An Unnoticed Strong Connection Between Algebraic-Based and Protograph-Based LDPC Codes, Part II: Nonbinary Case and Code Construction

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1 An Unnoticed Strong Connection Between Algebraic-Based and Protograph-Based LDPC Codes, Part II: Nonbinary Case and Code Construction Juane Li 1, Keke Liu 1, Shu Lin 1, William E Ryan 2, and Khaled Abdel-Ghaffar 1 1 Department of Electrical and Computer Engineering, University of California, Davis, CA 95616, USA 2 Zeta Associates, Eaton Place #500, Fairfax, VA {shulin, kkeliu, jueli, kaghaffar}@ucdavisedu, 2 ryan@ecearizonaedu Abstract This paper is the second part of a series of two papers on the investigation of the connection between two methods for constructing LDPC codes, namely the superposition and the protograph-based methods In the first part, by taking the graph point of view, we established a strong connection between these two construction methods and showed that, in the binary case, the protograph-based construction of LDPC codes is a special case of the superposition construction of LDPC codes Several special types of superposition construction of binary algebraic quasi-cyclic (QC) LDPC codes were presented In this second part, we continue the investigation of the connection between the superposition and the protograph-based methods for constructing nonbinary LDPC codes, especially non-binary QC-LDPC codes Several special superposition methods for constructing nonbinary QC-LDPC codes are presented The constructed codes can be regarded as algebraic-protograph QC-LDPC codes These codes not only have rich algebraic structures, but also perform well I INTRODUCTION This paper is the second part of a series of two papers The main objective of these two papers is to link the protograph-based method for constructing LDPC codes, proposed in 2003 by Thorpe [1], to an algebraic method, called superposition method, for constructing LDPC codes The superposition method of LDPC code construction was first introduced in 2002 under the name hybrid construction [2] and later renamed as the superposition construction in 2003 [3] and 2005 [4] In the first part [5], we established a strong connection between these two construction methods from the graph point of view We showed that, in the binary case, the protograph-based construction of LDPC codes is a special case of the superposition construction of LDPC codes From the graph point of view, the superposition method of LDPC code construction may be regarded as a generalized protographbased method for constructing LDPC codes Several special types of superposition constructions of binary algebraic QC- LDPC codes were presented In this second part, we continue our investigation of the connection between the superposition and the protographbased methods of nonbinary (NB) LDPC code construction, especially NB-QC-LDPC codes Several special superposition methods for constructing NB-QC-LDPC codes are presented All these construction methods are the protograph-based construction method from the graph point of view In our presentation, we follow the definitions and notations given in Part-I of our papers closely In many aspects, the developments are parallel to the developments for the binary case given in the companion paper (Part-I) [5] to this paper The rest of the paper (Part-II) is organized as follows In Section II, we present a generalized superposition method for constructing NB-LDPC codes and give a graph-theoretic interpretation of this construction method in terms of a generalized protograph Section III gives the superposition construction of NB-QC-LDPC codes in two different forms, one with sectionwise QC-structure and the other with block-cyclic structure Also presented in this section is a way of constructing the protograph of an NB-QC-LDPC code in which the edge labels give the precise information about how to connect the copies of the protograph to give the Tanner graph of the code Two special types of superposition construction of NB-QC-LDPC codes are given in Section IV Section V presents a family of binary and NB QC-LDPC codes constructed based on the Reed-Solomon (RS) codes with two information symbols Section VI gives a family of doubly QC-LDPC codes which have both section-wise QC-structure and block-cyclic structure [5] Also presented in this section are the constructions of two types of spatially coupled (SC) QC-LDPC code, convolutional and block Section VII concludes the paper with some remarks II SUPERPOSITION CONSTRUCTION OF NB-LDPC CODES In this section, we present the general framework of the superposition construction of NB-LDPC codes from both the matrix-theoretic and the graph points of view In our presentation, we will skip some of the definitions and explanations that were given in the companion paper [5] to this paper Similar to the binary case, superposition construction of an NB-LDPC code begins with a binary base matrix and a set of NB sparse matrices of the same size for replacing the 1- entries in the base matrix Let B b = [b i,j ] 0 i<m,0 j<n be an

