Article An Efficient Method to Construct Parity-Check Matrices for Recursively Encoding Spatially Coupled LDPC Codes

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1 entropy Artice An Efficient Method to Construct Parity-Check Matrices for Recursivey Encoding Spatiay Couped LDPC Codes Zhongwei Si, Sijie Wang and Junyang Ma Key Laboratory of Universa Wireess Communications, Ministry of Education, Beijing University of Posts and Teecommunications (BUPT), No. Xitucheng Road, Haidian District, Beijing 876, China; (S.W.); (J.M.) * Correspondence: sizhongwei@bupt.edu.cn; Te.: This paper is an extended version of our paper pubished in the 25 IEEE Internationa Symposium on Persona, Indoor and Mobie Radio Communications, Hong Kong, China, 3 August 2 September 25. Academic Editor: Raú Acaraz Martínez Received: 5 June 26; Accepted: 5 August 26; Pubished: 7 August 26 Abstract: Spatiay couped ow-density parity-check (LDPC) codes have attracted considerabe attention due to their promising performance. Recursive encoding of the codes with ow deay and ow compexity has been proposed in the iterature but with constraints or restrictions. In this manuscript we propose an efficient method to construct parity-check matrices for recursivey encoding spatiay couped LDPC codes with arbitrariy chosen node degrees. A genera principe is proposed, which provides feasibe and practica guidance for the construction of parity-check matrices. According to the specific structure of the matrix, each parity bit at a couping position is jointy determined by the information bits at the current position and the encoded bits at former positions. Performance anaysis in terms of design rate and density evoution has been presented. It can be observed that, in addition to the feature of recursive encoding, seected code structures constructed by the newy proposed method may ead to better beief-propagation threshods than the conventiona structures. Finite-ength simuation resuts are provided as we, which verify the theoretica anaysis. Keywords: channe coding; spatiay couped LDPC codes; protograph codes; recursive encoding; density evoution. Introduction Low-density parity-check (LDPC) codes [] were the most researched channe coding in the ast decade due to the good performance and the reasonabe compexity. Their convoutiona counterparts, the LDPC convoutiona codes, were invented in [2] and further deveoped in, e.g., [3,4]. Different constructions and anaytica resuts have been provided in [3 ]. For exampe, Lentmaier et a. [3] proposed a reaization of LDPC convoutiona codes and conjectured the capacity-achieving performance under beief-propagation (BP) decoding. The LDPC convoutiona codes were considered in a ream of spatiay couped codes [4], and the capacity-achieving property of the code famiy has been proved theoreticay [ 3]. The good features have triggered the appication of spatiay couping in many scenarios, such as compressed sensing [4], rate-compatibe coding [5,6], reay channes [7,8], wiretap channes [9], mutipe access channes [2], mutiuser detection [2], source coding [22], etc. To reaize the promising performance in practice, the impementation of spatiay couped LDPC codes is worth investigation. The best known impementation of spatiay couped LDPC codes is the one introduced in [6]. Based on the particuar structure of the parity-check matrix, a recursive Entropy 26, 8, 35; doi:.339/e8835

2 Entropy 26, 8, 35 2 of 4 encoder using a shift-register was proposed. The encoder requires ony a sma number of memory units and the computationa cost per bit is proportiona to the check node degree. This is a cear advantage compared with the encoding of LDPC bock code regarding storage requirements and computationa costs. Combined with the pipeine decoding or the siding-window decoding [6,23], spatiay couped LDPC codes provide a ow-deay and ow-compexity reaization of channe coding with capacity-approaching performance. The structure in [6] is suitabe for recursive encoding; however, severa constraints need to be satisfied when constructing the parity-check matrix. Aternativey, Sridharan et a. [8] proposed a simper way to generate the parity-check matrix using randomy permuted identity matrices on condition that the check node degree d c and the variabe node degree d v are not reativey prime. If we further require that the ratio of d c and d v is an integer, then the structure in [8] enabes recursive encoding. These constraints and requirements imit the practica appication of the codes. To extend the construction to the codes with arbitrary rates, we propose in this paper an efficient method to construct the parity-check matrices with permuted identity matrices regardess of the choice of d c and d v. By rearranging the edge connections, each parity bit at a couping position is jointy determined by the information bits at the current position and the encoded bits at former positions [24]. A generaized principe is proposed, which provides feasibe and practica guidance for constructing the recursive structures. The requirement of memory units and the computationa compexity retain simiar to the recursive encoder in [6]. Performance anaysis in terms of design rate and density evoution is provided. It can be observed that, with propery designed pattern, the modified code structure may converge faster in the beief-propagation (BP) decoding than the existing structure [8]. Consequenty, the corresponding spatiay couped LDPC ensembe is abe to provide better