Design of Spatially Coupled LDPC Codes over GF(q) for Windowed Decoding

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1 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 1 Design of Spatially Coupled LDPC Codes over GF(q) for Windowed Decoding Lai Wei, Student Meber, IEEE, David G. M. Mitchell, Meber, IEEE, Thoas E. Fuja, Fellow, IEEE, and Daniel J. Costello, Jr., Life Fellow, IEEE arxiv: v1 [cs.it] 17 Nov 2014 Abstract In this paper we consider the generalization of binary spatially coupled low-density parity-check (SC-LDPC) codes to finite fields GF(q), q 2, and develop design rules for q-ary SC-LDPC code ensebles based on their iterative belief propagation (BP) decoding thresholds, with particular ephasis on low-latency windowed decoding (WD). We consider transission over both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC) and present results for a variety of (J, K)-regular SC-LDPC code ensebles constructed over GF(q) using protographs. Thresholds are calculated using protograph versions of q-ary density evolution (for the BEC) and q- ary extrinsic inforation transfer analysis (for the BIAWGNC). We show that WD of q-ary SC-LDPC codes provides significant threshold gains copared to corresponding (uncoupled) q-ary LDPC block code (LDPC-BC) ensebles when the window size W is large enough and that these gains increase as the finite field size q = 2 increases. Moreover, we deonstrate that the new design rules provide WD thresholds that are close to capacity, even when both and W are relatively sall (thereby reducing decoding coplexity and latency). The analysis further shows that, copared to standard flooding-schedule decoding, WD of q-ary SC-LDPC code ensebles results in significant reductions in both decoding coplexity and decoding latency, and that these reductions increase as increases. For applications with a near-threshold perforance requireent and a constraint on decoding latency, we show that using q-ary SC-LDPC code ensebles, with oderate q > 2, instead of their binary counterparts results in reduced decoding coplexity. This work was supported by the U.S. National Science Foundation under grant CCF Soe of the aterial in this paper was presented at the Inforation Theory and Applications Workshop, San Diego, CA, Feb. 2014, and at the IEEE International Syposiu on Inforation Theory, Honolulu, HI, July L. Wei, D. G. M. Mitchell, T. E. Fuja, and D. J. Costello, Jr. are with the Departent of Electrical Engineering, University of Notre Dae, Notre Dae, IN, 46556, U.S. (e-ail: {lwei1, david.itchell, tfuja, dcostel1}@nd.edu).

2 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 2 Index Ters q-ary spatially coupled low-density parity-check codes, protographs, edge spreading, iterative decoding thresholds, binary erasure channel, q-ary density evolution, binary-input additive white Gaussian noise channel, q-ary extrinsic inforation transfer analysis, flooding-schedule decoding, windowed decoding, decoding coplexity, decoding latency I. INTRODUCTION Low-density parity-check block codes (LDPC-BCs) constructed over finite fields GF(q) of size q > 2 outperfor coparable binary LDPC-BCs [1], in particular when the block length is short to oderate. However, this perforance gain coes at the cost of an increase in decoding coplexity. A direct ipleentation of the q-ary belief-propagation (BP) decoder, originally proposed by Davey and MacKay in [1], has coplexity Ø(q 2 ) per sybol. More recently, an ipleentation based on the fast Fourier transfor [2] was shown to reduce the coplexity to Ø(q log q). Beyond that, a variety of siple but sub-optial decoding algoriths have been proposed in the literature, such as the extended in-su (EMS) algorith [3] and the trellisbased EMS algorith [4]. For coputing iterative BP decoding thresholds, a q-ary extrinsic inforation transfer (EXIT) analysis was proposed in [5] and was later developed into a version suitable for protograph-based code ensebles in [6]. A protograph [7] is a sall Tanner graph, which can be used to produce a structured LDPC code enseble by applying a graph lifting procedure [8] with lifting factor M, such that every code in the enseble is M ties larger and aintains the structure of the protograph, i.e., it has the sae degree distribution and the sae type of edge connections. In this way, the coputation graph [9] is aintained in the lifted graph [7], so BP threshold analysis can be perfored on the protograph. A protograph consisting of (c b) check nodes and c variable nodes has design rate R = b/c and can be represented equivalently by a (c b) c base (paritycheck) atrix B consisting of non-negative integers, in which the (i, j)-th entry (1 i c b and 1 j c) is the nuber of edges connecting check node i and variable node j. Fig. 1 illustrates a (3, 6)-regular protograph and its corresponding base atrix, which can be used to represent a (3, 6)-regular LDPC-BC enseble. To calculate the BP threshold of a protograph-based code enseble, conventional tools are adapted to take the edge connections into account [7], [10]. Although soe freedo is lost in the code design when the protograph structure is adopted,

3 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 3 B = [ 3 3 ] Fig. 1. A (3, 6)-regular protograph and its corresponding base-atrix representation. Black circles correspond to variable nodes and crossed boxes correspond to check nodes. one can use these odified protograph-based analysis tools to find good protograph-based ensebles with better BP thresholds than corresponding unstructured ensebles with the sae degree distribution [10], [11]. Spatially coupled LDPC (SC-LDPC) codes, also known as terinated LDPC convolutional codes [12], are constructed by coupling together a series of L disjoint, or uncoupled, LDPC-BC Tanner graphs. Binary SC-LDPC code ensebles have been shown to exhibit a phenoenon called threshold saturation [13], [14], [15], in which, as the coupling length L grows, the BP decoding threshold saturates to the axiu a-posteriori (MAP) threshold of the corresponding uncoupled LDPC-BC enseble, which, for the (J, K)-regular code ensebles considered in this paper, approaches channel capacity as the density of the parity-check atrix increases [16]. This threshold saturation phenoenon has been reported for a variety of code ensebles (e.g., (J, K)-regular SC-LDPC code ensebles [17], accuulate-repeat-by-4-jaggedaccuulate (AR4JA) irregular SC-LDPC code ensebles [18], bilayer SC-LDPC code ensebles [19], and MacKay-Neal and Hsu-Anastasopoulos spatially-coupled code ensebles [20]) and channel odels (e.g., channels with eory [21], ultiple access channels [22], intersybolinterference channels [23], and erasure relay channels [24]), thus aking SC-LDPC codes attractive candidates for practical applications requiring near-capacity perforance. For a ore coprehensive survey of the literature on SC-LDPC codes, refer to the introduction of [25]. BP decoding threshold results on the BEC for q-ary SC-LDPC code ensebles have been reported by Uchikawa et al. [26] and Pieontese et al. [27], and the corresponding threshold saturation was proved by Andriyanova et al. [28]. In each of these papers, the authors assued that decoding was siultaneously carried out across the entire parity-check atrix of the code; for siplicity, this will be referred to as flooding schedule decoding (FSD) in this paper. Eploying