2 m n binary matrix which serves as the base matrix for the superposition construction of an NB-LDPC code [2], [4], [5] Let R = {A 1,A 2,,A r } be a set of sparse k t matrices, with k t, over an NB field GF(q) where q is a prime or a power of a prime and q 2 Form an m n array H sp,q of k t matrices over GF(q) by replacing each 1-entry in B b by a member matrix in R and each 0-entry by a k t zero matrix (ZM) The array H sp,q is an mk nt sparse matrix over GF(q) The null space of H sp,q gives a q-ary LDPC code, denoted by C sp,q This construction is referred to as the superposition construction of NB-LDPC codes The subscripts sp and q of H sp,q and C sp,q stand for superposition and q-ary, respectively The base matrix B b and the set R are called the base matrix and the replacement (or superposition) set of matrices for the superposition construction of the code C sp,q, respectively As an m n array, H sp,q consists of n column-blocks, labeled by H 0,H 1,,H n 1, and m row-blocks, denoted by M 0,M 1,,M m 1 For 0 j < n,0 i < m, each column-block H j consists of t consecutive columns of H sp,q, and each row-block M i consists of k consecutive rows of H sp,q Let G b be the Tanner graph of the base matrix B b which consists of n variable-nodes (VNs) and m check-nodes (CNs), labeled by v 0,v 1,,v n 1 and w 0,w 1,,w m 1, respectively The Tanner graph associated with the matrix H sp,q, denoted by G sp,q (k,t), consists of nt VNs and mk CNs Partition the nt VNs of G sp,q (k,t) into n disjoint sets, denoted by Φ 0,Φ 1,,Φ n 1 For 0 j < n, Φ j consists of t VNs of G sp,q (k,t) which correspond to the t columns of the j-th column-block H j The t VNs in Φ j are referred to as the type-j VNs Partition the mk CNs of G sp,q (k,t) into m disjoint sets, denoted by Ω 0,Ω 1,,Ω m 1 For 0 i < m, Ω i consists of k CNs of G sp,q (k,t) which correspond to the k rows of the i-th row-block M i The k CNs in Ω i are referred to as the type-i CNs If an edge (j,i) connects a type-j VN and a type-i CN, it is called a type-(j,i) edge Each edge in G sp,q (k,t) is labeled by a nonzero element in GF(q) which is the nonzero entry in H sp,q that corresponds to the edge The Tanner graphg sp,q (k,t) associated with the arrayh sp,q is basically an expansion of the Tanner graph G b of the base matrix B b Each VN in G b is expanded into t VNs in G sp,q (k,t) and each CN in G b is expanded into k CNs in G sp,q (k,t) If t > k, the VNs and CNs in G b are expanded by two different degrees, t and k, respectively In this case, G b may be regarded as a protograph for constructing the larger Tanner graph G sp,q (k,t) Suppose we start with the bipartite graph representation G b of the base matrix B b To expand G b, we need to label each edge in G b by a member matrix in R Suppose the j-th VN and the i-th CN in G b are connected by an edge (j,i) which is labeled by the member matrix A e in R Let τ e denote the number of nonzero entries in A e In expanding G b, the j-th VN is expanded into t VNs of typej, the i-th CN in G b is expanded into k CNs of type-i, and the edge (j,i) is expanded into τ e edges of type-(j,i) The t type-j VNs are connected to the k type-i CNs by the τ e edges of type-(j,i) This expansion of G b gives the Tanner graph G sp,q (k,t) associated with the parity-check array H sp,q of theq-ary LDPC codec sp,q constructed by the superposition method usingb b and R as the base matrix and the replacement set, respectively Therefore, from the graph point of view, a q-ary LDPC code constructed based on the superposition method is a generalized protograph LDPC code Consider the special case for which t = k and each member matrix of the replacement set R is a k k regular matrix (ie, the weights of columns and weights of rows are constant and they are equal, see Part-I [5]) In this case, the expansion of G b is done by taking k copies of G b, permuting and labeling the edges of individual copies of G b according to the NB member matrices in R to connect them into a larger Tanner graph G sp,q (k,k) The null space of the adjacency matrix of G sp,q (k,k) gives a protograph-based NB- LDPC code The performance of a q-ary LDPC code depends on the choice of the base matrx B b and the member matrices of the replacement set R As stated in Part-I of this paper, the member matrices in R should satisfy the pair-wise RC-constraint There are many algebraic methods for constructing pair-wise RCconstrained matrices [6]-[21] III SUPERPOSITION CONSTRUCTION OF NB-QC-LDPC CODES QC-LDPC codes are preferred in applications to communications and data storage systems over other types of LDPC codes due to their advantages in encoding and decoding implementations In this section, we focus on superposition construction of NB-QC-LDPC codes Again, let B b be an m n binary base matrix for superposition construction of NB-LDPC codes Consider the special case for which k = t Suppose each member matrix in the replacement set R is a circulant over GF(q) of size k k Replacing each nonzero entry in B b by a member circulant in R and each zero-entry in B b by a k k ZM, we obtain an m n array H sp,q,qc of member circulants in R and/or ZMs of size k k Define the weight of a circulant as the weight of its generator (or its top row) If the weight of each k k member circulant in R is much smaller than k, then the array H sp,q,qc is a sparse matrix over GF(q) and its null space gives a q-ary QC-LDPC code, denoted by C sp,q,qc The subscript qc in H sp,q,qc and C sp,q,qc stands for quasi-cyclic In the following, we look at the above superposition construction of the q-ary QC-LDPC code C sp,q,qc from the graph point of view The array H sp,q,qc consists of n column-blocks, labeled by H 0,H 1,,H n 1, and m row-blocks, denoted by M 0,M 1,,M m 1 For 0 j < n,0 i < m, each column-block H j consists of k consecutive columns of H sp,q,qc, and each row-block M i consists of k consecutive rows of H sp,q,qc Now, we permute the rows and columns of H sp,q,qc based on the inverses of the row and column permutations defined by (2) to (5) given in the companion paper (Part-I) [5] of this paper The row and column permutations of H sp,q,qc give the

