm like Grover earch [8] A in our (non-quantum cae the amlification factor can decreae we rooe a heuritic to retrigger the roce from time to time to avoid a too low convergence Our exeriment how a more controlled roce than belief roagation which achieve a comarable decoding erformance The ret of thi aer i organized a follow In Section II we introduce our notation, in Section III we decribe the decoding algorithm, and in Section IV we dicu ractical conideration In Section V contain our exeriment and we conclude in Section VI (a A T contructed from convolutional code (b A T rereentation of a LDP code Figure 1: Examle of (homogenou trelli-contrained code (The black and gray edge correond to +1 and 1 ymbol II PRELIMINARIES A binary code i a et of binary word of length n with S { 1, +1} n, and it can be rereented uing a trelli grah, in which each ath from the left to the right rereent a codeword c A homogenou trelli-contrained code ( i defined by ( : {c : c (1 and cπ (2 }, where (1, (2 S are two contituent code that have a low trelli comlexity, and where π i a ermutation matrix Some examle are rereented in Figure 1 In non-homogenou T ome ymbol are not contrained To kee the decrition imle we aume homogenou T hereinafter We conider memoryle binary channel where P (r e L(r e L(r + e γ r L(r γ r + γ r γr,

2 with the log-likelihood ratio L(r : 1 2 ln P (r + 1 P (r 1 o that the likelihood that ( 1,, n S wa ent after receiving r (r 1,, r n R n i P (r : i P (r i i i The code-contrained likelihood are then (r : γ rt (1, γ iri γ rt P (2 (r : γ rt π (2, and P ( (r : γ rt (1 π (2, with Iveron bracket [9] defined a fale : 0 and true : 1 More detail on the channel can be found in the Aendix Alo note that the normalization factor omitted in thi decrition are alway uch that the robabilitie um u to 1 III AMPLITUDE AMPLIFIATION DEODING The maximum-likelihood decoding roblem i about finding which i equivalent to P ( (r, (r π (2 or (1 P (2 (r More generally, for w (1, w (2 R n and w (1 + w (2 r, (w (1 P (2 (w (2 Further, we can lit into (1 and (2 and write (1, (2 (w (1 (1 P (2 (w (2 (2 i (1 i (2 i, to how the ymbol contraint (1 i (2 i of word in ( During the decoding roce we can iteratively otimize w (1 and w (2 according to thee ymbol contraint To thi effect, we conider the ymbol-wie rojection Ξ u : (w (1 (1 P (2 (w (2 (2 (1 i (2 i 2u, (1, (2 which i uch that Ξ 1 i the likelihood to have (1 i 1 and (2 i +1, Ξ 0+ i the likelihood to have (1 i (2 i, and i the likelihood to have (1 i +1 and (2 i 1 All word (1, (2 S for which (1 i (2 i do not fulfill the contraint and we want to otimize w (1 and w (2 uch that the likelihood of thee word decreae, while the likelihood (a before otimization (b after otimization (λ Figure 2: The relative robability of Ξ 0 i larger or equal after the otimization of the ymbol robabilitie with λ of the word where (1 i (2 i remain contant We define the relative likelihood of the correct word č a P (r č ρ (1 Ξ 1 + Ξ 0 + Auming Ξ 1 0, we udate w (1,w (2 according to w (1 i w (1 i + log γ λ w (2 i w (2 i log γ λ to obtain a new relative likelihood P (r č ρ λ λ 1 Ξ 1 + Ξ 0 + λ The choice of λ that maximize ρ λ i λ max arg max λ (ρ λ Ξ 1 The relative likelihood of č i augmented by a factor ρ λ ρ Ξ 1 + Ξ 0 + λ 1 Ξ 1 + Ξ 0 + λ The idea i that if the oeration of the reviou ection i erformed over and over again on the variou ymbol contraint, the relative likelihood P (č r hould eventually become o imortant that one can eaily ditinguih č from the ret of the codeword (eg, uing the Viterbi algorithm [13] The effect of a ingle otimization i hown in Figure 2 With Ξ 1 + Ξ 0 + we can write ρ λmax ρ Ξ+1 Ξ 1 Ξ 1 + Ξ 0 + Ξ 1 2 Ξ 1 + Ξ 1 ( Ξ 1 2 and ince > ( Ξ it follow that ρ λ ρ 1, which mean the decoder will alway converge or to The likelihood of every word in ( (in articular the likelihood of the correct codeword č remain contant during the