4 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 4 FSD for SC-LDPC codes can result in large latency, since a large coupling length L is needed to achieve near-capacity thresholds [25]. To resolve this issue, a ore efficient technique, called windowed decoding (WD), was proposed in [29], [30] for binary SC-LDPC codes. Copared to FSD, WD exploits the convolutional nature of the SC parity-check atrix to localize decoding and thereby reduce latency. Under WD, the decoding window contains only a sall portion of the parity-check atrix, and within that window, BP decoding is perfored. In this paper, assuing that the binary iage of a codeword is transitted, we analyze the WD thresholds of a variety of (J, K)-regular protograph-based q-ary SC-LDPC code ensebles constructed fro the corresponding uncoupled q-ary (J, K)-regular LDPC-BC ensebles via the edge-spreading procedure [17], [25], where the finite field size is q = 2 and is a positive integer. In particular, 1) For the BEC, we extend the q-ary density evolution (DE) analysis proposed in [31] to a protograph version and apply this analysis in conjunction with WD to obtain windowed decoding thresholds for q-ary SC-LDPC code ensebles; 2) For the binary-input additive white Gaussian noise channel (BIAWGNC) with binary phaseshift keying (BPSK) odulation, we obtain windowed decoding thresholds for q-ary SC- LDPC code ensebles by applying a protograph-based EXIT analysis (originally proposed for q-ary LDPC-BC ensebles [6]) in conjunction with WD. In both cases, our priary contribution is to deterine how uch the decoding latency of WD can be reduced without suffering a loss in threshold. We observe that 1) Copared to FSD of the corresponding uncoupled q-ary LDPC-BC ensebles, WD of q- ary SC-LDPC code ensebles provides a threshold gain. This gain increases as the finite field size increases. 2) Copared to FSD of a given q-ary SC-LDPC code enseble, WD provides significant reductions in both decoding latency and decoding coplexity, and these reductions increase as the finite field size increases. 3) By carefully designing the protograph structure, using what we call a type 2 edgespreading forat, WD provides near-capacity thresholds for q-ary SC-LDPC code ensebles, even when both the finite field size and the window size are relatively sall. 4) When there is a constraint on decoding latency and operation close to the threshold of a

5 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 5 binary SC-LDPC code enseble is required, using the non-binary counterpart can provide a significant reduction in decoding coplexity. The rest of the paper is organized as follows. Section II describes the construction of protographbased q-ary SC-LDPC code ensebles and reviews the structure of WD. Then Sections III and IV present the WD thresholds of various q-ary SC-LDPC code ensebles for the BEC and the BIAWGNC, respectively, as the finite field size and/or the window size vary. The WD threshold is evaluated fro two perspectives: first, as the window size increases, whether it achieves its best nuerical value when the window size is sall to oderate; second, as the finite field size increases, whether this achievable value approaches capacity. Also, the effects of different protograph constructions on the WD threshold are evaluated and discussed. Finally, Section V studies the decoding latency and coplexity of q-ary SC-LDPC code ensebles and exaines the latency, coplexity, and perforance tradeoffs of WD. In suary, by exaining various q-ary SC-LDPC code ensebles, we bring additional insight to three questions: 1) Why spatially coupled codes perfor better than the corresponding uncoupled block codes, 2) Why windowed decoding is preferred to flooding schedule decoding, and 3) When non-binary codes should be used instead of binary codes. The results of this paper provide theoretical guidance for designing and ipleenting practical q-ary spatially coupled LDPC codes suitable for windowed decoding [32]. II. WINDOWED DECODING OF PROTOGRAPH-BASED q-ary SC-LDPC CODE ENSEMBLES A. Protograph-based q-ary SC-LDPC Code Ensebles A (J, K)-regular SC-LDPC code enseble can be constructed fro a (J, K)-regular LDPC-BC enseble using the edge-spreading procedure [17], [25], described here in ters of protograph representations of the code ensebles. Take J = 3, K = 6 as an exaple. As shown in Fig. 2, instead of transitting a sequence of codewords fro the (3, 6)-regular LDPC-BC enseble independently at tie instants t = 1, 2,..., L, edges fro the variable nodes at tie instant t, originally connected only to the check node at tie instant t, are now spread to also connect to check nodes at tie instants t, t+1,..., t+w; in this way, eory is introduced and the different tie instants are coupled together, i.e., a terinated convolutional, or spatially coupled, coding

6 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) tie t (a) w=2, B 0 = B 1 = B 2 =[1 1] (b) w=1, type 1:B 0=[1 1], B 1 =[2 2] (c) w=1, type 2:B 0=[1 2], B 1 =[2 1] (d) w=1, type 3:B 0=[2 2], B 1 =[1 1] (e) Fig. 2. (a) A sequence of L = 8 uncoupled (3, 6)-regular LDPC-BC protographs, and (b)-(e) various (3, 6)-regular SC-LDPC protographs constructed following the edge-spreading procedure with coupling length L = 8. structure is introduced. The paraeter w is referred to as the coupling width, and L is called the coupling length. Fig. 2 shows three different types of edge-spreading forats for w = 1 and one type for w = 2, all for the case J = 3, K = 6, and L = 8. The above edge-spreading procedure can be described in ters of the base (parity-check) atrix representation of protographs as well. Let B be a (c b) c block base atrix representing an LDPC-BC enseble with design rate R = b/c. Then the base atrix of an SC-LDPC code enseble can be constructed fro B as follows. First, B is spread into a set of (w + 1) coponent base atrices following the rule w B i = B, (1) i=0 so that each B i has the sae size as B. Next, an SC base atrix B SC is generated by stacking and shifting the base coponent atrices {B i } w i=0 at each tie instant t = 1, 2,..., L, thereby