BP threshods and smaer gaps to the Shannon imits. Bit erasure rate (BER) performance with finite code engths is provided as we, which verifies the anaysis of the BP threshods. In this manuscript we restrict ourseves ony to the convoutiona-ike LDPC codes with neary reguar structures. More advanced spatiay couped protograph codes [25] have been proposed with better decoding threshods, minimum distances and finite-ength trade-offs [26], such as spatiay couped repeat-accumuate (RA) codes [27], spatiay couped accumuate-jagged-accumuate (ARJA) codes [28] and spatiay couped MacKay Nea (MN) codes [29]. The easy encoding of these codes is definitey worth investigations, and we wi ook into it in the future work. The remainder of the paper is organized as foows. An overview of spatiay couped LDPC codes and the recursive encoding proposed in [6,8] are given in Section 2. In Section 3, we propose a principe on constructing the parity-check matrices of spatiay couped LDPC codes and iustrate the recursive encoding using a shift-register. Performance anaysis based on design rate and density evoution is provided in Section 4. Section 5 presents the finite-ength performance evauation in the binary erasure channe (BEC). Section 6 concudes the paper. 2. Overview of Spatiay Couped LDPC Codes Spatiay couped LDPC codes are constructed by couping together a series of disjoint LDPC bock codes into a singe couped chain. As shown in Figure, the eft unit is the protograph of a LDPC code, and the structure on the right represents the protograph of the corresponding spatiay couped LDPC code. The spatiay couped LDPC ensembe is obtained by dupication and edge rearrangement from the LDPC ensembe. A (d v, d c ) spatiay couped LDPC code with couping chain L can aso be defined using the parity-check matrix H[,L ] T as foows: H T [,L ] = M T()... MT (m s)... M T ( + m)... M T L (L )... MT L (L + m s ),

3 codes and the recursive encoding proposed in [6,8] are given in Section 2. In Section 3, we propose a principe on constructing the parity-check matrices of spatiay couped LDPC codes and iustrate the recursive encoding using a shift-register. Performance anaysis based on design rate and density evoution is provided in Section 4. Section 5 presents the finite-ength performance evauation in the Entropy binary 26, 8, erasure 35 channe (BEC). Section 6 concudes the paper. 3 of 4 2. Overview of Spatiay Couped LDPC Codes where each submatrix M T ( + m) is a c (c b) binary matrix. The parameter c corresponds to the Spatiay couped LDPC codes are constructed by couping together a series of disjoint LDPC number of variabe nodes at each position, and (c b) is the corresponding number of check nodes. bock codes into a singe couped chain. As shown in Figure, the eft unit is the protograph of a LDPC In the notation,, < L, denotes the position in the chain, and m, m m s, indicates the code, and the structure on the right represents the protograph of the corresponding spatiay couped position of edge connections. The parameter m s is caed the syndrome former memory, and (m s + )c LDPC code. The spatiay couped LDPC ensembe is obtained by dupication and edge rearrangement is regarded as the constraint ength of the code. from the LDPC ensembe. V, V, V, V, V2, V2, V3, V3, V4, V4, V5, V5, Figure Figure. Protograph. Protograph of a (3, of a 6, (3, L) 6, spatiay L) spatiay couped couped ow-density LDPC ensembe. parity-check The rectanges (LDPC) ensembe. represent The check rectanges nodesrepresent whie thecheck circesnodes denote whie variabe the circes nodes. denote The gray variabe circesnodes. correspond The gray to information circes correspond bits whie to information the white circes bits whie associate the white with circes parity bits. associate with parity bits. Once the node degrees (d v, d c ) have been given, the parity-check matrix H[,L ] T needs to satisfy the foowing constraints:. The number of ones in each row is d v. 2. The number of ones in each coumn is d c, except the beginning and the terminating ends of the chain. 3. M T( + m) =, for m < and m > m s,. 4. There exists such that M T( + m s) =. 5. M T () = has fu rank. To impement the systematic and recursive encoding according to [6], M T ( + m) must additionay satisfy: 6. The ast c b rows of M T () is a (c b) (c b) identity matrix. Note that, in this paper we assume that the encoder is systematic, i.e., the information bits is part of the codeword. We define the recursive encoding as that each parity bit to be encoded can be obtained through the modue-2 addition of the known information bits and parity bits. Once a parity bit is determined, it wi be utiized in the forthcoming cacuation of other unknown parity bits. It usuay takes some efforts to simutaneousy satisfy a the above constraints. Instead, if using the structure iustrated in [8], a syndrome matrix meeting a the requirements can be constructed simpy. We now describe the structure proposed in [8]. According to [8], the submatrix M T ( + m) is constructed as M T ( + m) = P, ( + m) P,d v ( + m). P i,j ( + m). P d c, ( + m) P d c,d ( + m) v where P i,j ( + m) is an M M random permutation matrix in the i-th row and the j-th coumn of M T ( + m). Let m b = gcd(d v, d c ) denote the greatest common divisor of d v and d c, then there exist positive integers d v = d v /m b, d c = d c /m b, and gcd(d v, d c) =. In this case, we have c = d cm, b = (d c d v)m, and the syndrome former memory m s = m b. The satisfaction of constraints 5 is obvious. When d c /d v is an integer, M T( + m) turns out to be a d c matrix. The constraint 6 is easy to access by setting P d c, ( + m) as an identity matrix. The recursive encoding of spatiay couped LDPC code for the case that d c /d v is an integer is discussed in the foowing. The protograph of the (3, 6, L) spatiay couped LDPC ensembe has been provided in Figure, where the gray circes correspond to information bits and the white circes,

