Cyclic and Quasi-Cyclic LDPC Codes: New Developments

Transcription

1 1 Cyclic and Quasi-Cyclic LDPC Codes: New Developments Qin Huang, Qiuju Diao, Shu Lin and Khaled Abdel-Ghaffar {qinhuang,qdiao, shulin, Department of Electrical and Computer Engineering University of California, Davis, CA Abstract This paper presents a technique to decompose a cyclic code given by a parity-check matrix in circulant form into descendant cyclic and quasi-cyclic codes of various length and rates Based on this technique, cyclic finite geometry (FG) LDPC codes are decomposed into a large class of cyclic FG-LDPC codes and a large class of quasi-cyclic FG-LDPC codes I INTRODUCTION In this paper, it is shown that a cyclic code given by a parity-check matrix in circulant form can be decomposed, through specific column and row permutations, into families of cyclic or quasi-cyclic (QC) codes, called descendant codes If the roots of the generator polynomial of a given cyclic code, called the mother code, are given, the roots of the generator polynomial of a cyclic descendant code can be characterized If the mother cyclic code is a cyclic LDPC code, such as a cyclic finite geometry (FG) LDPC code [1], then decomposition results in a family of cyclic descendant LDPC codes and a family of QC-LDPC codes Decomposition of cyclic FG-LDPC codes enlarges the repertoires of cyclic and QC-LDPC codes LPDC codes with cyclic or QC structure have advantages over other types of LDPC codes in hardware implementation of encoding and decoding A cyclic LDPC code can be efficiently and systematically encoded with a single feedback shift-register with complexity linearly proportional to the number of paritycheck symbols (or information symbols) of the code [2] Encoding of a QC-LDPC code can also be efficiently implemented but requires multiple shift-registers [3] It is in general more complex than encoding of a cyclic code but still enjoys linear complexity However, QC-LDPC codes enjoy some advantages in hardware implementation of decoding in terms of wire routing [4] Furthermore, the QC-structure allows partially parallel decoding [5] which offers a tradeoff between decoding complexity and decoding speed, while cyclic structure allows either full parallel or serial decoding A cyclic LDPC code can be put in QC form through column and row permutations As a result, a cyclic LDPC code enjoys both encoding and decoding implementation advantages Encoding is carried out in cyclic form while decoding is carried out in QC form This research was supported by NSF under the Grants CCF and CCF , NASA under the Grant NNX09AI21G and gift grants from Northrop Grumman Space Technology, Intel and Denali Software Inc II DECOMPOSITION OF A CIRCULANT Consider the following n n circulant W over the field GF(q) where q is a power of a prime: w 0 w 1 w n 1 w n 1 w 0 w n 2 W = Ψ(w) = (1) w 1 w 2 w 0 This circulant is uniquely specified by its first row w = (w 0,w 1,,w n 1 ) which is commonly called the generator of the circulant For simplicity, we denote the circulant W by its generator as follows: W = Ψ(w) = Ψ(w 0,w 1,,w n 1 ) From (1), we see that the rows and columns of W are labeled from 0 to n 1 Let Ψ (1) (w) denote the circulant obtained by simultaneously cyclically shifting all the rows of Ψ(w) one place to the right Note that Ψ(w) and Ψ (1) (w) have identical set of rows and identical set of columns except that all the columns are cyclically shifted one place to the right and all the rows are cyclically shifted upward one place Therefore, the null spaces of Ψ(w) and Ψ (1) (w) are identical Suppose n can be factored as a product of two positive integers, c and l, such that c 1 and l 1, ie, n = c l and c and l are proper factors of n Let I = {0,1,2,,c l 1} be the set of indices (or labels) for the rows and columns of the n n circulant Ψ(w) given by (1) Define the following index sets: π (0) = [0,c,2c,,(l 1)c], (2) π = [π (0),π (0) + 1,,π (0) + c 1] (3) Then, π gives a permutation of the indices in I Suppose that we first permute the columns and then the rows of W based on π These column and row permutations based on π result in the following c c array of circulants of size l l over GF(q): Φ(w) = Ψ(w 0 ) Ψ(w 1 ) Ψ(w c 2 ) Ψ(w c 1 ) Ψ (1) (w c 1 ) Ψ(w 0 ) Ψ(w c 3 ) Ψ(w c 2 ) Ψ (1) (w 2 ) Ψ (1) (w 3 ) Ψ(w 0 ) Ψ(w 1 ) Ψ (1) (w 1 ) Ψ (1) (w 2 ) Ψ (1) (w c 1 ) Ψ(w 0 ), (4) where, for 0 i < c, w i = (w i,w c+i,,w (l 1)c+i ) is the generator of the circulant Ψ(w i ) Each l l circulant Ψ(w i ) in Φ(w) is called a descendant of the circulant