7 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 7 foring a convolutional structure: B 0 B 1 B 0. B 1... B SC =. B w... B0 B w B 1...., (2) where the design rate of B SC is R L = 1 (L + w)(c b) Lc = B w (L+w)(c b) Lc Lb w(c b). (3) Lc Due to the terination of B SC after Lc coluns, there is a loss in the SC-LDPC code enseble design rate R L copared to the rate R = b/c of B. However, this rate loss diinishes as L increases and vanishes as L, i.e., li L R L = R = b/c. Next, a finite-length q-ary SC-LDPC code is constructed fro B SC = [b i,j ] by following the procedure for constructing a finite-length q-ary LDPC-BC fro B: 1) Lifting [7]: Replace the nonzero entries b i,j in B SC with an M M perutation atrix (or a su of b i,j non-overlapping M M perutation atrices if b i,j > 1), and replace the zero entries with the M M all-zero atrix, where M is called the lifting factor. 2) Labeling : Randoly assign to each non-zero entry in the lifted parity-check atrix a non-zero eleent uniforly selected fro GF(q), where q = 2 is the finite field size. After the lifting step, the parity-check atrix is still binary, i.e., the non-binary feature does not arise until the labeling step. 1 The total code length is n = LcM, and we define the constraint length as the axiu width of the non-zero portion of the parity-check atrix ν = (w+1)cm. Both the perutation atrices and the q-ary labels can be carefully chosen to obtain good codes with desirable properties. But constructing specific codes is not the ephasis of this paper; rather, we are interested in a threshold analysis of general q-ary ensebles consisting of all possible cobinations of liftings and labelings of a given protograph, where the diension of the essage odel used in the analysis depends on the size of the finite field [5], [31]. 1 Note that labeling can coe before lifting, resulting in a constrained protograph-based q-ary code as defined in [6].

8 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 8 cw _ (cjjb)w L _ 1 tie t L Fig. 3. WD exaple with window size W = 3: at t = 1 (solid red), t = 2 (dotted blue), and t = 3 (dashed green). J = 3, K = 6, w = 1; B 0 = [1, 1] and B 1 = [2, 2], both of size (c b) c = 1 2, for the B SC given by the protograph construction of Fig. 2(c). For each window position/tie instant, the first c = 2 colun blocks are target sybols. B. Windowed Decoding (WD) In this subsection, we briefly review the structure of WD. By construction, any two variable nodes (coluns of the parity-check atrix) in the graph of an SC-LDPC code cannot be connected to the sae check node if they are ore than a constraint length ν = (w + 1)cM (of coluns) apart. As previously entioned, copared to FSD, where iterative decoding is carried out on the entire parity-check atrix, WD of SC-LDPC code ensebles takes advantage of the convolutional structure of the parity-check atrix and localizes the decoding process to a sall portion of the atrix, i.e., the BP algorith is carried out only for those checks and variables covered by a window. Consequently, WD is an efficient way to reduce the eory and latency requireents of SC-LDPC codes [29], [30]. The WD algorith can be described as follows (see [29] for further details): In ters of the SC base atrix B SC, the window is of fixed size (c b)w cw (recall that the size of the coponent base atrices B i s in B SC is (c b) c) easured in sybols, and slides fro tie instant t = 1 to tie instant t = L, where W, called the window size, is defined as the nuber of colun blocks of size c in the window. An exaple of WD with W = 3 is illustrated in Fig. 3 for the SC-LDPC code enseble whose protograph is shown in Figure 2(c). At each tie instant/window position, the BP algorith runs until a fixed nuber of

9 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 9 iterations has been perfored or soe stopping rule [29], [30], [32] is satisfied, after which the window shifts c colun blocks and those c colun block sybols shifted out of the window are decoded. The first c colun blocks in a window are called the target sybols. We assue that all the variables and checks in a window are updated during each iteration and that, after the window shifts, the final essages fro the previously decoded target sybols are passed to the new window. Clearly, the largest possible W is equal to (L + w), in which case the whole parity check atrix is covered and akes WD equivalent to FSD, and the sallest possible W is (w+1), i.e., the window length (easured in variables) when decoding an SC-LDPC code ust be at least one constraint length. We are interested in searching for q-ary SC-LDPC code ensebles for which a sall window size W can provide WD with a good threshold, which iplies that the coupling width w should be kept sall. Indeed, our results for q-ary SC-LDPC codes together with those in the literature for binary SC-LDPC codes [29], [30] show that ensebles with w = 1 provide the best latency-constrained perforance with WD. C. Code Enseble Construction In this paper, we restrict our attention to (J, K)-regular LDPC code ensebles. 1) (J, K)-regular LDPC-BC ensebles: Let [ ] B = J J J denote the block base atrix corresponding to the protograph representation of a (J, K)-regular LDPC-BC enseble, where K = kj, k = 1, 2,..., and the design rate of the code enseble is R = (k 1) /k. That is, in the reainder of the paper, we let c b = 1 and c = k. We denote the (J, K)-regular LDPC-BC enseble constructed over GF(2 ) as B(J, K, ). 2) Edge spreadings of B: Given a variable node degree J, for a particular coupling width w, define E(J, w) = { [ J 0 J 1 J w ] 1 k (4) } w J i = J, J i {1, 2,..., J w}, (5) i=0

10 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 10 i.e., E(J, w) is the set of all possible colun vectors of length (w + 1) satisfying the constraint w i=0 J i = J, where J i {1, 2,..., J w}. Moreover, define B w 0 as i.e., B w 0 B w 0 = B 0 B 1. B w (w+1) k, (6) is the stack of all the coponent base atrices {B i } w i=0. Then an edge-spreading forat can be generated by selecting k eleents (with replaceent) fro E(J, w) as the k coluns of B w 0. Recall fro Section II-B that our ajor interest lies in q-ary SC-LDPC code ensebles for which windowed decoding (WD) achieves good thresholds under tight latency constraints, i.e., for a sall window size W, which iplies that the coupling width w should be sall. Therefore, we do not allow w to exceed (J 1), i.e., the block base atrix B should be spread into at ost J coponent base atrices B i. In other words, for E(J, w) in (5), we consider only values of w in the range 1 w J 1. The edge-spreading forat B w 0 deterines the SC base atrix B SC, and the q-ary WD thresholds depend on B SC. For a given B w 0, colun perutations do not affect the WD threshold, but row perutations do. Consequently, for each cobination of J and w, there will be E(J, w) (1 + E(J, w) ) /2 possible edge-spreading forats that can result in diffferent WD thresholds. For exaple, consider the (4, 8)-regular degree distribution with J = 4 and w = 2. Then E(4, 2) = {[ ] [ ] [ ] } 1 1 2, 1 2 1, 2 1 1, (7) and the E(4, 2) (1 + E(4, 2) ) /2 = 6 possible edge-spreading forats that can give different WD thresholds are given by B w 0 1 1, 1 2, 1 1, 2 2, 2 1, 1 1. (8) ) (J, K)-regular SC-LDPC code ensebles: We now detail the particular constructions of SC-LDPC code ensebles considered in the reainder of the paper. The first construction we

