Trapping Set Enumerators for Repeat Multiple Accumulate Code Ensembles

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1 Trapping Set Enumerators for Repeat Mutipe Accumuate Code Ensembes Christian Koer, Aexandre Grae i Amat, Jörg Kiewer, and Danie J. Costeo, Jr. Department of Eectrica Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Emai: {dcoste, ckoer}@nd.edu Department of Eectronics, Institut TELECOM-TELECOM Bretagne, 98 Brest, France Emai: aexandre.grae@teecom-bretagne.eu Kipsch Schoo of Eectrica and Computer Engineering, New Mexico State University, Las Cruces, NM 88, USA Emai: jkiewer@nmsu.edu Abstract The seria concatenation of a repetition code with two or more accumuators has the advantage of a simpe encoder structure. Furthermore, the resuting ensembe is asymptoticay good and exhibits minimum distance growing ineary with bock ength. However, in practice these codes cannot be decoded by a maximum ikeihood decoder, and iterative decoding schemes must be empoyed. For ow-density parity-check codes, the notion of trapping sets has been introduced to estimate the performance of these codes under iterative message passing decoding. In this paper, we present a cosed form finite ength ensembe trapping set enumerator for repeat mutipe accumuate codes by creating a treis representation of trapping sets. We aso obtain the asymptotic expressions when the bock ength tends to infinity and evauate them numericay. I. INTRODUCTION Turbo-ike codes [], as we as LDPC codes [], can perform cose to the Shannon imit using suboptima iterative decoding schemes. However, these codes typicay exhibit an error foor at medium to high signa-to-noise ratios (SNRs. In [], the height of the error foor of LDPC codes was inked to so-caed near codewords. Later, in [4], this concept was generaized to trapping sets, substructures in the Tanner graph of a code that may cause the iterative message passing decoder to fai. For certain LDPC codes, sma trapping sets, rather than the minimum distance of the code, dominate the error foor performance. Asymptotic spectra of trapping sets in LDPC code ensembes were computed in [5] for reguar and irreguar LDPC codes and in [6] for protograph-based codes. It was shown that there exist LDPC codes that exhibit a minimum trapping set size growing ineary with bock ength, for certain types of trapping sets. In turbo-ike codes, the concatenation of simpe component codes through intereavers can ead to powerfu code constructions. The simpest exampes are repeat mutipe accumuate (RMA codes. These codes have a ow encoding compexity of O( and can be decoded using reativey few iterations. This work was party supported by NSF grants CCF5-5, CCF8-666, NASA grant NNX7AK56, and the Marie Curie Intra- European Feowship within the 6th European Community Framework Programme. Furthermore, it has been shown in [7] and [8] that the doube seriay concatenated repeat accumuate accumuate (RAA code of rate / or smaer is asymptoticay good and exhibits minimum distance growing ineary with bock ength. Like LDPC codes, turbo-ike codes are decoded in an iterative fashion. Commony, the component codes are decoded with a maximum a posteriori probabiity (MAP decoding agorithm and the extrinsic information provided by a component decoder functions as a priori information for another. For RMA codes, the turbo decoder can be represented as a message passing decoder [9], simiar to the beief propagation decoder, abeit with a different message passing schedue. Thus the turbo decoder may aso be susceptibe to trapping sets. To predict the error foor of a code one generay needs to have fu knowedge of the trapping sets that dominate the error foor, i.e., one needs to know their graph structure and enumerate their mutipicities, and to find the probabiity that the decoder gets trapped in a particuar set. The atter not ony depends on the graph structure of the trapping set but aso on the channe mode, the decoding agorithm, and the particuar decoder impementation that is used. In this paper we address the first part of the probem, the enumeration of subgraphs in an RAA code. We derive a cosed form trapping set enumerator (TSE for genera (a, b trapping sets, as defined in [4] and [5]. A genera (a, b trapping set for a given Tanner graph is a set of a variabe nodes that induces a subgraph containing b odd degree check nodes, which can be thought of as unsatisfied checks, and an arbitrary number of even degree check nodes. If there are ony a few unsatisfied check nodes and a sufficienty arge number of erroneous variabe nodes, the iterative message passing decoder may not be abe to correct the erroneous nodes. The TSE is the average number of (a, b trapping sets in the ensembe composed of a possibe intereaver reaizations. We aso derive asymptotic expressions for the TSE and anayze them. II. TRAPPING SET ENUMERATORS FOR REPEAT ACCUMULATE ACCUMULATE CODE ENSEMBLES The encoder structure of an RAA code C RAA is shown in Fig.. It is a seria concatenation of a repetition code