4 Entropy 26, 8, 35 4 of 4 associate with parity bits. As shown in the protograph, the parity bits v, in each position are determined according to the information bits v, and the reated bits v,, v,, <, from the previous positions. In this way, a recursive encoder is impemented. However, it is disappointing that for the case d c /d v is not an integer, we cannot perform recursive encoding as above. Let us consider the case d v = 4 and d c = 6. The protograph of the LDPC bock code, which acts as the smaest couping unit, is given on the eft of Figure 2. The protograph of a (d v, d c, L) = (4, 6, L) spatiay couped LDPC ensembe constructed according to [8] is provided in Figure 2a. The notations are the same as in Figure. The protograph consists of d c = 3 variabe nodes and d v = 2 check nodes at each position. The variabe node degree equas d v, and the check node degree is d c. At each position in Figure 2a, v, denotes the information bits whie v, and v,2 represent the parity bits, for < L. It can be seen that the parity bits v, (and v,2 ) can not be cacuated ony depending on v, and the encoded bits at former positions, as the two variabe nodes associated with v, and v,2 connect to the same check node. Therefore, recursive encoding cannot be appied for this case.... V, V, V,2 V, V, V,2 V2, V2, V2,2 V3, V3, V3,2 V4, V4, V4,2 V5, V5, V5, (a) V, V, V,2 V, V, V,2 V2, V2, V2,2 V3, V3, V3,2 V4, V4, V4,2 V5, V5, V5, (b) Figure 2. Protograph of a (4, 6, L) spatiay couped LDPC ensembe with (a) the structure in [8]; (b) a modified structure. Fortunatey, the probem may have a soution using a modified protograph structure as iustrated in Figure 2b. Simiary, v, represents the information bits, and v,, v,2 are the parity bits, at each position. The edge connection on the variabe node associated with v,2 is rearranged. It avoids connecting to both the check nodes at the current position but connects to a check node one position away. By doing this, each check node connects with ony one unknown variabe nodes and d c known variabe nodes. The parity bit v, can be cacuated depending on v, and the encoded bits at former positions. Afterwards, v,2 can be obtained in turn through v,, v, and the encoded bits at former positions. After the modification, a the parity bits at the current position can be cacuated depending on the information bits and the previousy encoded parity bits. The recursive encoding is reaized through a structure modification. The recursive encoding significanty reduces the deay and the compexity compared to the encoding of LDPC bock codes, and it promotes the appication of spatiay couped LDPC codes in practice. Therefore, it is important to find an efficient and effective method to impement the recursive encoding for arbitrariy chosen node degrees. In the next section, we propose a simpe but generaized principe for constructing the parity-check matrix and impement the recursive encoding for the proposed structures.

5 Entropy 26, 8, 35 5 of 4 3. Recursive Encoding of Spatiay Couped LDPC Codes with Arbitrary Node Degrees In the foowing we first propose a genera method which guides the construction of the parity-check matrix for recursive encoding. Afterwards, the detais of the encoding procedure is provided. 3.. Genera Principe on Constructing Parity-Check Matrices Instead of using the d c d v structure as the smaest couping unit, we redefine a structure consisting of d c d v permuted identity matrices as: B T = P (), () P(d v ),d v ().. P (n). i,j ()., () P () d c, () P(d v ) d c,d v () where P (n) i,j () is an M M random permutation matrix in the i-th row and the j-th coumn of B T, and n ( n < d v ) represents that P (n) i,j () is the n-th non-zero permutation matrix in the i-th row. By varying the offsets between adjacent rows, different types of couping units are obtained. We define a pattern a = (a, a,, a i,, a dc ), where a i {, } for i = {,,, d c }, to represent the couping unit H T after row offsetting as: H T = a = P (), () P(), () P(j+),j+ () P(dv ),d () v a i P () i,j () P() i,j+ () P(dv ) i,j+d () v a i = P () i,j+ () P() i,j+2 () P(dv 2) i,j+d () P(dv ) v i,j+d v () a dc P () d () c,d v P() d c,d v () P (dv ) d c,2d v 2 () (2) The eement a i = means that the beginning of the non-zero matrices in the i-th row shifts by one coumn compared to that in the (i )-th row, whie a i = indicates that the non-zero matrices in the i-th row are aigned with those in the (i )-th row, for < < d c. Note that a = means that the beginning of the first row in H T shifts by one coumn compared to the ast row of H T. Figure 3 iustrates how to obtain H T from B T according to pattern a, and then how to assembe H T [,L ] accordingy. In the foowing we propose a theorem which seects a the parity-check matrix structures suitabe for recursive encoding. Theorem. For a couping unit represented by a = (a, a,, a i,, a dc ), if the pattern a satisfies d c a i = d v i= a =, (3) the spatiay couped LDPC ensembe {d v, d c, L} constructed by couping the unit of pattern a aows recursive encoding.