3 following k k array of m n matrices over GF(q) (the same array given by (1) in [5]): H sp,q,cyc = G 0 G 1 G k 1 D 0 D 1 D k 1 D = k 1 D 0 D k 2, (1) D 1 D 2 D 0 which has the block-cyclic structure Each row-block G i consists of k sections, each being an m n matrix over GF(q) The subscript cyc in H sp,q,cyc stands for block-cyclic The null space of H sp,q,cyc gives a q-ary QC-LDPC code C sp,q,cyc with block-cyclic structure C sp,q,qc and C sp,q,cyc are combinatorially equivalent For 0 e < k, we label the rows and columns of each matrix D e in H sp,q,cyc from 0 to m 1 and 0 to n 1, respectively, as in (1) Next, we combine the k matrices, D 0,D 1,,D k 1, into an m n matrix, denoted by B ptg,q, in which, for 0 i < m and 0 j < n, the entry at the location (i,j) is a k-tuple v i,j = (v i,j,0,v i,j,1,,v i,j,k 1 ) over GF(q) where, for 0 e < k, v i,j,e is the entry at the location (i,j) of the matrix D e Then, the Tanner graph G ptg,q of B ptg,q consists of n VNs and m CNs The edge connecting the j-th VN of G ptg,q to the i-th CN of G ptg,q is labeled by the k-tuple v i,j = (v i,j,0,v i,j,1,,v i,j,k 1 ) If v i,j is the allzero k-tuple, ie, all the components in v i,j are zeros, then the j-th VN and the i-th CN are not connected in G ptg,q We call the k-tuple v i,j the label vector Following the replacements of the 1-entries of B b by the member matrices in R and the row and column permutations performed onh sp,q,qc, it is easy to see that the label vector v i,j for the edge (j,i) is simply the generator of the member circulant A e in R that is used to replace the 1-entry in B b at the location (i,j) In the protograph-based construction of a q-ary QC-LDPC code C ptg,q,qc, the graph G ptg,q is used as the protograph To expand the protographg ptg,q by a factor ofk, we takek copies of G ptg,q, and then connect the copies of G ptg,q by permuting the edges of the individual copies of G ptg,q among the k copies ofg ptg,q Thek VNs of type-j are connected tok CNs of typei based on the label vector v i,j = (v i,j,0,v i,j,1,,v i,j,k 1 ) and its k 1 cyclic-shifts Let λ i,j be the number of nonzero components in v i,j (ie, the weight of v i,j ) Then, there are λ i,j VNs of type-j which are connected to a CN of type-i The locations of the nonzero components in v i,j give the copies of G ptg,q for which the VNs of type-j are connected to a CN of type-i Connecting the copies ofg ptg,q in this way, we obtain a larger graph G ptg,q (k,k) which is the Tanner graph associated with the array H sp,q,qc (or H sp,q,cyc ) The subscript ptg of G ptg,q, G ptg,q (k,k) and C ptg,q,qc stands for protograph The above shows that the q-ary QC-LDPC code C sp,q,qc constructed by superposition with the replacement set R in which each member matrix is a circulant of size k k over GF(q) is a protograph-based q-ary QC-LDPC code and C sp,q,qc is identical to C ptg,q,qc This says that a protographbased q-ary QC-LDPC code can be constructed algebraically by using the superposition method without computer aid to construct a protograph and connect its copies The protographbased NB-LDPC codes have been recently studied in [22], [23] Based on the above analysis, we see that the Tanner graph G b of the base matrix B b and the protograph G ptg,q are structurally identical, one without edge labels and the other with edge labels Weight-one circulants are the most commonly used circulants for constructing QC-LDPC codes, binary or NB For the binary case, such a circulant is called circulant permutation matrix (CPM) A q-ary CPM, denoted by Q, can be obtained from a binary CPM A by replacing all the 1-entries in A by the same nonzero element chosen from GF(q) Two different q-ary CPMs of the same size are mutually disjoint if the single nonzero components of their generators are at two different locations There are k pair-wise RC-constrained q- ary CPMs and they can be used as the member matrices in the replacement set R for superposition construction of NB- QC-LDPC codes Example 1 Suppose we choose the following matrix as the base matrix for superposition construction of an NB-QC- LDPC code: [ B b = Let α be the primitive element of GF(2 2 ) constructed based on the primitive polynomial 1 + X + X 2 over GF(2) The field GF(2 2 ) consists of 4 elements, 0,1,α,α 2 Let R be the replacement set for superposition construction which consists of the following 4 circulants of size 3 3 over GF(2 2 ), Q 1 = Q 3 = 1 α α α α2 α α 2 0,Q 2 =,Q 4 = ] 0 α α2 α 2 0 α α α 2 0 α α α Construct a 2 3 array H sp,q,qc by replacing the 1-entries of B b with the above circulants and the 0-entries by 3 3 ZMs as follows: 1 α 0 0 α α [ ] 0 1 α α 2 0 α Q1 Q H sp,q,qc = 2 0 α 0 1 α α = Q 3 0 Q α α 0 0 α α 0 0 α α Permuting the rows and columns of H sp,q using the inverses of the permutations defined by (2) to (5) in [5], we obtain the following 3 3 array of 2 3 submatrices over GF(2 2 ): H sp,q,cyc =, α α 0 0 α α α α α α 0 α α α α 0 0 α α α,