2 v k Layer Fig.. code. v k x k- x k x k- x k Bock diagram and factor graph representation of an accumuate Fig.. Bock diagram and factor graph (q = of an RAA encoder. The circes represent variabe nodes and the boxes check nodes, respectivey. C rep of rate R rep = /q and two identica rate-, memory-, accumuate codes C acc, = {, }, with generator poynomias g(d = /( + D, connected by intereavers π and π. For the repetition code C rep, we denote the binary input sequence of ength K by u = [u,..., u K ] and the binary output sequence of ength N by x rep = [x rep,...,xrep N ]. Likewise, for encoder C acc, v = [v,..., v N ] and x = [x,...,x N ] denote the input sequence and the codeword, respectivey, where both are of ength N. Note that v = π (x rep and v = π (x. The overa code rate is R = K/N. The factor graph of an RAA code is aso depicted in Fig. for a repetition factor of q =. The circes represent variabe nodes whie the boxes represent check nodes. The information symbos u correspond to the variabe nodes of the first ayer and their degree is equa to the repetition factor q. The variabe nodes of the second ayer correspond to the output of the first accumuator x, and the variabe nodes of the third ayer to the output of the second accumuator x, respectivey. The input bits of the accumuators are represented by the variabe nodes of the next higher ayer. Ony the variabe nodes in the third ayer are transmitted through the channe. This impies that, initiay, ony variabe nodes in the third ayer can be in error, whie the others have a neutra initia vaue. However, in the first decoding iteration, the variabe nodes in ayers and get vaues assigned based on the received sequence. If there are trapping sets containing variabe nodes in those ayers, erroneous vaues that were assigned during the first iteration may never be corrected and may cause the iterative decoder to fai. Therefore, we consider the whoe graph when enumerating for trapping sets. Let Āa,b CRAA be the ensembe-average TSE of an RAA code ensembe, i.e., the average number of (a, b trapping sets. With reference to Fig., we denote by w the number of information bits that participate in an (a, b trapping set of C RAA. Aso, et a i, ao, and b be the number of variabe nodes corresponding to input bits, the number of variabe nodes corresponding to code bits, and the number of unsatisfied checks, respectivey, of code C acc invoved in an (a, b trapping set of C RAA. To proceed, we must define the trapping set enumerators of the component codes A Crep w,qw and ACacc a i,ao,b, for = {, }. Since there are no check nodes in ayer of the factor graph, A Crep w,qw = ( K w is the input-output weight enumerator (IOWE of the repetition code, giving the number of codewords in C rep of input weight w and output weight qw, whie A Cacc a i,ao,b is the input-output trapping set enumerator (IOTSE of code C acc, denoting the number of trapping sets in C acc consisting of a i input variabe nodes (i.e., variabe nodes corresponding to information bits, a o output variabe nodes (i.e., variabe nodes corresponding to code bits, and b unsatisfied checks. With these definitions, the ensembe average TSE ĀCRAA a,b can be computed using the uniform intereaver concept [] as: Ā CRAA a,b = = w,a o,ao : w+ao +ao =a b,b : b +b =b w,a o,ao : w+ao +ao =a b,b : b +b =b A Crep w,qwa Cacc,bACacc qw,a o a o,ao (,b N qw ( N a o Ā CRAA w,a o,b,ao,b, ( where ĀCRAA w,a is caed the ensembe-average conditiona TSE. o,b,ao,b The evauation of ( requires the computation of A Cacc a i,ao,b, which wi be presented in the next section. The extension of ( to more than two seriay concatenated accumuators is straightforward. III. INPUT-OUTPUT TRAPPING SET ENUMERATOR FOR THE ACCUMULATE CODE In the foowing, we address the computation of the IOTSE A Cacc a i,ao,b of an accumuate code by considering an equivaent treis representation of trapping sets in the factor graph. In Fig., the bock diagram of an accumuate code and a singe section of the corresponding factor graph are depicted. From the figure we obtain the foowing reation: v k = x k + x k. ( Four different -tupes (v k, x k, x k are possibe, namey (,,, (,,, (,,, and (,,, such that the parity check is satisfied. Their factor graph representations are shown in Fig. (a, where back circes represent non-zero symbos and empty circes represent zero symbos. Now consider an (a, b trapping set of C RAA, and assume that (some of the variabe nodes of accumuate code C acc corresponding to (v k, x k, x k participate in the trapping set and cause an unsatisfied check. Again, ony four different configurations are possibe. They are depicted in Fig. (b, where back circes