11 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 11 consider is the classical edge spreading [13] of the (J, K)-regular LDPC-BC base atrix B given by (4), where K = kj and w = J 1: B 0 = B 1 = = B w = [ ] (9) 1 k Unless noted otherwise, the coupling length for all the q-ary SC-LDPC code ensebles in this paper is taken to be L = 100, in order to keep the rate loss sall. Consequently, we do not include L in the enseble notation, and we denote as C J 1 (J, K, ) the SC-LDPC code enseble constructed over GF(2 ) using the coponent atrices B 0, B 1,..., B w given by (9) in the base atrix B SC given by (2), with coupling width w = J 1. As noted previously, under tight latency constraints, the WD threshold can be iproved by using sall w; in fact, excellent WD perforance has been shown for binary SC-LDPC code ensebles using repeated edges in the protograph and w = 1 [29], [30]. In the case of q- ary SC-LDPC code ensebles, we have also found that the case w = 1, i.e., the set of edge spreadings E(J, w = 1) = 1, 2,..., J 1 J 1 J 2 1, (10) results in the best thresholds for low latency WD. Moreover, if we further restrict our attention to the edge-spreading pair E A = 1, E B J 1 E(J, 1), (11) J 1 1 we obtain the ost interesting and representative constructions copared to the other possible selections of colun vectors fro E(J, 1). Cobining E A and E B, there are (k+1) possible choices for B w=1 0. An edge-spreading forat is called type-p if there are (k p + 1) coluns of E A in B 1 0 followed by (p 1) coluns of E B, i.e., B 1 0 = = [ ] E A E A E B E B }{{}}{{} p 1 k p J 1 } {{ J 1 } k p+1 J 1 J 1 = B 0, (12) 1 1 B 1 }{{} p 1

12 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 12 where 1 p k + 1. Again, note that the ordering of coluns is not iportant, because this siply results in colun perutations of the resulting base atrix B SC and does not change the code or graph properties. We again oit L fro the enseble notation and denote as C 1 (J, K,, p) the type-p SC-LDPC code enseble constructed over GF(2 ) using coponent atrices B 0 and B 1 to for B SC, with coupling width w = 1, where 1 p k + 1. For a particular (J, K) pair and Galois field GF(2 ), we refer inforally to the collection of ensebles {B(J, K, ), C J 1 (J, K, ), C 1 (J, K,, p) p = 1, 2,..., k + 1} (13) as the (J, K, ) ensebles, and we further refer to the collection of ensebles { C J 1 (J, K, ), C 1 (J, K,, p) p = 1, 2,..., k + 1} (14) as the (J, K, ) SC ensebles. For exaple, for an arbitrary, let (J, K) = (3, 6). In this case k = 2, and we consider the classical edge spreading with w = J 1 = 2 along with k + 1 = 3 types of edge spreading with w = 1, viz.: [ ] C 2 (3, 6, ): B 0 = B 1 = B 2 = [ ] 1 1 ; [ ] C 1 (3, 6,, 1): B 0 = 1 [ 1, B 1 = 2 ] [ 2 ; ] C 1 (3, 6,, 2): B 0 = 1 [ 2, B 1 = 2 ] [ 1 ; ] C 1 (3, 6,, 3): B 0 = 2 2, B 1 = 1 1. These four ensebles for the (3, 6, ) SC ensebles, and together with B(3, 6, ) they for the (3, 6, ) ensebles. Fig. 2 shows each of the (3, 6, ) ensebles with coupling length L = 8 and arbitrary. III. THRESHOLD ANALYSIS OF q-ary SC-LDPC CODE ENSEMBLES ON THE BEC A. Protograph Density Evolution (DE) for q-ary LDPC Code Ensebles on the BEC The q-ary DE algorith presented in [31] was originally derived for randoized uncoupled q-ary LDPC-BC ensebles where 1) the sybol set is the vector space GF 2 over the binary field, and 2) the edge labeling set is the general linear group GL 2 of diension over the binary field, which is the set of all invertible atrices whose entries are in {0, 1}. The thresholds of these code ensebles, as pointed out by the authors of [31], are very good approxiations to those of q-ary LDPC-BC ensebles defined over GF(2 ), since the nuerical difference is on the order of 10 4.

13 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 13 Consider an ordered list of the eleents of GF 2, and assue that the zero eleent is in the 0th position of the list. For a specific code, a probability doain essage vector in q-ary BP decoding is of length 2, where the entry at position i corresponds to the a posteriori probability that the sybol is the i-th eleent fro GF 2. Since transission is on the BEC and it can be assued that the all-zero codeword is transitted without affecting decoding perforance [31], all the non-zero eleents in the essage vector ust be equal; in fact, the set of sybols (eleents fro GF 2 ) whose a posteriori probabilities are non-zero fors a subspace of GF 2, and the essage vector is said to have diension n if it contains 2 n non-zero eleents, n = 0, 1,...,. Consequently, for the purpose of q-ary DE, which is concerned only with asyptotic ensebleaverage properties rather than decoding a specific finite-length code, only the diension of the BP decoding essage vector needs to be tracked by the algorith. As a result, a q-ary DE essage vector for the BEC can be represented by a vector of length ( + 1), whose n-th entry, n = 0, 1,...,, indicates the a posteriori probability that the BP decoding essage vector has diension n. Siilar to the procedure used to extend q-ary EXIT analysis to a protograph version in [6], we now extend the q-ary DE algorith to a protograph version, which we refer to as q-ary protograph DE (PDE). Since the edge connections are taken into account and the coputation graph is equal for all ebers of the enseble, PDE reduces to the BP algorith perfored on the protograph. We use notation siilar to that in [6] and [28]. Let b i,j denote a non-zero entry in the base atrix and recall that, fro the perspective of the protograph, the value of b i,j is the nuber of edges connecting check node i (the row index in the atrix) to variable node j (the colun index), rather than an edge label. Let N(i) (resp. M(j)) denote the neighboring variables (resp. checks) of check i (resp. variable j). Let p (l) C (i, j) (resp. p(l) V (i, j)) denote the check-i-to-variable-j (resp. variable-j-to-check-i) q-ary DE essage vector during iteration l. Finally, let the erasure probability of the BEC be ɛ. Then the q-ary PDE algorith consists of four steps as follows: Initialization: for each b i,j > 0, let p (0) V (i, j) = p(0) V (j) = p(0) (ɛ), (15) where p (0) (x) is a vector of length ( + 1) in the probability doain, whose n-th entry is