4 v 0 v 1 v 2 c 0 c 1 (a) v 0 v 1 v 2 v0,0 v 0,1 v 1,0 c 0 c 1 Fig 1 (a) The protograph G b associated with the base matrix B b given in Example 1; (b) The Tanner graph G ptg,q associated with the base matrix B ptg,q given in Example 1, where v 0,0 = (1,α,0), v 0,1 = (0,α,α 2 ), v 1,0 = (0,0,α 2 ), and v 1,2 = (α,0,0) Combining the three 2 3 matrices in the first row-block of H sp,q,cyc, we obtain the following 2 3 matrix in which each entry is a 3-tuple over GF(2 2 ): B ptg = (b) [ (1,α,0) (0,α,α 2 ) (0,0,0) (0,0,α 2 ) (0,0,0) (α,0,0) The Tanner graphs G b and G ptg,q of B b and B ptg,q are shown in Fig 1(a) and 1(b), respectively Example 2 Consider the following 2 4 base matrix for the superposition construction of an NB-QC-LDPC code: [ ] B b = Choose a replacement set R which consists of the 8 member CPMs of size 8 8 over GF(2 8 ), denoted by Q 1,Q 2,,Q 8 Let α be a primitive element of GF(2 8 ) The generators of these ary CPMs are: g 1 = (0,0,0,0,0,0,α 61,0), g 2 = (0,0,0,0,0,α 106,0,0), g 3 = (0,0,0,α 240,0,0,0,0), g 4 = (0,0,0,0,0,0,0,α 125 ), g 5 = (0,0,0,α 229,0,0,0,0), g 6 = (0,0,0,0,0,0,0,α 94 ), g 7 = (0,0,0,0,0,0,α 199,0), g 8 = (0,0,0,0,0,α 99,0,0) Replacing 1-entries in B b by the member 256-ary CPMs in R according to the replacement constraint [4], [5], we obtain a 2 4 array of 256-ary CPMs of size 8 8 One such replacement gives the following 2 4 array: [ ] Q1 Q H sp,q,qc = 2 Q 3 Q 4 Q 5 Q 6 Q 7 Q 8 The null space of H sp,q,qc gives a (2,4)-regular 256-ary (32,16) QC-LDPC code C sp,q,qc The Tanner graph of the code C sp,q,qc has girth 8 and contains 20 cycles of length 8, no cycle of length 10, and 160 cycles of length 12 The block error performances of the code C sp,q,qc decoded with 50 and 100 iterations of the Fast-Fourier-Transform q-ary sumproduct algorithm (FFT-QSPA) [17] are included in Fig 3 The decoding converges fast The gap between 50 and 100 iterations is less than 01 db From the figure, at the block error rate (BLER) around 10 6, we see that the code C sp,q,qc performs 075 db away from the sphere packing bound (SPB) For comparison, we include two other 256-ary codes reported in [22], [23] in Fig 3, denoted by C C-NBPB and ] v 1,2 C U-NBPB, respectively The subscripts C-NBPB and U- NBPB stand for constrained nonbinary protograph-based and unconstrained nonbinary protograph-based, respectively The code C C-NBPB is constructed through scale-copypermute operations based on a scaled protograph, and the code C U-NBPB is constructed through copy-scale-permute operations based on an unlabeled protograph The Tanner graphs of these two codes have the same cycles distributions and contain the same number of cycles of length 8, 10, and 12 as the proposed code C sp,q,qc The code C C-NBPB is in QC form, but the code C U-NBPB is not in QC form The block error performances of the two codesc C-NBPB andc U-NBPB were decoded with 100 iterations of FFT-QSPA in [22], [23] From Fig 3, we see that the code C sp,q,qc slightly outperforms the codes C C-NBPB and C U-NBPB in the simulation region IV SUPERPOSITION CONSTRUCTION OF NB-QC-LDPC CODES THROUGH CPM-DISPERSION In the companion paper [5] to this paper, we investigated the superposition construction of binary QC-LDPC codes using the binary CPM-dispersion of a base matrix over a finite field In this section, we present the superposition construction of NB-QC-LDPC codes using the CPM-dispersion over an NB field The presentation is in parallel to the presentation of the binary case given in [5] A CPM-Dispersion for Constructing NB-QC-LDPC Codes Let α be a primitive element of the finite field GF(q) where q is either a prime or a power of a prime and q 2 Then, the powers of α, α 0 = 1,α,α 2,,α q 2, give all the nonzero elements of GF(q) For 0 l < q 1, we represent the element α l by a q-ary CPM of size (q 1) (q 1) (with rows and columns labeled from 0 to q 1, respectively) whose generator (or top row) has α l as its single nonzero component at the position l We denote this q-ary CPM by Q(α l ) All the nonzero entries in Q(α l ) are α l It is clear that the representation of the element α l by the q-ary CPM Q(α l ) of size (q 1) (q 1) is unique and the mapping between α l and Q(α l ) is one-to-one This matrix representation of α l is referred to as the q-ary CPM-dispersion (CPM-qD) of α l [20], [21] Notice that if we replace all the nonzero entries in Q(α l ) by the unit element 1 of GF(q), we obtain a binary CPM of size (q 1) (q 1) which is called the binary CPM-dispersion (CPM-bD) of α l, denoted by A(α l ) LetB q = [b i,j ] 0 i<m,0 j<n be anm n matrix over GF(q) We form an m n array H q,qc of q-ary CPMs and/or ZMs of size (q 1) (q 1) from B q as follows: (1) if b i,j 0 and b i,j = α l, we replace b i,j with its q-ary CPM-dispersion Q(α l ); and (2) if b i,j = 0, we replace b i,j by a ZM of size (q 1) (q 1), ie, Q(0) is a (q 1) (q 1) ZM The array H q,qc is called the CPM-qD of the base matrix B q The null space of H q,qc gives a q-ary QC-LDPC code C q,qc of length n(q 1), called a q-ary CPM-QC-LDPC code The above method for constructing q-ary CPM-QC-LDPC code is called the CPM-qD construction The CPM-qD construction is basically a superposition construction with a deterministic

5 q-ary CPM replacement of each nonzero entry in the base matrix B q Notice that the base matrix for the superposition construction is now a matrix over a finite field Let G q denote the Tanner graph of B q The edge of G q are labeled by their corresponding nonzero entries in B q The q- ary CPM-dispersions of the nonzero entries in B q expand the Tanner graph G q of B q by a factor of q 1 into a large bipartite graph, denoted by G q,qc (q 1,q 1), with n(q 1) VNs and m(q 1) CNs Each VN in G q is expanded into q 1 VNs in G q,qc (q 1,q 1) and each CN in G q is expanded into q 1 CNs in G q,qc (q 1,q 1) In terms of the protograph-based code construction, G q is used as the protograph This protograph is duplicated q 1 times The edges of the q 1 copies of G q are permuted to connect the copies into a large Tanner graph based on the constituent q-ary CPMs in the array H q,qc The above CPM-qD construction of a q-ary CPM-QC- LDPC code reduces to the construction of its base matrix B q over GF(q) For such a q-ary CPM-QC-LDPC code, the girth of the Tanner graph of the code is related to the choice of the base matrix B q In the companion