2 2 W = Ψ(w) Since Ψ(w i ) and Ψ (1) (w i ) are isomorphic for 0 i < c, there are at most c distinct descendant circulants of Ψ(w) in Φ(w), namely Ψ(w 0 ),Ψ(w 1 ),,Ψ(w c 1 ) Since the array Φ(w) is obtained by applying the permutation π to the columns and rows of the ciruclant Ψ(w), we write Φ(w) = π(ψ(w)) Let π 1 be the inverse permutation of π Then Ψ(w) = π 1 (Φ(w)) From the structure of displayed by (4), we see that each row of l l circulants is a right cyclic-shift of the row above it, however, when the last circulant on the right is shifted around to the left, all its rows are cyclically shifted one place to the right within the circulant This structure is referred to as the doubly cyclic structure which is pertinent to the construction of new cyclic codes from a given cyclic code For 0 i < c, the ith descendant circulant Ψ(w i ) of Ψ(w) appears c i times and its shift Ψ (1) (w i ) appears i times Summarizing the above results on circulant decomposition, we have the following theorem Theorem 1 Given an n n circulant W = Ψ(w) over a field with generator w, if n can be properly factored, then there is a permutation π which puts W into an array of circulants of the same size in the form of (4) Conversely, if an array Φ(w) of circulants of the same size has the doubly cyclic structure in the form given by (4), then there is a permutation π 1 which puts the array Φ(w) into a circulant W with generator w Theorem 1 gives a basis for decomposing a cyclic code into families of cyclic and QC codes or putting a group of cyclic codes into a longer cyclic code III CYCLIC AND QC DESCENDANTS OF A CYCLIC CODE Let C c be a cyclic code over GF(q) of length n given by the null space of an n n circulant parity-check matrix H circ = Ψ(w) where w is the generator of the circulant Suppose n can be properly factored as the product of two integers, c and l Then, the circulant parity-check matrix H circ = Ψ(w) of C c can be decomposed into a c c doubly cyclic array H qc of circulants of size l l in the form given by (4) Then, the null space of H qc = Φ(w) gives a QC code C qc over GF(q) which is combinatorially equivalent to C c From the array H qc = Φ(w), we can construct new cyclic codes of three different types These new cyclic codes are called cyclic descendant codes (simply descendants) of the cyclic code C c The cyclic code C c itself is called the mother code For 0 i < c, if Ψ(w i ) is a nonzero circulant, then the null space over GF(q) of H (1) i = Ψ(w i ) gives a cyclic descendant of C c, denoted by C (1) i, of length l This descendant code is referred to as a type-l descendant of C c Since there are at most c distinct non-isomorphic descendant circulants of H circ = Ψ(w) in the array H qc = Φ(w) There are at most c distinct type-1 cyclic descendants of C c From (4), we see that each column of the array H qc = Φ(w) consists of the circulants in the first row of H qc For 0 i < c, each circulant Ψ(w i ) or its cyclic shift Ψ (1) (w i ) appears once and only once in each column Since a circulant Ψ(w i ) and its cyclic shift Ψ (1) (w i ) differ only in permutation of their rows and hence their null spaces are identical Consequently, the null spaces of all the columns of H qc = Φ(w) are the same In fact, the null space of each column of H qc = Φ(w) is identical to the null space of the cl l matrix [Ψ (1) (w 0 ) T Ψ (1) (w 1 ) T Ψ (1) (w c 1 ) T ] T For 1 k < c, let i 1,i 2,,i k be k distinct integers such that 0 i 1,i 2,,i k < c Let H (2) col,k = [Ψ(1) (w i1 ) T Ψ (1) (w i2 ) T Ψ (1) (w ik ) T ] T (5) H (2) col,k is a kl l matrix over GF(q) whose null space gives a cyclic code of length l, denoted by C (2), which is referred to k as a type-2 cyclic descendant code of the mother cyclic code C c For 1 k < c, let i 1,i 2,,i k be a set of distinct integers such that 0 i 1,i 2,,i k < c Suppose we replace the descendant circulants, Ψ(w i1 ),Ψ(w i2 ),,Ψ(w ik ) of H circ = Ψ(w) and all their cyclic shifts in the array H qc = Φ(w) by zero matrices of size l l By doing this, we obtain a c c array H (3) qc,mask = Φ(w) mask of circulants and zero matrices of size l l Since cyclically shifting all the rows of a zero matrix still gives a zero matrix, the array Φ(w) mask is still in the form given by (4) with doubly cyclic structure Applying the inverse permutation π 1 to the columns and rows of the c c array H (3) qc,mask = Φ(w) mask, we obtain a new circulant matrix H (3) circ,mask = Ψ(w) mask = π 1 (Φ(w) mask ) (6) Then the null space of H (3) circ,mask = Ψ(w) mask gives a cyclic code C (3) mask which is referred to as a type-3 cyclic descendant code of the cyclic code C c The replacement of a set of circulants in the array H qc = Φ(w) is called masking H (3) circ,mask = Ψ(w) mask and H (3) qc,mask = Φ(w) mask are called masked circulant and masked array of H circ = Ψ(w) and H qc = Φ(w), respectively Different masking patterns result in different cyclic descendant codes of C c The results developed above show that decomposition of a circulant parity-check matrix of a cyclic code C c gives a family of cyclic descendant codes For any pair (s,t) of integers with 1 s,t c, let H qc (s,t) be an s t sub-array of H qc = Φ(w) Since H qc (s,t) is an array of circulants, its null space gives a QC code This QC code is called a QC descendant code of C c (or C qc ) It is clear that the null space of H (3) qc,mask = Φ(w) mask also gives a QC code The null space of the transpose [H (2) col,k ]T of H (2) col,k also gives a QC code Therefore, decomposition of a circulant parity-check matrix of a cyclic code C c gives a family of QC descendant codes If the circulant parity-check matrix H circ = Ψ(w) of C c is a sparse circulant over GF(q), then the null space of H circ = Ψ(w) gives a cyclic LDPC code over GF(q) Decomposition of this cyclic LDPC code, families of descendant cyclic and QC-LDPC codes can be obtained The only known classes of cyclic LDPC codes are the classes of LDPC codes constructed based on the incidence structures of Euclidean and projective geometries [1], called EG-LDPC and PG-LDPC codes These two classes of cyclic

3 3 FG-LDPC codes have large minimum weights and their Tanner graphs [6] have girth at least 6 They perform well with various iterative message-passing algorithms Cyclic-LDPC codes constructed based on two-dimensional Euclidean and projective geometries have been proved that their Tanner graphs do not have trapping sets [7] of sizes smaller than their minimum weights [8], [9] As a result, their error-floors are primarily determined by their minimum weights Since they have large minimum weights, their error-floors are expected to be very low Unfortunately, cyclic FG-LDPC codes form a small class of cyclic LDPC codes However, using circulant decomposition, we can construct large classes of cyclic and QC descendant LDPC codes from cyclic FG-LDPC codes These cyclic and QC descendant LDPC codes of cyclic FG- LDPC codes also have good trapping set structures [8] IV GENERATOR POLYNOMIALS OF CYCLIC DESCENDANT CODES For any positive integer m, let GF(q m ) be an extension field of GF(q) Let C c be an (n,k) cyclic code over GF(q) where n is a factor of q m 1 and (n,q) = 1 Let g(x) = g 0 + g 1 X + + g n k 1 X n k 1 + X n k be its generator polynomial which divides X n 1 A polynomial v(x) of degree n 1 or less over GF(q) is a code polynomial if and only if it is divisible by g(x) The generator polynomial g(x) has n k roots in GF(q m ) The condition (n,q) = 1 ensures that all the roots of X n 1 are distinct elements of GF(q m ) In construction of a cyclic code, its generator polynomial is often specified by its roots This is the case for BCH and RS codes [2] Let h(x) = (X n 1)/g(X) = h 0 + h 1 X + + h k X k, where 0 j k,h j GF(q) and h k = 1 The polynomial h(x) is called the paritycheck polynomial Form the following n-tuple over GF(q): w = (h k,h k 1,,h 1,h 0,0,0,,0) The n n circulant generated by this n-tuple gives a circulant parity-check matrix H circ of the cyclic code C c generated by g(x) [10] The n- tuple w = (h k,h k 1,,h 1,h 0,0,0,,0) is called the paritycheck vector of the cyclic code C c The rank of H circ is n k Suppose n can be factored as the product of two proper factors c and l Then based on H circ, we can decompose C c into families of cyclic and QC descendant codes In code construction, the generator polynomial g(x) of