14 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 14 defined as ( ) x n (1 x) n. (16) n Check-to-variable update: the essage vector fro check i to variable j is [ ( )] ( ) p (l) C (i, j) = s N(i)\j b i,s p (l 1) V (i, s) bi,j 1 p (l 1) V (i, s), (17) where the notation (see Appendix A of [28] for details) is described as follows. For two q-ary DE essage vectors p 1 and p 2, p 1 p 2 has n-th eleent n n Ci,j,np 1,i p 2,j, (18) i=0 j=n i where p 1,i is the i-th eleent of p 1, p 2,j is the j-th eleent of p 2, C i,j,n = G i, ng i,n j 2 (n i)(n j) G, j (19) is the probability of choosing a subspace (of GF 2 ) of diension j whose su with a subspace of diension i has diension n, and 1 if k = or k = 0, k l G,k = if 0 < k <, 2 k 2 l l=0 0 otherwise, is the Gaussian binoial coefficient, the nuber of different subspaces of diension k of GF 2. Finally, b i,j 1 p = p p... p, with (b i,j 1) occurrences of p. Variable-to-check update: the essage vector fro variable j to check i is [ ( )] ( ) p (l) V (i, j) = p(0) V (j) s M(j)\i b s,j p (l) C (s, j) bs,j 1 p (l) C (s, j), (21) where p 1 p 2 has n-th eleent and i=n i+n j=n (20) V i,j,np 1,i p 2,j, (22) V i,j,n = G i,ng i,j n 2 (i n)(j n) G,j (23) is the probability of choosing a subspace of diension j whose intersection with a subspace of diension i has diension n (again, see Appendix A of [28] for details).

15 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 15 Convergence check: the a-posteriori essage vector for variable j is [ ( )] p (l) V, APP (j) = p(0) V (j) i M(j) b i,j p (l) C (i, j). (24) The q-ary PDE algorith ends when Either a decoding success is declared: for all the variables to be decoded, the 0th entry of each p (l) V, APP (j) (denoted as p(l) V, APP (j)[0]) is at least (1 δ), i.e., p(l) V, APP (j)[0] 1 δ, where δ [0, 1] is a preset erasure rate, Or a decoding failure is declared: the algorith reaches soe axiu nuber of iterations. The paraeter δ should be chosen sall enough so that it is essentially certain that q-ary PDE has converged if the condition is satisfied. 1) Flooding-Schedule Decoding (FSD) Thresholds for q-ary SC-LDPC Code Ensebles: Given characterizing the sybol set and ɛ characterizing the BEC, if q-ary PDE is perfored over the entire base atrix B SC of an SC-LDPC code enseble, then the algorith deterines asyptotically (i.e., for coupling length L and lifting factor M ) whether FSD can be successful on an enseble average basis for that specific BEC. Thus, q-ary PDE can be used to calculate the FSD threshold, denoted ɛ (, δ), which is the largest channel erasure rate such that all transitted sybols can be recovered successfully with probability at least (1 δ), as the nuber of iterations l goes to infinity, i.e., { } ɛ (, δ) = sup ɛ [0, 1] p (l) V, APP (j)[0] 1 δ for 1 j kl, as l. (25) The following nuerical FSD threshold results on the BEC are obtained for δ = 10 6, and fro this point forward ɛ (, δ) will be denoted siply as ɛ (). 2) Windowed Decoding (WD) Thresholds for q-ary SC-LDPC Code Ensebles: We also apply q-ary PDE to WD in order to calculate the WD threshold of an SC-LDPC code enseble defined over GF(2 ). The q-ary WD-PDE algorith consists of perforing q-ary PDE for all the window positions/tie instants t = 1, 2,..., L, as illustrated in Fig. 3. For each window position, q-ary PDE is perfored within the W kw window; however, unlike the case of FSD, now the convergence check involves only the target sybols, i.e., the first k sybols in the window. Starting fro t = 1, if q-ary PDE declares a decoding failure, then the whole q-ary WD-PDE terinates and

16 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 16 declares a decoding failure; otherwise, the window slides forward and q-ary PDE is perfored for the next window position. The q-ary WD-PDE algorith declares a decoding success for a specific BEC if and only if its coponent q-ary PDE declares decoding successes for all the window positions. Thus, given, ɛ, and W, q-ary WD-PDE can be used to calculate the WD threshold of an SC-LDPC code enseble. We now define ɛ WD(, W, t, δ) = sup ɛ [0, 1] p (l) V, APP (j)[0] 1 δ for tk k + 1 j tk, (26) as l as the largest channel erasure rate such that all the target sybols in window position t can be recovered successfully with probability at least (1 δ), as l goes to infinity, given that all the target sybols in the previous (t 1) windows have already been recovered successfully with probability at least (1 δ). Then the WD threshold ɛ WD(, W, δ) is the infiu of {ɛ WD(, W, t, δ)} L t=1, i.e., ɛ WD(, W, δ) = inf 1 t L ɛ WD(, W, t, δ), (27) guaranteeing that all the transitted sybols consisting of all the target sybols in all the windows can be recovered successfully with probability at least (1 δ), as l goes to infinity. It was proved in Proposition 1 of [29] that the WD thresholds of binary SC-LDPC code ensebles on the BEC are non-decreasing with increasing W, i.e., ɛ WD(1, W, δ) ɛ WD(1, W + 1, δ) for any δ [0, 1] and all W, W = w + 1, w + 2,..., w + L. By cobining this proof with the onotonicity of q-ary variable and check node updates, proved in Appendix B of [28], we can state the following theore. Theore 1 (Monotonicity of ɛ WD(, W, δ) with increasing W ): For a fixed 1, any δ [0, 1], and all W, W = w + 1, w + 2,..., w + L, ɛ WD(, W, δ) ɛ WD(, W + 1, δ). (28) As in the case of FSD thresholds, we choose δ = 10 6, and fro this point forward ɛ WD(, W, δ) will be denoted siply as ɛ WD(, W ). B. Nuerical results: k = 2 (R = 1/2) In this subsection we focus on the BP thresholds of the rate R = 1/2 q-ary SC-LDPC code ensebles with k = 2: in particular, we consider the (2, 4)-, (3, 6)-, (4, 8)-, and (5, 10)-regular