6 Entropy 26, 8, 35 6 of 4 * contains i rows... T H - a = i = * i *, 2 * T contains( dc-i) rowsin H - - H T T contains dc rows inh[, L-]... T B H T T [, L-] H Figure 3. The construction of the parity-check matrix for the (4, 6, L) spatiay couped LDPC ensembe. Proof of Theorem. We first prove that any matrix H[,L ] T couping from a structure H T satisfying condition (3) is the parity-check matrix of a spatiay couped LDPC ensembe. Each P (n) i,j () is a permutation matrix, which impies that each coumn or each row of P (n) i,j () ony contains singe one. Considering d c a i = d v and a =, the structure H T must have a form as in (2). i= It is obvious that H T is a d c (2d v ) matrix, and each P (n) i,j () is an eement of it. Apparenty each row in H T contains d v non-zero coumns, and so does H[,L ] T. Then, as Figure 3 shows, the ast d v coumns of H T is aigned with the first d v coumns of H T if a =. Assuming that a i represents the second (a must be ) in a, a coumns in row i, i i, has been shifted to the right by at east one coumn. Thus the first coumn in H T contains ony i rows and the d v -th coumn contains d c i rows. Note that the d v -th coumn of H T corresponds to the first coumn of H T. After assembing H T and H T, the corresponding coumn in HT [,L ] contains d c rows. The same anaysis can be appied to other coumns except the beginning and terminating ends of H T [,L ]. Therefore the matrix H[,L ] T is the parity-check matrix of a spatiay couped LDPC ensembe with node degrees d v and d c. We then prove why recursive encoding can be appied with this structure. We have assumed that the couping unit has d c variabe nodes and d v check nodes. We use C(, k) to denote the k-th check node in the -th couping unit in the chain, k < d v. In each couping unit, the number of variabe nodes associated with the parity bits equas d v, and the number of variabe nodes corresponding to information bits is d c d v. For convenience, we name the variabe nodes associated with the parity bits as parity nodes, and we denote the k-th parity node in the -th couping unit as P(, k). The nodes reated to the information bits are denoted as I(, j), j < d c d v. To guarantee recursive encoding, we set the foowing conditions on the edge connections from the check node C(, k): (i) there is one and ony one edge connecting the check node C(, k) and the parity node P(, k); (ii) the remaining (d c ) edges are ony aowed to connect with the nodes in the set {P(, k ) : d v + k < d v + k} {I(, j) :, j < d c d v }. A protograph which iustrates the above connections for d v = 4 and d c = 6 is given in Figure 4. Without the oss of generaity, we assume the information bits are associated with the first two variabe nodes in each couping unit.

7 Entropy 26, 8, 35 7 of 4 C(,) C(,) C(,2) C(,3) I(,) I(,) P(,) P(,) P(,2) P(,3) Figure 4. The connections for (4, 6) spatiay couped LDPC codes assuming the information bits are associated with the first two variabe nodes. In the foowing we describe the above conditions in form of parity-check matrix. Each couping unit is described by a d c d v structure of permutation matrices. Without disarranging the edge connections, we change the order of rows so that the first d v rows reate to parity nodes P(, k), k < d v. The indexes of the coumns represent the check nodes C(, k). According to the condition (i), the permutation matrix at the k-th row and the k-th coumn for k < d v is non-zero. Meanwhie, based on the condition (ii), the permutation matrix at the k-th row and the k -th coumn, k > k, k, k < d v, must be a zero-matrix. We have defined that the eement a i = means that the beginning of the non-zero permutation matrices in the i-th row shifts by one coumn compared to that in the (i )-th row. According to the conditions, there is obviousy a number of d v s in the pattern a. As a =, the above theorem seects a number of ( d c d v ) structures which aow recursive encoding. However, some different structures may actuay ead to the same edge connections in the chain. In order to screen out the structures which have truy different edge connections, we take into account the foowing restriction. The submatrices indexed by a a and a b are considered to resut in different edge connections in the couping chain ony if they satisfy: (a a ) N = a b (4) where (a a ) N indicates a sequence obtained by circuar shifting the eements in a a for N times, N {,, d c }. Once the pattern a is obtained, the positions for the information bits are determined. If a i =, < i < d v, the (i )-th variabe node is designated for one information bit. The structures seected by (3) and (4) for recursive encoding spatiay couped LDPC ensembes with d v = 4 and d c = 6 are a a = (,,,,, ), a b = (,,,,, ), and a c = (,,,,, ). The corresponding matrix structures are iustrated in Figure 5. The various structures ead to different performance which wi be discussed ater. (a) (b) (c) Figure 5. The couping unit of (4, 6, L) spatiay couped LDPC ensembes with (a) a a = (,,,,, ); (b) a b = (,,,,, ) and (c) a c = (,,,,, ). The spatiay couped LDPC codes need to be initiaized and terminated propery for finite L. In order to ensure that the spatiay couped LDPC codes with the modified structures exhibit good