3 whole decoding roce, and only the um of the likelihood of the word not contained in ( decreae Therefore one can ae the rogre of the decoding roce by oberving the evolution of Ξ 1 + Ξ 0 + during the iteration Remark 1 In the ecial cae where Ξ 1 0 we know that either w (1 i or w (2 i, and we can et w (1 i and w (2 i to w (1 i + w (2 i for thoe ymbol IV PRATIAL ONSIDERATIONS Uing the BJR algorithm [1] one can efficiently comute Ξ 1 i (w (2 + 1, Ξ 0+ i (w (2 1 + i (w (2 + 1, and i (w (2 1 and obtain the value for all ymbol (i {1,, n} at once In Section III we have een how to otimize w (1 and w (2 for ingle ymbol contraint, but it i not clear in which order thee ymbol contraint hould be otimized Therefore we invetigate two aroache, a equential and a arallel one In the equential aroach (AmSEQ in every iteration only the bet ymbol contraint (in term of convergence i otimized Thi mean the ymbol contraint which ha the highet ρ λ /ρ value Thi ha the advantage that one doe not have to conider any oible correlation between contraint, but the diadvantage i that it can oibly take many iteration to find to the correct codeword In the arallel aroach (AmPAR we aume ome degree of indeendence between the ymbol and we otimize all the contraint in arallel In order to avoid too big te we ue a caling factor 0 < κ < 1 and udate the value according to w (1 i w (1 i + κ log γ λ w (2 i w (2 i κ log γ λ in order to avoid too large te During the decoding roce the convergence rate can rogreively low down o that ρ λmax /ρ tend to 1 A heuritic to retrigger the decoding roce i then to ue the ymbol likelihood in Ξ 0 a new inut robabilitie, which mean to udate w (1 and w (2 according to w (l i 1 2 log γ i (w (2 + 1 i (w (2 1 (2 A imle trategy that i alo ued in the following exeriment i to choe a contant t and to erform (2 every t iteration The roblem i that thi oeration i not necearily converging and may caue the decoding roce to fail V EXPERIMENTS In thi ection we examine the behavior of our amlitude amlification and we comare it with looy belief roagation, for tranmiion over a binary ymmetric channel (BS and a binary eraure channel (BE More reciely, we conider decoding with Looy belief roagation (BP, Am amlification decoding, equential (AmSEQ, and Am amlification decoding, arallel (AmPAR During Am amlification decoding the decoding roce i retriggered according to (2 every t iteration In a firt exeriment we comare the evolution of the relative likelihood ρ, defined in (1, when decoding with belief roagation and with amlification decoding We conider a T code of rate 1/3 and length n 108, contructed uing two convolutional code imilar to the one in Figure 1a, where the generator matrix i built by reeating the block B [ In Figure3 we ee a elected decoding roce, where a word received via a BS i uccefully decoded with AmSEQ but not with BP We ee in Figure 3a that the value of ln(ρ i not continuou with belief roagation, while it i continuouly growing with amlitude decoding, excet for the retriggering according to (2 that i erformed every t 100 iteration Here every time the value ln(ρ i ignificantly droing Further, we ee in Figure 3b and 3c the evolution of Ξ 0 /(Ξ 1 +Ξ 0 + for every ymbol during the decoding roce In the light area the robabilitie of Ξ 1 + are negligible comared to Ξ 0, which mean that the correonding ymbol have (1 i (2 i with a high robability In Figure 3c one can ee the effect of retriggering AmSEQ on the ymbol level In a econd exeriment, we look at the number of iteration In Figure 4 we ee a decoding roce of a word from a (4,6 LDP code with n 108 ent over a BS While all decoder find the correct codeword, we ee that BP decode in only 7 iteration, while AmPAR with κ 1/16 take around 140 iteration and AmSEQ need more than 250 iteration We again oberve that value dro during the retriggering which i done every t 100 iteration, but in AmPAR the value are not droing a much a in AmSEQ In a third exeriment, we comare the error-decoding erformance of BP and AmPAR decoding with t 10 and t 100, with a maximum of 1000 iteration for each decoder We reue the T code contruction from the two reviou exeriment with n 1008 A hown in Figure 5a, on the BS channel the BP decoder exhibit the bet erformance, followed by AmPAR with t 100 and then AmPAR with t 10, which mean that here it i better not to retrigger too frequently In Figure 5b on the BE channel we ee that BP and AmPAR with t 10 erform equally well, followed by AmPAR with t 100 In thi cae the decoder with the le frequent retriggering with t 100 eem too lead to a too low convergence ]