17 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 17 code ensebles. Our ephasis is on the scenario when WD is used, and the q-ary WD-PDE algorith described in the previous subsection is adopted to calculate the corresponding BP thresholds. Recall fro Section II-C that, for k = 2, the SC-LDPC code ensebles we consider are the following: [ ] C J 1 (J, K, ) : B 0 = B 1 = = B J 1 = 1 1 ; (29) [ ] [ ] C 1 (J, K,, 1) : B 0 = 1 1, B 1 = J 1 J 1 ; (30) [ ] [ ] C 1 (J, K,, 2) : B 0 = 1 J 1, B 1 = J 1 1 ; (31) [ ] [ ] C 1 (J, K,, 3) : B 0 = J 1 J 1, B 1 = 1 1. (32) The classical edge spreading results in the axiu coupling width w = J 1 by choosing [ ] each B i in B w 0 equal to 1 1. When w = 1, the type 1 and type 3 ensebles, C 1 (J, K,, 1) and C 1 (J, K,, 3), will have the sae FSD threshold ɛ (), since their SC base atrices are equal up to row perutations and the q-ary PDE algorith is perfored over the entire base atrix B SC. However, their WD thresholds are different. Type 2 has one colun of B w 0 that is the sae as type 1 and the other colun that is the sae as type 3, so it is expected that its WD threshold will be between those of types 1 and 3. 1) The (2, 4) ensebles: When (J, K) = (2, 4), all four types of edge spreading for q-ary SC-LDPC code ensebles are the sae. For = 1, 2,..., 10, the FSD and WD thresholds are shown in Fig. 4: Coparing C 1 (2, 4, ) to B(2, 4, ), the iproveent in the FSD threshold ɛ () introduced by the spatially coupled structure is negligible for sall. However, as increases, ɛ () for C 1 (2, 4, ) increases and approaches the BEC capacity of a rate R = 1/2 code enseble. 2 This is consistent with the observations ade in [26]. We note that the B(2, 4, ) ensebles do not display this behavior; in particular, their thresholds diverge fro capacity as increases, 5. For WD of C 1 (2, 4, ) with fixed, the threshold ɛ WD(, W ) iproves as the window size W increases see Theore 1 in Section III-A and saturates nuerically to a (axiu) 2 Since L = 100, the design rate of C 1(2, 4, ) is R L = and capacity is ɛ Sh = = This gap to capacity vanishes as L, since the thresholds do not further decay and R L 1/2.

18 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) ǫ () or ǫ WD(,W) Block: FSD SC: FSD SC: 10 SC: 12 SC: 15 SC: W = Fig. 4. FSD thresholds ɛ () and WD thresholds ɛ WD(, W ) of C 1(2, 4, ). constant value ˆɛ WD (). Thus, we define W () = in {W ɛ WD(, W ) = ˆɛ WD ()} (33) as the sallest window size that provides the best threshold ˆɛ WD () for a fixed ; here, = is used to denote a nuerically indistinguishable equality. 3 We now ake three observations regarding the enseble C 1 (2, 4, ): For all, ˆɛ WD () = ɛ (), i.e., when the window size W is large enough, the WD threshold equals the FSD threshold. As increases, W () is non-increasing, i.e., increasing the finite field size can speed up the saturation of ɛ WD(, W ) to ˆɛ WD () as W increases. The saturation of ɛ WD(, W ) to ˆɛ WD () is relatively slow as W increases, especially when is sall. For exaple, when = 1, we need a window size of W (1) = 30 to obtain the threshold ˆɛ WD (1). Moreover, even for a fairly large window, say W = 20, the WD threshold of C 1 (2, 4, ) is worse than the FSD threshold of B(2, 4, ) for = 1, 2, and 3. This indicates that C 1 (2, 4, ) does not perfor well unless W and/or are large. 3 For our purposes, two thresholds are nuerically indistinguishable if their absolute difference is no ore than 10 6.

19 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) ǫ () B(3,6,) C 1 (3,6,,1) C 1 (3,6,,2) Fig. 5. FSD thresholds ɛ () coparison of the (3, 6, ) ensebles. As a result, we conclude that C 1 (2, 4, ) is not a good candidate for WD, since a desirable q-ary SC-LDPC code enseble should provide a near-capacity threshold when both the finite field size and the window size are relatively sall, resulting in both sall decoding latency and sall decoding coplexity details will be discussed later in Section V. We will see in the reainder of this section, however, that increasing the node degrees in the code graph speeds up the saturation of ɛ WD(, W ) to ˆɛ WD (). 2) The (3, 6) ensebles: As a benchark, Fig. 5 copares the FSD thresholds of ensebles C 1 (3, 6,, 1) (and thus C 1 (3, 6,, 3)) and C 1 (3, 6,, 2) to that of B(3, 6, ) for various. 4 It is observed that: For all finite field sizes 2, the introduction of the spatially coupled structure provides all four q-ary SC-LDPC code ensebles with significant iproveent in the FSD threshold copared to the corresponding q-ary LDPC-BC enseble. In fact, the gap between B(3, 6, ) and the (3, 6, ) SC ensebles increases as increases. Again, this is consistent with the observations ade in [26] and [27]. Note that, like the B(2, 4, ) ensebles discussed above, the B(3, 6, ) thresholds diverge 4 Code ensebles C 2(3, 6, ) are not included in Fig. 5 because their thresholds are alost indistinguishable (although slightly different) fro those of C 1(3, 6,, 1).