3 (a (b Fig.. Factor graph representations of an accumuate code. Fig. 4. // // // // // // // // Extended Treis Section. correspond to erroneous symbos and a back box means that the check is unsatisfied. Note that a possibe trapping sets can be obtained by propery combining the eight factor graph sections of Fig.. For enumeration purposes, it is simper to refer to an equivaent treis representation. Assign to the variabe nodes and the check nodes in Fig. that participate in a trapping set (the back circes and boxes the vaue. Then the eight factor graph sections in Fig. can be convenienty represented by the equivaent treis section of Fig. 4. We ca this the extended treis section since it extends the standard treis section of an accumuate code to incude a possibe trapping sets. Each edge between two treis states is abeed with a binary -tupe s i /c/s o, where s i denotes the input symbo, s o denotes the output symbo, and c is if the check node in the corresponding equivaent factor graph representation is unsatisfied. The four abes in back correspond to the four configurations of Fig. (a and define the standard treis section of an accumuate code, whie the four abes in red correspond to the four configurations of Fig. (b. Now the IOTSE of the accumuate code can be computed from the treis representation of Fig. 4 by considering a treis consisting of N concatenated treis sections ike the one in Fig. 4 and enumerating a possibe paths. The IOTSE is given in cosed form in the foowing Theorem. Theorem. Let (a i, a o, b be a trapping set with a i information variabe nodes, a o code variabe nodes, and b unsatisfied checks. The input-output trapping set enumerator (IOTSE for the rate-, memory-, convoutiona encoder C acc with generator poynomia g(d = /( + D, terminated to the a-zero state at the end of the treis, and with input and output bock ength N, can be given in cosed form as: A a i,a o,b = ( ( N a o a o m m m n ( a o ( m N a o ( m m a n m n i b, + m ( where m and n must satisfy the constraints m ai b, m min{a o, N a o }, (4 n ai + b a o, n N a o m. Proof: Consider the extended treis section of the encoder g(d = /( + D in Fig. 4. Denote by n the number of ength-one error events // from the zero state to the zero state, caed type- error events, and by m the number of error events that eave the zero state, and remerge ater to the zero state, caed type- error events. Further, et a i, a o, and b be the number of information variabe nodes, the number of code variabe nodes, and the number of unsatisfied checks, respectivey, participating in the trapping set. Aso, et w t denote the tota input weight associated with the transitions (from state zero to state one and (from state one to state zero in the m type- error events. Ony type- error events are responsibe for the weight at the output of the accumuator. From [], we know that the number of permutations of m type- error events resuting in an output weight of a o is ( ( N a o a o. m m (Here, the transitions away from and back to the zero state are not ony caused by the input weight w t but aso by m w t unsatisfied check nodes. The m type- error events incude a o m transitions from the one state to the one state (, and the input weight associated with the transitions is w = a i n w t. (5 Moreover, the foowing equaity hods: w t = ai b + m. (6 Aso, due to termination, a i + b is even. From (5 and (6 it now foows that w = ai +b n m. This weight can be ordered in ( a o m different ways, which gives the n m third binomia coefficient in (. On the other hand, there are N a o m transitions from the zero state to the zero state with an associated input weight n. Therefore, we obtain the. The ast binomia coefficient in ( resuts term ( N a o m n from the ordering of the w t ones in the m transitions and, in ( m a i b ways. +m To summarize, the number of paths in the extended treis consisting of n type- error events and m type- error events is given by: ( ( N a o m a o m ( a o m n m ( N a o m n ( m a i b + m The resut for the encoder g(d = /( + D foows by summing over a possibe vaues of n and m. Coroary. For b =, the expression in ( reduces to the we-known IOWE for the rate-, memory-, accumuate code []..