8 Entropy 26, 8, 35 8 of 4 performance, we try to maintain the degree of variabe nodes for a the positions on the couping chain but reduce the degree of check nodes at the boundary positions. Take Figure 2b as an exampe. If we keep a the variabe node degree to be d v, there appears an additiona check node at the end of the chain compared to Figure 2a. This check node wi sighty reduce the design rate of the ensembe. On the other hand, if we remove this check node, the design rate wi be the same as that of Figure 2a. For this case, the ast variabe node has a reduced degree d v. In this manuscript, we aways choose to keep the variabe node degree d v on a the positions. The same as in [6], the state of the register is non-zero after encoding. A soution to this probem has been proposed in [6], but it has to sove a system of inear equation which is compex. To sove the probem more efficienty, we adopt the efficient termination proposed in [9] Impementation of Recursive Encoding In this section, we introduce the impementation of recursive encoding based on the modified structure of spatiay couped LDPC codes as above. To iustrate the recursive encoding, we first define the foowing notations. The subsequence of information bits at position is defined as u = [u,, u,,, u,dc d v ] for < L. The encoded subsequence at position is denoted as v = [v,, v,,, v,dc ], and c = [c,, c,,, c,dv ] is the parity-bit vector for < L. We have the entire code sequence as v [,L ] = [v, v,, v L ], and it satisfies the equation v [,L ] H [,L ] T =. (5) We divide (5) into severa sub-equations, and the -th sub-equation is written as v H T = p = [s, q ], < L. The vector p = [s, q ] is the partia syndrome from the -th sub-equation, where s = [s (), s (),, s (d v ) ] and q = [q (), q (),, q (d v 2) ]. From the structure of H T, for < L, we have p = { v H T, s (k) q (k) = = p(k+d v), k < d v, k = d v. If P () i,j () for a i+ = ( i < d c, a dc = a = ) is an identity matrix, then the parity bits can be cacuated simpy with the encoded bits obtained earier. For < L, the i-th ( i < d c ) encoded bit v,i is determined depending on the information bits, the encoded parity bits and the partia syndrome as u,g, a i+ = v,i = c,k = s (k) + i v,i P ( ) i i,j (), a i+ =, (6) = where u,g and c,k correspond to the coming information bits and the parity bits to be encoded, respectivey. The indexes g, k and j are determined by the reaization of H T. For =, s =, the encoder is initiaized by p = [, q ] = v H T, and the encoded bits are u,g, a i+ = v,i = c,k = i v,i P ( ) i i,j [], a i+ = = (7)

9 Entropy 26, 8, 35 9 of 4 According to (6) and (7), we can impement the recursive encoder of spatiay couped LDPC codes using shift-registers as in [6]. The encoder needs a number of (d v )M storage units, and the encoding compexity per bit is proportiona to d c. 4. Theoretica Anaysis In this section, the design rate of the terminated spatiay couped LDPC ensembe with the modified structure is cacuated. In addition, anaysis based on density evoution is provided. For a fair comparison with [8], in the foowing we assume an ideaized termination where the state of the shift-register comes back to zero. For the ease of presentation, we refer to the structure in [8] as the origina structure in the rest of the paper. 4.. Design Rate To get the design rate of a (d v, d c, L) spatiay couped LDPC ensembe, we cacuate the tota number of information bits and parity bits separatey. Since there are M nodes for each position, the tota number of variabe nodes equas d c LM, and the tota number of check nodes is (d v L + d v )M. Hence, the design rate of the spatiay couped LDPC code is R = d vl + d v d c L = d v L + dv d v. d c L As we pointed out earier, this rate is sighty ower than that of the origina ensembe when gcd(d v, d c ) < d v. When gcd(d v, d c ) = d v, the modified ensembe is equivaent to the origina structure, and the design rate is the same for both. For a arge L, the rate reduction is negigibe. When L tends to infinity, the design rate converges to d v /d c Density Evoution In the foowing we derive the density evoution of the proposed code structures. For comparison, we pot the remaining erasure probabiities after iterations for both the origina spatiay couped structure and the newy proposed structures. There are d c variabe nodes and d v check nodes at each position. Due to the variety of edge connections, we track the message passing on each individua node instead of each position. We use T (k, n) to denote the set of indices ( f, g) of the check nodes connected with variabe