20 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) Capacity 1 5 = Capacity 1 2 = ǫ () ǫ () L (a) C 1(3, 6,, 2) L (b) C 2(3, 6, ). Fig. 6. FSD thresholds ɛ () of the SC-LDPC code ensebles with different coupling lengths L. fro capacity as increases, while the FSD thresholds of C 1 (3, 6,, 1) and C 2 (3, 6, ) increase and approach the BEC capacity for rate R = 1/2 as increases. Surprisingly, this is not the case for C 1 (3, 6,, 2), whose FSD threshold increases and approaches capacity for = 5, but then decreases slowly and thus diverges slightly fro capacity as increases further. As a result, in Fig. 5, there exists a sall gap between the thresholds of C 1 (3, 6,, 1) and C 1 (3, 6,, 2) for large. We now briefly deonstrate the FSD threshold behavior of SC-LDPC code ensebles for varying coupling lengths L. Fig. 6 shows the FSD thresholds ɛ () for ensebles C 1 (3, 6,, 2) and C 2 (3, 6, ) with increasing L. For fixed and increasing L, the FSD thresholds initially decrease and then saturate to a constant value for sufficiently large L, which is consistent with results for binary protograph-based SC-LDPC code ensebles [13], [25]. Note that Figure 6 also illustrates the point ade above that the C 1 (3, 6,, 2) enseble does not have onotonically increasing thresholds with. Specifically, in Fig. 6(a), for C 1 (3, 6,, 2), we have ɛ (1) < ɛ (5) but ɛ (10) < ɛ (5), while in Fig. 6(b), for C 2 (3, 6, ), ɛ () increases uniforly as increases: this confirs our observation of the sall gap between the FSD thresholds of C 1 (3, 6,, 1) (alost indistinguishable fro C 2 (3, 6, )) and C 1 (3, 6,, 2) for

21 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 21 large noted in Fig. 5. With reference to Fig 6, given, let L () be the iniu L such that the threshold has saturated to its constant value, i.e., L () = in {L ɛ (, L) = ɛ (, L ), L = L + 1, L + 2,...}. (34) As shown in Figs. 6(a) and 6(b), L () is non-increasing as increases; for exaple, for C 1 (3, 6,, 2), L (1) = 15, L (3) = 10, and L () = 8 when 6. Thus, we see that increasing the finite field size speeds up the saturation of the FSD threshold as L increases. To avoid repetition, we oit the FSD thresholds obtained for other (J, K)-regular SC-LDPC code ensebles with varying L; however, it should be noted that the threshold saturation behavior described above is consistent over all considered code ensebles. We now consider the WD thresholds of the (3, 6, ) SC-LDPC code ensebles, again with L = 100. The WD thresholds of C 2 (3, 6, ) with the classical edge-spreading forat are shown in Fig. 7(a). As expected, for fixed, the WD thresholds iprove with increasing W, and we find that ˆɛ WD () = ɛ () for W W (), i.e., for a sufficiently large window, the WD threshold is equal to the FSD threshold for all. We note that W () is non-increasing as increases, i.e., the saturation of the WD thresholds ɛ WD(, W ) to ˆɛ WD () is faster for larger. For exaple, W (2) = 15, W (4) = 12, and for 7, W () = 8. Due to a cobination of the existence of degree-1 variable nodes in the window and the larger coupling width w = 2, C 2 (3, 6, ) does not perfor well using WD with a relatively sall window. Next, we consider the cases when w = 1: C 1 (3, 6,, 1), C 1 (3, 6,, 2), and C 1 (3, 6,, 3), shown in Figs. 7(b), 7(c), and 7(d), respectively. We observe that Siilar to the C 2 (3, 6, ) enseble, for each of the three ensebles, at a particular, the WD threshold ɛ WD(, W ) iproves as W increases and saturates nuerically to a constant value ˆɛ WD (). Again, increasing the finite field size speeds up the saturation as W increases; for exaple, W (2) = 10, W (4) = 8, and W (6) = 6 for C 1 (3, 6,, 1). Siply choosing W W () does not necessarily guarantee good WD thresholds, since ˆɛ WD () ay not equal ɛ () even when W is large. 5 In fact, ˆɛ WD () = ɛ () for all 5 Of course, as entioned earlier, by selecting W = L+w in WD, the decoding window covers the whole parity-check atrix and WD is equivalent to FSD. However, we are not considering this extree case here.

22 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) ǫ () or ǫ WD(,W) ǫ () or ǫ WD(,W) SC: FSD SC: 4 SC: 5 SC: 6 SC: W = SC: FSD SC: 4 SC: 5 SC: W = (a) C 2(3, 6, ) (b) C 1(3, 6,, 1) ǫ () or ǫ WD(,W) ǫ () or ǫ WD(,W) SC: FSD SC: 4 SC: 5 SC: W = SC: FSD SC: 3 SC: W = (c) C 1(3, 6,, 2) (d) C 1(3, 6,, 3) Fig. 7. bencharks. WD thresholds ɛ WD(, W ) of the (3, 6, ) SC-LDPC code ensebles. FSD thresholds ɛ () are included as only for C 1 (3, 6,, 1) and C 1 (3, 6,, 2); for C 1 (3, 6,, 3), on the other hand, ˆɛ WD () diverges fro ɛ () as increases, as shown in Fig. 7(d). We turn our attention now to the iplications of the WD thresholds on protograph design. Recall the three types of edge-spreading forats of the (3, 6, ) SC ensebles with w = 1

23 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 23 Strong check node (a) C 1(3, 6,, 1) Weak variable nodes Weak check node (b) C 1(3, 6,, 3) Strong variable nodes Fig. 8. The portion of the base atrix covered by the window when W = 4. defined in (30)-(32), where B 1 0 is given as ] ] ] [E A E A, [E A E B, and [E B E B, respectively. As we ove fro type 1 to type 2 to type 3, the q-ary SC-LDPC code enseble includes [ [ ore E B = 2 1] spreading and less EA = 1 2] spreading. As illustrated in Fig. 8(a) for C 1 (3, 6,, 1) with a window size W = 4, E A spreading has a strong (lower degree) check node at the beginning of the window and weak variable nodes (with degree 1) at the end of the window. As a result, for all, ˆɛ WD () = ɛ () when W is large enough, but ɛ WD(, W ) is not very good when W is relatively sall for exaple, W = 4 in Fig. 7(b). (See also the threshold behavior of the C 2 (3, 6, ) ensebles in Fig. 7(a) which have a siilar structure but larger w.) On the other hand, as illustrated in Fig. 8(b) for C 1 (3, 6,, 3), E B spreading provides strong (higher degree) variable nodes at the end of the window and a weak (higher degree) check node at the beginning of the window. As a result, copared to C 1 (3, 6,, 1) and C 1 (3, 6,, 2), C 1 (3, 6,, 3) has the sallest W () when is fixed, i.e., threshold saturation to ˆɛ WD () is fastest as W increases, but ˆɛ WD () itself does not converge to ɛ (), resulting in unsatisfactory WD thresholds, especially when is large. In fact, coparing Fig. 7(d) to Fig. 5, we observe that the WD threshold of C 1 (3, 6,, 3) becoes ore block-like as increases, i.e., the curve for the WD threshold of C 1 (3, 6,, 3) behaves siilarly to the curve for the FSD threshold of B(3, 6, ) for 4. This block-like behavior occurs for type 3 spreading because the edges of the block protograph have not been sufficiently spread, i.e., only one edge fro each variable node in a block protograph is spread to the adjacent block protograph.