4 IV. ASYMPTOTIC ENSEMBLE TRAPPING SET ENUMERATOR In order to determine the asymptotic spectra shape of the trapping sets associated with a particuar code ensembe, as the bock ength N tends to infinity, we define the normaized ogarithmic asymptotic TSE r C (, β of a code ensembe C as r C n (, β = imsup ĀC a,b N N, (7 where = a/n, β = b/n, and the supremum is taken over a intermediate variabes. We aso define the functions f Crep and f Cacc as the asymptotic behavior of the IOWE of a repeat code and the asymptotic behavior of the IOTSE of an accumuate code C acc, respectivey: n A Crep f Crep w,qw (ω = im N N (8 n A Cacc a i,ao,b f Cacc ( i, o, β = im, =,, N N where ω = w/k, i = ai /N, o = ao /N, and β = b /N. Using Stiring s approximation for binomia coefficients ( n n k e nh( n k, where H( is the binary entropy function with natura ogarithms, the functions in (8 can be written as: f Crep (ω = H(ω, (9 q and ( f Cacc ( i, o, β = sup ( o H µ µ,ν o + ( ( + o µ H o + ( o i µ H + β (ν + µ ( o µ + ( ( + ( o µ ν i H o µ + µ H β + µ 4µ ( where we have defined the normaized quantities µ = m /N and ν = n /N. Then, using (8- and ( in (7, the asymptotic TSE of a code ensembe C RAA can be written as: r CRAA (, β = sup =ω/q+ o +o β=+β C f rep (ω + f Cacc (ω, o, + + f Cacc ( o, o, β H(ω H( o, with the constraints = ω q + o + o and β = + β. V. NUMERICAL EVALUATION ( In this section, we present a numerica evauation of (. Foowing [6], in the curves for the asymptotic TSE that we present, we keep the ratio = β/ of unsatisfied check nodes to erroneous variabe nodes constant and compute r(, for varying vaues of. In Fig. 5, the unsatisfied checks in the RAA code ensembe are equay distributed between the midde and inner accumuator, i.e., = β = β/. For =, when no unsatisfied checks are present in the factor graph, the spectra shape r(, exhibits a zero stretch in the beginning and turns positive when the number of codewords with normaized weight starts to grow exponentiay in, r(, =. = β = β / =. =.5 =.. =. =. = Fig. 5. Asymptotic TSE for different vaues of and = β = β/. r(, x / β =. / β =. / β =. / β =.4 / β =.5 / β =.6 / β =.7 / β =.8 / β =.9 / β = =. = Fig. 6. Asymptotic TSE for different fractions /β. N with increasing. The presence of unsatisfied checks in the factor graph resuts in a positive initia sope, and we observe a quasi-inear increasing first section of the curve, unti there is a discontinuity in the sope. In the second section, the sope of the curve is simiar for a vaues of, and the curve shifts to the eft with increasing. Aso, as the fraction of unsatisfied checks increases, the sope in the first section aso increases. Because of the arge number of parameters invoved in taking the supremum in (, it is difficut to draw genera concusions about the trapping set structures that are most ikey to cause decoding faiures. The structure of a trapping set is greaty infuenced by the choice of these parameters. We are primariy concerned with trapping set configurations that ead to decoding errors and this requires ω >. The choice of the parameters and β determines how many unsatisfied checks are associated with the midde and inner accumuator, respectivey. For instance, in the extreme case of = β and β =, a the unsatisfied checks are associated with the midde accumuator, and there are no unsatisfied checks in the graph of the inner accumuator. In Fig. 6 we vary the ratio /β, the fraction of unsatisfied check nodes associated with the midde accumuator in the