paper (Part-I) [5] of this paper, two theorems (Theorems 1 and 2) are stated, which give the necessary and sufficient conditions on the base matrix B q for the Tanner graph of a binary CPM-QC-LDPC code to have girth at least 6 and 8, respectively These two theorems can also apply to q-ary CPM-QC-LDPC codes constructed using the CPM-qD construction We restate these two theorems here for the NB case Theorem 1 Let B q be the base matrix of a q-ary CPM- QC-LDPC code C q,qc whose parity-check matrix H q,qc is the CPM-qD of B q A necessary and sufficient condition for the Tanner graph of C q,qc to have girth at least 6 is that every 2 2 submatrix in the base matrix B q contains at least one zero entry or is non-singular Theorem 2 A necessary and sufficient condition for the Tanner graph of a q-ary CPM-QC-LDPC code C q,qc with base matrix B q to have girth at least 8 is that no 2 2 or 3 3 submatrix in the base matrix B q has two identical nonzero terms in its determinant expansion The conditions given in Theorems 1 and 2 are called the 2 2 and 3 3 submatrix (SM) constraints (the 2 2 and 3 3 SM-constraints), respectively In the CPM-qD construction of a q-ary CPM-QC-LDPC code, the 2 2 and 3 3 SMconstraints are used to construct a base matrix B q B CPM-bD Construction of NB-CPM-QC-LDPC Codes Through Binary-to-NB Replacement Using the dispersion method to construct NB-CPM-QC- LDPC codes directly as presented above, there is a constraint on the size of CPMs If we want a large CPM size, we have to use a large field This will result in high decoder complexity To resolve this problem, we can combine the dispersion method with the binary-to-nb (B-to-NB) replacement method [20], [21] to construct NB-CPM-QC-LDPC codes whose parity-check arrays consist of large CPMs over small fields This combined construction method is presented below First, a chosen base matrix B q = [b i,j ] 0 i<m,0 j<n over GF(q) is dispersed into an m n array H b,qc = [A i,j ] 0 i<m,0 j<n of binary CPMs and/or ZMs of size (q 1) (q 1), ie, a nonzero entry in B q is dispersed into its corresponding binary CPM Then, for 0 i < m,0 j < n, we replace all the 1-entries in the binary CPM A i,j by a nonzero element from a chosen code symbol field GF(q ) The field GF(q ) for code symbols can be smaller or larger than the dispersion field GF(q), or the same This binary to q - ary replacement operation transforms the binary CPM A i,j into a q -ary CPM Q i,j All the nonzero elements in Q i,j are equal Applying this binary to q -ary replacement to each binary CPM in the array H b,qc, we obtain an m n array H q,qc = [Q i,j ] 0 i<m,0 j<n of q -ary CPMs and/or ZMs of size (q 1) (q 1) The null space of H q,qc gives a q -ary CPM-QC-LDPC code C q,qc Using H b,qc as the binary base array and the B-to-NB replacement for code construction, we can construct an NB- CPM-QC-LDPC code over any finite field, large or small For simplicity, we call this combined construction method the CPM-bD/B-to-NB construction With the CPM-bD/B-to- NB construction method, we can construct a base array of binary CPMs of large size using a large field and then using a small code symbol field as the B-to-NB replacement field The replacement of the 1-entries of a binary CPM by a nonzero element from another field may be considered as another superposition operation Hence, the above construction of NB-CPM-QC-LDPC code may be regarded as double superposition construction In fact, if we disperse each nonzero entry in H q,qc into either a binary or q -ary CPM of size (q 1) q 1), we obtain a larger array of CPMs and/or ZMs of size (q 1) q 1) The CPM-bD/B-to- NB construction method is very flexible for constructing both binary and nonbinary CPM-QC-LDPC codes C Masking There are many algebraic methods for constructing base matrices that satisfy the 2 2 SM-constraint given in Theorem 1 The Tanner graph associated with the arrays obtained by CPM-dispersions of these base matrices have girth at least 6 The masking technique presented in [9], [10], [18], [19] can be performed on a 2 2 SM-constrained base matrix to obtain a new base matrix which also satisfies the 3 3 SM-constraint Then, the Tanner graph associated with the CPM-dispersion of the masked base matrix has girth at least 8 Properly masking a base matrix not only increases the girth but also significantly reduces the number of short cycles of the Tanner graph associated with the array obtained by CPMdispersion of the unmasked base matrix Let B q = [b i,j ] 0 i<m,0 j<n be an m n 2 2 SMconstrained base matrix over GF(q) Suppose we replace a nonzero entry in B q by the zero-element of GF(q) In the q-ary CPM-dispersion H q,qc of B q, this replacement results in replacing a (q 1) (q 1) q-ary CPM in H q,qc by a

6 (q 1) (q 1) ZM which is referred to as masking Let δ be a nonnegative integer less than the number of nonzero entries in B q The replacement of δ nonzero entries in B q by δ zeros amounts to replacing δ q-ary CPMs by δ ZMs at the locations inh q,qc corresponding to the locations of theδ nonzero entries in B q which are replaced by zeros Masking δ q-ary CPMs in H q,qc amounts to removing δ(q 1) edges from the Tanner graphg q,qc (q 1,q 1) associated withh q,qc Removing these edges in G q,qc (q 1,q 1) may break many short cycles in G q,qc (q 1,q 1) As a result, the resultant Tanner graph, denoted by G q,qc,mask (q 1,q 1), may have a much smaller number of short cycles, or a larger girth (larger than 6), or both The subscript mask of G q,qc,mask (q 1,q 1) stands for masking In choosing the nonzero entries in B q to be masked, we have to avoid disconnecting the Tanner graph of H q,qc