node (k, n) and J ( f, g) to represent the set of indices (k, n) of the variabe nodes reated to check node ( f, g). The check node ( f, g) is the g-th check node at position f, and variabe node (k, n) is the n-th variabe node at position k. In the t-th iteration of the decoding, we use x n,g k, f (t) to denote the probabiity that the message from variabe node (k, n) to check node ( f, g) is an erasure, whie the probabiity that the message from check node ( f, g) to variabe node (k, n) corresponds to an erasure is represented by y g,n f,k (t). As the variabe node (k, n) connects with the check nodes indexed by T (k, n), x n,g k, f (t) can be updated as Simiary, for check node ( f, g), we get x n,g k, f (t) = ε ( f,g ) T (k,n) s.t.( f,g ) =( f,g) y g,n f,k (t ). y g,n f,k (t) = [ x n,g k, f (t)]. (k,n ) J ( f,g) s.t.(k,n ) =(k,n)

10 Entropy 26, 8, 35 of 4 In Figure 6, the density evoution for (d v, d c, L) = (4, 6, 5) spatiay couped LDPC ensembe with iteration numbers n = {,, 2, 3,..., 9} is given for the binary erasure channe with erasure probabiity ε =.646. Figure 6a refers to the origina spatiay couped LDPC ensembe in [8], whie Figure 6b d correspond to the spatiay couped LDPC ensembes modified for recursive encoding. The horizonta axis is the index of the variabe nodes in the protograph, where every adjacent d c variabe nodes beong to the same position. The vertica axis presents the remaining erasure probabiity..8.8 Erasure Probabiit Erasure Probabiity Index of Variabe Node (a) Index of Variabe Nodes (c) Erasure Probabiit Erasure Probabiit Index of Variabe Node (b) Index of Variabe Node (d) Figure 6. Density evoution for (4, 6, 5) spatiay couped LDPC ensembes with structure (a) in [8]; (b) a j = (,,,,, ); (c) a j = (,,,,, ) and (d) a j = (,,,,, ). The curves of density evoution refect the irreguarity of check node degree at the boundary positions. In the initiaization, the degree of the eft four check nodes is respectivey {3, 3, 6, 6}, {2, 3, 5, 6}, {3, 4, 5, 6}, and {2, 3, 4, 6} for (a) to (d). In genera, ower check node degree eads to better decoding performance, which is verified in Figure 6. We can see that (b) and (d) show an advantage over (a) and (c) on the eft boundary of the protograph. Simiar superiority is observed for (b) and (c) on the right boundary due to their smaer check node degree. For the origina spatiay couped LDPC ensembe, a the variabe nodes at the same position has the same couping behavior. Therefore, (a) exhibits a stair for every d c variabe nodes. In the protograph of the modified structures, even for the same position, the edge connection may be different from one variabe node to another. Consequenty, the curves of remaining erasure probabiity in (b), (c), and (d) are smoother than that in (a). Meanwhie, the manner of edge connection ceary affects the convergence behavior in iterative decoding. For the erasure probabiity we chose, the codes are in the water-fa region. We can see from the distance between neighboring curves, (b) converges the fastest among a the ensembes, and (c) performs the worst. The BP decoding performance of spatiay couped LDPC ensembes is jointy determined by the boundary effect and the manner of edge connections. Combining the above observations, we expect that the ensembe associated with Figure 6b outperforms the others regarding the decoding threshod. The BP threshods wi be given in the next section.

11 Entropy 26, 8, 35 of 4 On top of BP threshod, the performance of a spatiay couped LDPC code can be anayzed through the distance property of the ensembe. For exampe, the minimum distance growth rate of a spatiay couped LDPC ensembe can be studied via ensembe weight enumerator [3], and the smaest stopping/trapping set size growth can be studied via stopping/trapping set enumerator [3]. This manuscript mainy discuss the impementation aspects of the recursive encoding after modifying the structure, and a comprehensive performance anaysis based on the distance property wi be addressed in the future work. 5. Simuation Resuts and Discussions In this section, we evauate the performance of the proposed structures in the binary erasure channe numericay. We compare the BP threshods of the spatiay couped LDPC codes after the modifications with the BP threshods of the corresponding origina structures in [8]. In addition, the bit erasure rates for different structures are provided. We first provide the comparison of BP threshods among different reaizations of the modification. We choose the parameters d v = 4, d c = 6 and L = 5 and carry out the density evoution for a the structures shown in Figure 5. The BP threshods of the modified spatiay couped LDPC ensembes and the origina impementation in [8] over the BEC are shown in Tabe, which are represented by ε BP. Besides, ε Sh denotes the Shannon imit corresponding to different design rate. As predicted in Section 4.2, the ensembe with