4 (a Value of ln(ρ during the decoding roce (BS (a Value of ln(ρ during the decoding roce (BS (b Symbol-wie Ξ 0 /(Ξ 1 + Ξ 0 + with BP (b Symbol-wie Ξ 0 /(Ξ 1 + Ξ 0 + with BP (c Symbol-wie Ξ 0 /(Ξ 1 + Ξ 0 + with AmSEQ Figure 3: Single decoding run of a codeword from a rate 1/3 T code (contructed from two convolutional code with n 108, ent over a BS channel (c Symbol-wie Ξ 0 /(Ξ 1 + Ξ 0 + with AmPAR Figure 4: Single decoding run of a codeword from a T code with n 108 correonding to a LDP (4,6 code, ent over a BS channel VI ONLUDING REMARKS We rooed a novel decoding algorithm that i baed on amlitude amlification For each ymbol we otimize the inut o that the likelihood each word where (1 (2 remain contant, while the overall likelihood of word where (1 (2 decreae So in articular the relative robability of the correct codeword i augmented in every iteration Like in quantum comuting the decoder doe not care whether the ymbol (1 (2 are both 0 or both 1, the relative robabilitie of both tate are increaed One can efficiently comute the required value uing the BJR algorithm and by otimizing the ymbol in arallel the number of iteration can be reduced A the convergence tend to low down, we rooed a heuritic to retrigger the roce from time to time Thi heuritic i not otimal and can caue the overall decoding roce can diverge, o that a challenge for future work i to find better way to handle thi Our exeriment how that we can achieve a imilar error-correction erformance than looy belief roagation

5 [9] DE Knuth Two note on notation American Mathematical Monthly, age , 1992 [10] R J McEliece, D J MacKay, and J-F heng Turbo decoding a an intance of Pearl belief roagation algorithm IEEE Journal on elected area in communication, 16: , Feb 1998 [11] J Pearl Probabilitic Reaoning in Intelligent Sytem: Network of Plauible Inference 1988 [12] E Shannon and W Weaver The mathematical theory of communication The mathematical theory of communication, 27(Setember, 1949 [13] Andrew Viterbi Error bound for convolutional code and an aymtotically otimum decoding algorithm IEEE tranaction on Information Theory, 13(2: , 1967 (a Frame error rate with a BS (b Frame error rate with a BE Figure 5: Error-correction erformance for a rate 1/3 T code baed on convolutional code with n 1008 REFERENES [1] Lalit Bahl, John ocke, Frederick Jelinek, and Joef Raviv Otimal decoding of linear code for minimizing ymbol error rate (corre IEEE Tranaction on information theory, 20(2: , 1974 [2] laude Berrou and Alain Glavieux Near otimum error correcting coding and decoding: Turbo-code IEEE Tranaction on communication, 44(10: , 1996 [3] Gille Braard and Peter Hoyer An exact quantum olynomial-time algorithm for imon roblem In Proceeding of the Fifth Iraeli Symoium on Theory of omuting and Sytem, age IEEE, 1997 [4] Gille Braard, Peter Hoyer, Michele Moca, and Alain Ta Quantum amlitude amlification and etimation ontemorary Mathematic, 305:53 74, 2002 [5] hritian Franck and Uli Sorger Some roertie of homogenou trellicontrained code In th International Symoium on Turbo ode and Iterative Information Proceing (IST, age IEEE, 2016 [6] Brendan J Frey, David J MacKay, et al Trelli-contrained code In Proc of annual Allerton conference on communication control and comuting, volume 35, age Univerity of Illinoi, 1997 [7] R Gallager Low-denity arity-check code Information Theory, IRE Tranaction on, 8(1:21 28, 1962 [8] Lov K Grover A fat quantum mechanical algorithm for databae earch In Proceeding of the twenty-eighth annual AM ymoium on Theory of comuting, age AM, 1996 APPENDIX We conider channel for which P (r γ r, where the value γ i deendent on the channel For examle, the Binary Symmetric hannel (BS where { 1 if r P (r : if r o that L(r r ln 1 + r ln 1 ( 1 2 ln 1 r and ( 1 γ BS ex 2 ln 1 (1 /; the Binary Eraure hannel (BE where 1 if r P (r : if r 0 0 if r o that L(r r ln r ln + r ln 0 1 r and (with 0 : 0 and 0 : 1 γ BE ex ( ; and the Additive White Gauian Noie hannel where P (r : 1 (r 2 ex 2πσ 2σ 2 o that L(r 1 2 ln ( ex and σ 2 r (r 12 2σ 2 12 (ex ln γ AW GN ex ( σ 2 (r σ 2