into disjoint graphs The operation of masking the base matrix B q = [b i,j ] 0 i<m,0 j<n can be modeled mathematically as a Hadamard matrix product [24] Let Z = [z i,j ] 0 i<m,0 j<n be an m n matrix with the zero and unit elements of GF(q) as entries Define the following product of Z and B q : B q,mask = Z B q = [z i,j b i,j ] 0 i<m,0 j<n where z i,j b i,j = b i,j if z i,j = 1 and z i,j b i,j = 0 if z i,j = 0 In this matrix product, some nonzero entries in B q at the locations corresponding to the locations of zero entries in Z are replaced (or masked) by zeros The q-ary CPM-dispersion of B q,mask gives an m n masked array, denoted by H q,qc,mask, of CPMs and ZMs of size (q 1) (q 1) We call Z and B q,mask the masking matrix and the masked base matrix, respectively The null space of H q,qc,mask also gives an NB-CPM-QC-LDPC code, denoted by C q,qc,mask Since B q satisfies the 2 2 SM-constraint, the masking matrix Z should be designed to obtain a masked base matrix B q,mask which satisfies the 3 3 SM-constraint In this case, the Tanner graph G q,qc,mask (q 1,q 1) associated with the masked array H q,qc,mask has girth at least 8 as well as a smaller number of short cycles CPM-dispersions of properly chosen base matrices in conjunction with masking can result in NB- CPM-QC-LDPC codes with good error performance From the viewpoint of graph, the Tanner graph G q,mask of B q,mask is the protograph for constructing q-ary CPM-QC- LDPC code C q,qc,mask The edges of G q,mask are labeled by their corresponding nonzero entries in the masked base matrix B q,mask V CPM-DISPERSION CONSTRUCTION OF NB-CPM-QC-LDPC CODES BASED ON RS CODES In this section, we use a known class of 2 2 SMconstrained base matrices for constructing both binary and NB CPM-QC-LDPC codes This class of 2 2 SM-constrained base matrices are constructed based on RS codes with two information symbols [7], [10], [19] Let α be a primitive element of GF(q) We form the following (q 1) (q 1) matrix over GF(q) (see [18, p487, (115)]): W= w 0 w 1 w q 2 α 0 1 α 1 α 2 1 α q 3 1 α q 2 1 = α q 2 1 α 0 1 α 1 α q 4 1 α q 3 1 α 1 α 2 1 α 3 1 α q 2 1 α 0 1 (2) Each row w f of W, with 0 f < q 1, is a minimumweight codeword of the (q 1,2,q 2) RS code over GF(q) with length q 1, dimension 2 and minimum distance q 2 Each row (or each column) of W contains a single 0-entry There are q 1 0-entries which lie on the main diagonal of W The matrix W has the cyclic structure Each row of W is the cyclic shift of the row above it one place to the right and the top row is the cyclic shift of the last row one place to the right W satisfies the 2 2 SM-constraint [16] For 1 m,n < q, any m n submatrix B q of W can be used to as a base matrix for constructing an NB-CPM- QC-LDPC code using either the CPM-qD construction or the CPM-bD/B-to-NB construction Example 3 Consider the field GF(2 6 ) with a primitive element α Using this field, we can construct a matrix W over GF(2 6 ) in the form given by (2) Take a 4 8 submatrix (avoiding the 0-entry) from W and use it as a base matrix B q for code construction We mask B q with the following 4 8 matrix: Z = (3) After masking, we obtain a 4 8 masked base matrix: B q,mask = α 35 0 α 23 0 α 47 α 27 α 56 α 59 0 α 35 0 α 23 α 33 α 47 α 27 α 56 α 25 α 2 α 35 α 52 α 23 0 α 37 0 α 61 α 25 α 2 α 35 0 α 23 0 α 47 (4) which has row and column weights 3 and 6, respectively This masked base matrix satisfies the 3 3 SM-constraint The 64- ary CPM-dispersion ofb q,mask gives a4 8 arrayh q,qc,mask of 64-ary CPMs and ZMs of size The null space of the array H q,qc,mask gives a 64-ary (3,6)-regular (504,252) CPM-QC-LDPC code C q,qc,mask The protograph G q,ptg of this code is shown in Fig 2 The label vectors v i,j with 0 i < 4,0 j < 8 are simply the generators of 64-ary CPMs in the array H q,qc,mask The Tanner graph of C q,qc,mask has girth 8 and contains 819 cycles of length 8 and 12,348 cycles of length 10 The block error performance of the code C q,qc,mask decoded with a maximum number of 50 iterations of the FFT- QSPA is included in Fig 4 From this figure, at the BLER of 10 6, we see that C q,qc,mask performs about 16 db away from the SPB Also included in this figure is the block error performance of a 64-ary LDPC code C q,peg constructed based on the PEG algorithm proposed in [25] The code C q,peg is an irregular

7 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1,0 v 3,0 v 3,1 v 0,1 v 0,3 v 2,1 v2,2 v 3,2 v 1,2 v 1,4 v2,3 c 0 c 1 c 2 c 3 v 0,4 v 2,0 v 0,5 v 2,4 v 1,5 v 1,6 v 2,6 v 3,3 v 0,1 v 0,7 Fig 2 The protograph G q,ptg of the 64-ary (504,252) CPM-QC-LDPC code given in Example 3 code which has column weight 3 and average row weight 6 Its Tanner graph also has girth 8 and contains 945 cycles of length 8 and 11,655 cycles of length 10 We decoded this PEG code using the FFT-QSPA with 50 iterations Fig 4 shows that the code C q,qc,mask performs just as well as the PEG code C q,peg One point necessary to mention here is that the PEG code C q,peg does not have QC structure Example 4 In this example, we will use the CPM-bD/B-to- NB construction method to construct two 8-ary (2376, 1188) CPM-QC-LDPC codes Based on the prime field GF(199), we can construct a base matrix W over GF(199) in the form given by (2) We take a 6 12 submatrix B q from W avoiding the 0-entries Next, we use the PEG algorithm given in [25] to construct a 6 12 binary matrix Z with average column weight 25 as follows: Z = v3,5 v 1,7 v 3,7 (5) Masking B q with the masking matrix Z given by (5), we obtain a 6 12 masked base matrix B q,mask which satisfies both the 2 2 and 3 3 SM-constraints given in Theorems 1 and 2 The binary