24 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 24 We suarize the above observations for WD thresholds with respect to the advantages and disadvantages of E A and E B spreading based on their effects on the portion of the parity-check atrix covered by the window: 1) The advantage of E A : Due to the strong check node at the start of the window, for a sufficiently large window size, the WD threshold saturates to the FSD threshold, which in turn approaches the channel capacity as the finite field size increases. 2) The disadvantage of E A : Due to the weak variable nodes at the end of the window, WD does not perfor well when the window size is relatively sall, so for sall finite field sizes, there are large gaps between the WD threshold and the FSD threshold. 3) The advantage of E B : Due to the strong variable nodes at the start of the window, for relatively sall window sizes, the WD threshold quickly saturates to its best achievable value, even for relatively sall finite field sizes. 4) The disadvantage of E B : Due to the weak check node at the end of the window, WD tends to provide ore block-like behavior, so that as the finite field size increases, the WD threshold diverges fro the FSD threshold of the q-ary SC-LDPC code enseble and approaches the FSD threshold of the corresponding uncoupled q-ary LDPC-BC enseble. Based on the advantages and disadvantages of these two antipolar spreading forats, we can develop design rules that cobine fast saturation and FSD-achieving thresholds by ixing E B spreading and E A spreading, resulting in the type 2 spreading C 1 (3, 6,, 2). For exaple, as shown in Fig. 7(c), we see that C 1 (3, 6,, 2) has good WD thresholds even when both and W are relatively sall, i.e., with = 5 and W = 5, the best perforance is already achieved and lies within 0.15% of channel capacity. These design rules are consistent with the design rules proposed in [29] for the binary case, but they are ore general in the sense that the effect of non-binary code alphabets is included. To suarize, given the (3, 6)-regular degree distribution, to achieve near-capacity thresholds with sall decoding latency and sall decoding coplexity (see Section V for further details), the q-ary SC-LDPC code enseble C 1 (3, 6,, 2) is recoended due to its excellent thresholds when the window size W and the finite field size q are both relatively sall. 3) The (4, 8) and (5, 10) ensebles: We now exaine the WD thresholds of the (4, 8)- regular q-ary SC-LDPC code ensebles with w = 1 and the (5, 10)-regular q-ary SC-LDPC code ensebles with w = 1 to explore how the advantages and disadvantages of E A and E B

25 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) Block ǫ () or SC ǫ WD(,W) Block ǫ () or SC ǫ WD(,W) B(3,6,) C 1 (3,6,,1) C 1 (3,6,,2) C 1 (3,6,,3) B(4,8,) C 1 (4,8,,1) C 1 (4,8,,2) C 1 (4,8,,3) (a) (3, 6, ) ensebles, W = 3 (b) (4, 8, ) ensebles, W = Block ǫ () or SC ǫ WD(,W) ǫ WD(,W) B(5,10,) C 1 (5,10,,1) C 1 (5,10,,2) C 1 (5,10,,3) C 1 (3,6,,2) C 1 (4,8,,2) C 1 (5,10,,2) (c) (5, 10, ) ensebles, W = 3 (d) Type 2 spreading, W = 5 Fig. 9. WD thresholds ɛ WD(, W ) of q-ary SC-LDPC code ensebles with w = 1 and W = 3: (a) the (3, 6, ) ensebles, (b) the (4, 8, ) ensebles, and (c) the (5, 10, ) ensebles. FSD thresholds ɛ () of the corresponding q-ary LDPC-BC ensebles are included for reference. WD thresholds ɛ WD(, W ) of the C 1(J, 2J,, 2) ensebles, J = 3, 4, and 5, with W = 5 are shown in (d). spreading are affected by the density (J, K) of the parity-check atrix, where we still have k = K/J = 2.

26 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 26 For coparison, the WD thresholds of the (3, 6, ) SC ensebles with w = 1 and W = 3 are shown in Fig. 9(a), and the WD thresholds of the (4, 8) and (5, 10) SC ensebles with w = 1 and W = 3 are shown in Figs. 9(b) and 9(c), respectively. In addition to several features that are siilar to the (3, 6) SC ensebles, soe further observations can be ade for the (4, 8) and (5, 10) SC ensebles: Recall that for E B spreading, the advantage results fro the strong variable nodes with degree (J 1) at the end of the window, and the disadvantage results fro the weak check node with degree 2(J 1) at the beginning of the window, as shown in Fig. 8(b) for J = 3. Thus, as the density J increases, both the positive and the negative effects are strengthened. On the one hand, the saturation of the WD threshold ɛ WD(, W ) to its best achievable value ˆɛ WD () as W increases is faster. For exaple, for = 3, we find that W (3) = 4 for C 1 (3, 6,, 3), W (3) = 4 for C 1 (4, 8,, 3), and W (3) = 3 for C 1 (5, 10,, 3), i.e., for fixed, W () is non-increasing for C 1 (J, 2J,, 3) as J increases. On the other hand, we observe fro Fig. 9 that: The WD thresholds of C 1 (J, 2J,, 3) onotonically decrease as increases ( 3 for C 1 (3, 6,, 3)), Their curves are alost parallel to the corresponding curves for the FSD thresholds of B(J, 2J, ) this effect is ore apparent for J = 4 and 5, and The gap between these two curves decreases as J increases. Thus, the denser the parity-check atrix is, the ore block-like the WD thresholds of type ] 3 spreading [E B E B becoe. As previously entioned, this is because only one edge fro each variable node in a block protograph is spread to the adjacent block protograph in type 3 edge spreading. The disadvantage of E B spreading also affects the WD thresholds of type 2 edge spreading. Fig. 9(d) copares the WD thresholds of the C 1 (3, 6,, 2), C 1 (4, 8,, 2), and C 1 (5, 10,, 2) ensebles with W = 5. We see that, as J increases, the thresholds of C 1 (J, 2J,, 2) diverge ore significantly fro channel capacity as increases, consistent with the observation that the disadvantage of E B spreading is strengthened as J increases. Moreover, the divergence occurs sooner as J increases, e.g., the WD threshold of C 1 (5, 10,, 2) increases only up to = 2 and then starts to decrease as increases further, whereas the divergence for both