the pattern () performs the best among a the structures. The gap between the Shannon imit and the BP threshod is smaer than that of the origina structure. The structure modification does not necessariy resut in better BP threshod. However, a propery designed pattern can enabe both easy encoding and better threshod. Tabe. BP threshods of (4, 6, 5) spatiay couped LDPC ensembes with different structures. Pattern a Design Rate ε Sh ε BP Gap Origina () () () Based on the observations in Tabe, we reaize the modifications with propery seected patterns for different d v and d c and provide the BP threshods in Tabe 2. The seected patterns are isted in the tabe, and the couping ength is L = 5. The gaps between the BP threshods and the Shannon imits for the modified spatiay couped LDPC ensembes and for the origina ensembes in [8] are represented by Gap and Gap, respectivey. Comparing Gap with Gap, we can find that a the gaps are narrowed after proper modifications. For the modified structures, a the gaps are very sma and can be further narrowed if we increase the parameters. Propery modified spatiay couped LDPC ensembes are abe to provide capacity-approaching performance over the BEC. A simiar concusion can be extended to the genera binary memoryess symmetric channe. Tabe 2. BP threshods of modified spatiay couped LDPC ensembes with different node degrees. (d v, d c ) Pattern a Design Rate ε Sh ε BP Gap Gap (4, 6) () (6, 9) () (8, 2) () (6, ) () (9, 5) () (2, 2) ()

12 Entropy 26, 8, 35 2 of 4 Bit erasure rates for the modified spatiay couped LDPC codes and the origina code over BECs are provided in Figure 7. Here, we set M = 5 and L = 5, and we use the seected patterns as in Tabe 2. We aso incude the corresponding BP threshods as the dashed ines and the Shannon imits as the dotted ines. Comparing the BER curves, we can find that the spatiay couped LDPC codes with propery modified structures ceary outperforms the origina spatiay couped LDPC code in [8]. Note that, in addition to the good decoding performance, the proposed structure aows recursive encoding which faciitates the appication of spatiay couped LDPC codes in practice. Bit Erasure Rate 2 3 (6,9)MSCLDPC (6,)MSCLDPC (6,9)SCLDPC (6,)SCLDPC Shannon imit BP threshod Erasure Probabiity Figure 7. Bit erasure rates of spatiay couped LDPC codes over binary erasure channes. In the egend, MSCLDPC represents the modified code whie SCLDPC stands for the origina code. 6. Concusions In this manuscript, structure modification for the spatiay couped LDPC code has been proposed, which enabes the recursive encoding using a shift-register for arbitrary node degrees. The proposed principe provides feasibe and practica guidance for the code construction and significanty reduces the computationa compexity and memory requirement for the encoding. Performance anaysis in terms of design rate and density evoution has been provided. The beief-prorogation decoding threshod is improved if proper edge rearrangement is appied in the modified spatiay couped LDPC ensembe. Finite-ength simuation resuts are provided, which verify the theoretica anaysis. Acknowedgments: This work was supported in part by the Nationa Natura Science Foundation of China (No. 6437), the 863 project (No. SS25AA33), and the NSFC-RS joint funding grant (Nos and IE4387). Author Contributions: Zhongwei Si and Junyang Ma deveoped the first proposa of the study for the recursive encoding of spatiay couped LDPC codes with arbitrary rates. Zhongwei Si and Sijie Wang extended the study by proposing a generaized design principe and proving the theorem. The anaysis and simuations were carried out by a the three authors. A authors have participated in writing the manuscript. A authors have read and approved the fina manuscript. Conficts of Interest: The authors decare no confict of interest. The founding sponsors had no roe in the design of the study; in the coection, anayses, or interpretation of data; in the writing of the manuscript, and in the decision to pubish the resuts.

13 Entropy 26, 8, 35 3 of 4 References. Gaager, R.G. Low Density Parity Check Codes; MIT Press: Cambridge, MA, USA, Jimenez-Fetström, A.; Zigangirov, K.S. Time-varying periodic convoutiona codes with ow-density parity-check matrix. IEEE Trans. Inf. Theory 999, 45, Lentmaier, M.; Sridharan, A.; Costeo, D.J., Jr.; Zigangirov, K.S. Iterative decoding threshod anaysis for LDPC convoutiona codes. IEEE Trans. Inf. Theory 2, 56, Kudekar, S.; Richardson, T.; Urbanke, R. Threshod saturation via spatia couping: Why convoutiona LDPC ensembes perform so we over the BEC. IEEE Trans. Inf. Theory 2, 57, Mitche, D.G.M.; Pusane, A.E.; Zigangirov, K.S.; Costeo, D.J., Jr. Asymptoticay good LDPC convoutiona codes based on protograph. In Proceedings of the 28 IEEE Internationa Symposium on Information Theory, Toronto, ON, Canada, 6 Juy 28; pp Pusane, A.E.; Jimenez-Feström, A.; Sridharan, A.A.; Lentmaier, M.M.; Zigangirov, K.S.; Costeo, D.J., Jr. Impementation aspects of LDPC convoutiona codes. IEEE Trans. Commun. 