CPM-dispersion of B q,mask gives a 6 12 array H b,qc,mask of binary CPMs and ZMs of size This array H b,qc,mask is an matrix over GF(2) with average column weight 25 Since the base matrix B q,mask satisfies the2 2 and3 3 SM-constraints, the Tanner graph G b,qc,mask associated with H b,qc,mask has girth at least 8 Actually, the graph G b,qc,mask has girth 8 and contains 396 cycles of length eight, 594 cycles of length 10 and 3762 cycles of length 12 Using the arrayh b,qc,mask as the binary base array and the field GF(2 3 ) as the code symbol field for the CPM-bD/B-to-NB construction, we can construct an 8-ary (2376, 1188) CPM- QC-LDPC code C q,qc,mask with rate 1/2 The Tanner graph G q,qc,mask of the code C q,qc,mask has the same girth and cycle distributions as the Tanner graph G b,qc,mask associated with the binary array H b,qc,mask The parity-check array H q,qc,mask of the 8-ary CPM-QC- LDPC code C q,qc,mask is a 6 12 array of CPMs and ZMs of size over GF(2 3 ) Each row-block of H q,qc,mask consists of 5 CPMs and 7 ZMs The generators of the CPMs in each row-block of H q,qc,mask are given in Table I, each specified by a pair (j,α s ) with 0 j < 198 and 0 s < 7 The first component j of (j,α s ) is a number which specifies the location of the single nonzero component of the generator of a CPM and the second component α s gives the value of this single nonzero component which is a nonzero element of GF(2 3 ) where α is a primitive element of GF(2 3 ) Suppose we take a different 6 12 submatrix B q from W Applying the CPM-bD/B-to-NB construction method together with the masking matrix Z given by (5), we can construct another 8-ary (2376,1188) CPM-QC-LDPC code Cq,qc,mask The Tanner graph of the code Cq,qc,mask has girth 10 and contains 396 cycles of length 10 and 4653 cycles of length 12 The parity-check array H q,qc,mask of the 8-ary CPM-QC- LDPC code Cq,qc,mask is also given in Table I in terms of the generators of its CPMs The block error performances of the codes C q,qc,mask and Cq,qc,mask decoded with 80 iterations of the FFT-QSPA are shown in Fig 4 At the BLER of3 10 5, the codesc q,qc,mask and Cq,qc,mask performs about 075 db and 07 db away from the SPB, respectively The code Cq,qc,mask outperforms C q,qc,mask by about 005 db This may be due to the fact that the Tanner graph of Cq,qc,mask has a larger girth (girth 10) than that of C q,qc,mask (girth 8) For comparison, the block error performance of an 8-ary (2496,1248) CPM-QC-LDPC code C q,qc,hlzz given in [26, p3441, Figure 2] is included in Fig 4, where the subscript hlzz stands for the initial letters of the last names of the authors The Tanner graph of this code has girth 12 and contains 23,504 cycles of length 12 The code C q,qc,hlzz is longer than the code C q,qc,mask by 120 symbols In [26], the code C q,qc,hlzz was decoded with 80 iterations of the FFT- QSPA From Fig 4, we see that the code C q,qc,mask slightly outperforms the C q,qc,hlzz below the BLER of 10 2 even though the code C q,qc,mask is shorter and the Tanner graph of the code C q,qc,hlzz has a larger girth The code Cq,qc,mask outperforms C q,qc,hlzz by about 01dB This shows that a relatively large girth can give a better error performance to some extent, but a too large girth may not help In this following example, we will construct a binary highrate CPM-QC-LDPC code Example 5 Choose the prime field GF(127) for the code construction We can construct a matrix W in the form of (2) Taking the 0-th, 21-st, 42-nd, 63-rd, 84-th, and 105-th rows from W (the rows and columns of W are labeled from 0 to 125), we obtain a submatrix B q and use B q as the base matrix The way in which we choose the 6 rows from W gives a submatrix B q which has block-cyclic structure Every row in B q is the cyclic shift of the row above it 21 positions to the right and the top row is the cyclic shift of the last row 21 positions to the right The binary CPM-dispersion of B q gives a array H b,qc of CPMs and ZMs of size The array H b,qc

8 is a matrix over GF(2) and its null space gives a binary nearly (6, 126)-regular (15876, 15120) CPM-QC- LDPC C b,qc with rate The Tanner graph of the code has girth 6 and contains 37,817,640 cycles of length 6 The error performances of the codec b,qc decoded with 5, 10 and 50 iterations of the min-sum algorithm (MSA) are shown in Fig 5 At the BLER of 10 6, the code decoded with 50 iterations of the MSA performs less than 09 db away from the SPB At the BER of 10 8, the code decoded with 50 iterations of the MSA performs about 11 db away from the Shannon limit From this figure, we can see that the decoding of the code C b,qc converges very fast The performance gap between the 5 and 10 iterations is about 02 db and the gap between the 10 and 50 iterations is less than 01 db Notice that since the base matrix B q has block-cyclic structure, the null space of H b,qc has both the section-wise and the block-cyclic structures, called the doubly QC-structure [5] Thus, the code C b,qc given by the null space of H b,qc is a doubly QC-LDPC code Therefore, we can decode the code C b,qc with either the reduced-complexity iterative decoding scheme proposed in [19], [20] based on its section-wise cyclicshift structure or the one proposed in [27] based on its blockcyclic structure For a given field, the combination of the CPM-bD construction method, masking and submatrices of W given by (2) can give us a family codes with different lengths and rates The bit error performances of a sequence of codes constructed based on GF(223) are shown in Fig 6, where n represents the code length and r represents the code rate From Fig 6, we can see