5 r(, =. =. =. = β, β = q = 5 q = 4 q = Fig. 7. Asymptotic TSE of the RAA code ensembe for different repetition factors q. Fig. 8. r(, 8 x RAA RAAA RAAAA =.5 q=, = β, β = = Asymptotic TSE for the RAA, RAAA, and RAAAA code ensembes. RAA code ensembe. For arger /β, when reativey more unsatisfied check nodes are present in the midde accumuator, the sope in the first section is smaer and the infuence of varying /β on the sope becomes greater as /β gets coser to one. However, the infuence that varying has on the sope is much greater than the infuence of varying /β. Aso, for arger vaues of, the variance of the curves with /β is greater. In Figs. 7 and 8 we dispay the infuence of the repetition factor q and the number of concatenated accumuators, respectivey, on the shape of the asymptotic TSE. In both cases the greatest effect on the sope in the first section was observed when = β, i.e., when a unsatisfied check nodes are associated with the outermost accumuator. In the other extreme case, when a the unsatisfied check nodes are associated with the inner accumuator, the sope in the first section did not change. In Fig. 7, the reduction in sope in the first section caused by increasing the repetition factor q is ony margina and the curves amost ie on top of each other. However, increasing the number of seriay concatenated accumuators decreases the sope in the first section, as can be seen in Fig. 8. Finay, we note that increasing the repetition factor q or adding more accumuators increases the minimum distance of the code, and thus the transition from the quasiinear section of the asymptotic TSE to the more steepy increasing section takes pace at higher vaues of. VI. CONCLUSIONS We have presented a simpe cosed form method to enumerate genera (a, b trapping sets for RAA code ensembes. The trapping set enumerator is first obtained for finite bock engths N and its asymptotic expression is derived by etting N go to infinity. Simiar to [5] and [6], we observe that, when unsatisfied check nodes are present in the factor graph, the asymptotic TSE ies stricty above the asymptotic spectra shape for the case when no unsatisfied check nodes exist in the graph. Athough the RAA code ensembe is asymptoticay good and exhibits minimum distance growing ineary with bock ength, in contrast to reguar and some protographbased irreguar LDPC codes, there exists no region where the minimum trapping set size grows ineary with bock ength. It can, at best, grow ony subineary in the bock ength, since the asymptotic TSE of the RAA code ensembe is aways positive if unsatisfied check nodes are present in the graph. Whie the method presented in this paper aows us to enumerate a genera (a, b trapping sets, the infuence that these trapping sets have on the error foor must sti be evauated separatey. As noted earier, the probabiity that the decoder gets stuck in particuar types of trapping sets depends on the channe, the decoding agorithm, and the particuar decoder impementation. In future work we hope to evauate this probabiity for the turbo decoder and the beief propagation decoder, in order to obtain a reiabe estimate of the height of the error foor for RMA codes. REFERENCES [] C. Berrou, A. Gavieux, and P. Thitimajshima, Near Shannon imit error-correcting coding and decoding: Turbo-codes, in Proc. IEEE Int. Conf. Commun. (ICC, (Geneva, Switzerand, pp. 64 7, May 99. [] R. G. Gaager, Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, 96. [] D. J. C. MacKay and M. Posto, Weaknesses of Marguis and Ramanujan-Marguis ow-density parity-check codes, Eectronic Notes in Theoretica Computer Science, vo. 74,. [4] T. J. Richardson, Error foors of LDPC codes, in Proc. 4st Annua Aerton Conf. on Commun., Contr., and Comp., pp ,. [5] O. Mienkovic, E. Sojanin, and P. 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