28, 56, Tanner, R.M.; Sridhara, D.; Sridharan, A.; Fuja, T.E.; Costeo, D.J., Jr. LDPC bock and convoutiona codes based on circuant matrices. IEEE Trans. Inf. Theory 24, 5, Sridharan, A.; Truhachev, D.V.; Lentmaier, M.; Costeo, D.J.; Zigangirov, K.S. Distance bounds for an ensembe of LDPC convoutiona codes. IEEE Trans. Inf. Theory 27, 53, Tazoe, K.; Kasai, K.; Sakaniwa, K. Efficient termination of spatiay-couped code. In Proceedings of the 22 IEEE Information Theory Workshop, Lausanne, Switzerand, 3 7 September 22; pp Yeda, A.; Jian, Y.-Y.; Nguyen, P.S.; Pfister, H.D. A simpe proof of threshod saturation for couped scaar recursions. In Proceedings of the 22 IEEE Internationa Symposium on Turbo Codes and Iterative Information Processing, Gothenburg, Sweden, 27 3 August 22; pp Yeda, A.; Jian, Y.-Y.; Nguyen, P.S.; Pfister, H.D. A simpe proof of threshod saturation for couped vector recursions. In Proceedings of the 22 IEEE Information Theory Workshop, Lausanne, Switzerand, 3 7 September 22; pp Kudekar, S.; Richardson, T.; Urbanke, R. Spatiay couped ensembes universay achieve capacity under beief propagation. In Proceedings of the 22 IEEE Internationa Symposium on Information Theory, Cambridge, UK, 6 Juy 22; pp Kumar, S.; Young, A.J.; Macris, N.; Pfister, H.D. Threshod saturation for spatiay couped LDPC and LDGM codes on BMS channes. IEEE Trans. Inf. Theory 24, 6, Kudekar, S.; Pfister, H.D. The effect of spatia couping on compressive sensing. In Proceedings of the Aerton Conference on Communications, Contro, and Computing, Monticeo, NY, USA, 29 September October 2; pp Si, Z.; Thobaben, R.; Skogund, M. Rate-compatibe LDPC convoutiona codes achieving the capacity of the BEC. IEEE Trans. Inf. Theory 22, 58, Nitzod, W.; Lentmaier, M.; Fettweis, G.P. Spatiay couped protograph-based LDPC codes for incrementa redundancy. In Proceedings of the Internationa Symposium on Turbo Codes and Iterative Information Processing, Gothenburg, Sweden, 27 3 August 22; pp Si, Z.; Thobaben, R.; Skogund, M. Biayer LDPC convoutiona codes for decode-and-forward reaying. IEEE Trans. Commun. 23, 6, Schwandter, S.; Amat, A.G.; Matz, G. Spatiay couped LDPC codes for two-user decode-and-forward reaying. In Proceedings of the Internationa Symposium on Turbo Codes and Iterative Information Processing, Gothenburg, Sweden, 27 3 August 22; pp Rathi, V.; Urbanke, R.; Andersson, M.; Skogund, M. Rate-equivocation optimay spatiay couped LDPC codes for the BEC wiretap channe. In Proceedings of the 2 IEEE Internationa Symposium on Information Theory, Saint Petersburg, Russia, 3 Juy 5 August 2; pp Kudekar, S.; Kasai, K. Spatiay couped codes over the mutipe access channe. In Proceedings of the 2 IEEE Internationa Symposium on Information Theory, Saint Petersburg, Russia, 3 Juy 5 August 2; pp Schege, C.; Truhachev, D. Mutipe access demoduation in the ifted signa graph with spatia couping. IEEE Trans. Inf. Theory 23, 59,

14 Entropy 26, 8, 35 4 of Yeda, A.; Pfister, H.D.; Narayanan, K.R. Universaity for the noisy Sepian-Wof probem via spatia couping. In Proceedings of the 2 IEEE Internationa Symposium on Information Theory, Saint Petersburg, Russia, 3 Juy 5 August 2; pp Iyengar, A.R.; Papaeo, M.; Siege, P.H.; Wof, J.K.; Vanei-Corai, A.; Corazza, G.E. Windowed decoding of protograph-based LDPC convoutiona codes over erasure channes. IEEE Trans. Inf. Theory 22, 58, Ma, J.; Si, Z.; He, Z.; Niu, K. Recursive encoding of spatiay couped LDPC codes with arbitrary rates. In Proceedings of the 25 IEEE Internationa Symposium on Persona, Indoor and Mobie Radio Communications, Hong Kong, China, 3 August 2 September 25; pp Mitche, D.; Lentmaier, M.; Costeo, D.J. Spatiay couped LDPC codes constructed from protographs. IEEE Trans. Inf. Theory 25, 6, Stinner, M.; Omos, P.M. On the waterfa performance of finite-ength SC-LDPC codes constructed from protographs. IEEE J. Se. Areas Commun. 26, 34, Johnson, S.; Lechner, G. Spatiay couped repeat-accumuate codes. IEEE Commun. Lett. 23, 7, Divsaar, D.; Doinar, S.; Jones, C.R.; Andrews, K. Capacity approaching protograph codes. IEEE J. Se. Areas Commun. 29, 27, Kasai, K.; Sakaniwa, K. Spatiay-couped MacKay Nea codes and Hsu Anastasopouos codes. In Proceedings of the 2 IEEE Internationa Symposium on Information Theory, Saint Petersburg, Russia, 3 Juy 5 August 2; pp Divsaar, D. Ensembe weight enumerators for protograph LDPC codes. In Proceedings of the 26 IEEE Internationa Symposium on Information Theory, Seatte, MA, USA, 9 4 Juy 26; pp Abu-Surra, S.; Divsaar, D.; Ryan, W.E. Enumerators for Protograph-based ensembes of LDPC and generaized LDPC codes. IEEE Trans. Inf. Theory 2, 57, c 26 by the authors; icensee MDPI, Base, Switzerand. This artice is an open access artice distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) icense (