that all the codes can achieve a bit error rate of 10 8 without visible error-floors VI DOUBLY AND SPACIALLY COUPLED QC-LDPC CODES In this section, we present two special types of QC-LDPC codes constructed based on the 2 2 SM-constrained matrix W over GF(q) given by (2) A Doubly QC-LDPC Codes Label the rows and columns of W from 0 to q 2 Suppose q 1 can be factored as the product of two positive integers l and r, ie, lr = q 1 Partition W into r submatrices of size l (q 1), denoted byw 0,W 1,,W r 1, where for0 i < r, the submatrix W i consists of the il-th to the ((i+1)l 1)- th rows of W Next, we partition the submatrix W 0 into r submatrices of sizel l, denoted byw 0,0,W 0,1,,W 0,r 1 Then, W 0 = [W 0,0 W 0,1 W 0,r 1 ] Based on the cyclic structure of W, we readily see that, for 0 i < r, the submatrix W i can be obtained by cyclically shifting all the rows of W 0 = [W 0,0 W 0,1 W 0,r 1 ] il positions to the right, ie, W i = [W 0,r i W 0,r 1 W 0,0 W 0,r i 1 ] Hence, the matrix W can be put as an r r array of l l submatrices in the following form: W 0,0 W 0,1 W 0,r 1 W 0,r 1 W 0,0 W 0,r 2 W =, (6) W 0,1 W 0,2 W 0,0 which has a block-cyclic structure For 0 j < r and 1 m,n < l, we take an m n submatrix R 0,j from W 0,j The submatrices R 0,0,R 0,1,,R 0,r 1 are taken from W 0,0,W 0,1,,W 0,r 1 under the following restriction: for j j, the locations of the entries of R 0,j in W 0,j are identical to the locations of the entries of R 0,j in W 0,j Form the following r r array B q,cyc of m n submatrices over GF(q): R 0,0 R 0,1 R 0,r 1 R 0,r 1 R 0,0 R 0,r 2 B q,cyc = (7) R 0,1 R 0,2 R 0,0 B q,cyc is a submatrix of W and satisfies the 2 2 SMconstraint This simply follows from the block-cyclic structure of W 0,W 1,,W r 1 and the location constraint on the entries of the submatices R 0,0,R 0,1,,R 0,r 1 B q,cyc has block-cyclic structure In the code construction, the array B q,cyc is used as the base matrix for CPM-dispersion Dispersing each nonzero entry of B q,cyc into a CPM of size (q 1) (q 1) and each zero entry into a ZM of size (q 1) (q 1), we obtain an r r array H qc,cyc of m n subarrays of CPMs and/or ZMs of size (q 1) (q 1) Then the null space of H qc,cyc gives a doubly CPM-QC-LDPC code C qc,cyc With doubly QC-structure, the code C qc,cyc can be decoded with either the reduced-complexity iterative decoding scheme proposed in [19], [20] based on its section-wise cyclic-shift structure or the one proposed in [27] based on its block-cyclic structure For 0 j < r, let CPM(R 0,j ) denote the CPM-dispersion of the m n submatrix R 0,j CPM(R 0,j ) is an m n array of CPMs and/or ZMs of size (q 1) (q 1) and it is an m(q 1) n(q 1) matrix We combine the matrices CPM(R 0,0 ), CPM(R 0,1 ),, CPM(R 0,r 1 ) into an m(q 1) n(q 1) matrix B ptg with vector entries as described in Section III, each vector being an r-tuple Then, from the graph point of view, the Tanner graph G ptg of B ptg is a protograph for constructing the doubly QC-LDPC code C qc,cyc The edges ofg ptg are labeled with the vector entries in B ptg In the protograph-based construction of the codec qc,cyc, we taker copies ofg ptg and connect then into a bipartite graph G ptg (r,r) as described in Section III Then, G ptg (r,r) is the Tanner graph of the doubly QC-LDPC code C qc,cyc B Spatialy Coupled QC-LDPC Codes The base array B q,cyc given by (7) consists of r columnblocks, each containing r submatrices of size m n Each column-block of B q,cyc is a cyclic-shift of the column-block on its left one submatrix downward Based on this downward cyclic-shift structure, we construct two types of spatially

9 coupled (SC) QC-LDPC codes, convolutional and block types We use two special cases to illustrate the constructions Using the two submatrices R 0,0 and R 0,r 1 in the first column-block of B q,cyc, we form the following semi-infinite matrix: B q,sc,con = R 0,0 R 0,r 1 R 0,0 R 0,r 1 R 0,0 R 0,r 1 R 0,0 (8) SinceB q,cyc satisfies the2 2 SM-constraint, we can readily see that the semi-infinite array B q,sc,con also satisfies the 2 2 SM-constraint The CPM-dispersion of the array B q,sc,con gives a semi-infinite array H sc,con of CPMs and ZMs of size (q 1) (q 1) which specifies a convolutional SC-QC-LDPC code, denoted by C sc,con The subscripts sc and con in B q,sc,con, H sc,con and C sc,con stand for spatially coupled and convolutional, respectively The base array for constructing the second type of SC-QC- LDPC code is given in the following form: B q,sc,blk = R 0, R 0, R 0, R 0,0 R 0,1 R 0,2 R 0,3 R 0,r 1 (9) The CPM-dispersion of the matrix B q,sc,blk gives an array of CPMs and ZMs of size (q 1) (q 1) whose null space gives a block SC-QC-LDPC code, denoted by C sc,blk The subscript blk in B q,sc,blk and C sc,blk stands for block Clearly, the above two constructions of the SC-QC-LDPC codes can be generalized by using more than two of the submatrices R 0,0,R 0,1,,R 0,r 1 VII CONCLUSION AND REMARKS In this paper, we presented the superposition method for constructing NB-LDPC codes and provided a graph-theoretic interpretation of this method in terms of the protograph We showed that there is a strong connection between the superposition and protograph-based methods for constructing NB-LDPC codes Various specific superposition methods for constructing NB-QC-LDPC codes were presented Also in this paper, we presented the constructions of doubly QC-LDPC codes and spatially coupled QC-LDPC codes, convolutional and block types, based on a class of base matrices that have cyclic structure This class of base matrices are constructed based on RS codes with two information 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