an (n,k) cyclic code C c over GF(q) is specified by its roots Let β 0,β 1,,β n k 1 be the roots of g(x) which are nonzero elements of GF(q m ) Then g(x) = (X β i ) Let α 0 i<n k be a primitive nth root of unity Then, for 0 i < n k,β i is a power of α Since α n = 1, (β i ) n = 1 for 0 i < n k A polynomial v(x) of degree n 1 or less over GF(q) is a code polynomial if and only if v(x) has β 0,β 1,,β n k 1 as roots, ie, v(β i ) = 0 for 0 i < n k For 0 i 1,i 2 < n k, suppose there exists an integer t with 0 < t < c such that β i2 = α tl β i1 In this case, since α cl = α n = 1, we must have β c i 1 = β c i 2 We say that β i1 and β i2 are equal in cth power We partition the n k 1 roots, β 0,β 1,,β n k 1, of g(x) into m equal classes in cth power where 1 m c For 0 e < m, let Ω e = {β e0,β e1,,β ere 1 } be the eth class of equal roots in cth power where each β ef in Ω e is one of the roots, β 0,β 1,,β n k 1, and r e is the number of equal roots in Ω e Then βe c 0 = βe c 1 = = βe c re 1 The first element β e0 in Ω e is called the leader of Ω e (any element in Ω e can be used as the leader) For 0 e < m,0 f < r e and 0 j < c, let λ ef = β n k 1 e f σ e,j = r e 1 f=0 n k 1 (β ef β s ) s=0,s e f λ ef β j e f Then the following theorem characterizes the roots of the generator polynomial of the type-1 jth cyclic descendant of the cyclic code C c The proof of this theorem is lengthy and is omitted in this paper It can be found in [8] Theorem 2 Let GF(q m ) be an extension field of GF(q) where q is a power of prime Let n be a factor of q m 1 and (n, q) = 1 Let α be the primitive nth root of the unity Suppose that n can be factored as product of two integers c and l such that c 1 and l 1 Let C c be an (n, k) cyclic code whose generator polynomial g(x) has β 0,β 1,,β n k 1 as roots which are powers of α Then, the generator polynomial g (1) j 1 (X) of the type-1 cyclic descendant code C (1) j of C c given by the null space of the l l decendant circulant Ψ(w j ) of Ψ(w) in the array Φ(w) given by (4) with w = (h k,h k 1,,h 1,h 0,0,0,,0) and 0 j < c has βe c 0 as a root if and only if σ e,j 0 with 0 e < m Example 1 Let α be a primitive element of GF(2 11 ) Consider the binary primitive (2047,2025) BCH code whose generator polynomial g(x) has α, α 2, α 3, α 4 and their conjugates as roots The length 2047 can be factored as a product of c = 89 and l = 23 Then the circulant parity-check matrix H circ can be decomposed into a array of circulants The null space of each descendant circulant gives a (23,12) Golay code, a perfect triple-error-correction code The generator polynomial is g(x) = 1 + X + X 5 + X 6 + X 7 + X 9 + X 11 [2] The next theorem characterizes the roots of the generator polynomial of a type-2 cyclic descendant of a decomposable cyclic code C c Theorem 3 For 1 k < c, let i 1,i 2,,i k be a set of distinct integers such that 0 i 1,i 2,,i k < c For 1 t k, let g (1) i t (X) be the generator polynomial of i t - th type-1 cyclic descendant code C (1) i t of C c given by the null space of i t -th descendant circulant Ψ(w it ) of H circ = Ψ(w) Then the generator polynomial g (2) k (X) of the type-2 cyclic descendant code C (2) k of C c given by the null space of the parity-check matrix H col,k of (5) is the least common multiple of g (1) i 1 (X),g (1) i 2 (X),,g (1) i k (X), ie, g (2) k (X) = LCM{g(1) i t (X),0 t < k} The roots of g (2) k (X) is the union of the roots of,

4 4 g (1) i 1 (X),g (1) i 2 (X),,g (1) i k (X) V DECOMPOSITION CYCLIC LDPC CODE ON TWO-DIMENSIONAL EUCLIDEAN GEOMETRIES Cyclic finite geometry (FG) LDPC codes form the only class of cyclic LDPC codes [1], [2], [10] These codes are constructed based on two types of finite geometries, namely Euclidean and projective geometries [2], [11], [12] The parity check matrix of a cyclic FG-LDPC code consists of a single circulant or multiple circulants of the same size in a column Decomposition of a cyclic FG-LDPC code results in a family of cyclic FG-LDPC codes and a family of QC-LDPC codes In this and next sections, we only consider decomposition of cyclic Euclidean geometry codes constructed based on the incidence structure of lines on points Decomposition of cyclic projective geometry codes is similar We begin with a two-dimensional Euclidean geometry EG(2,q) over the field GF(q), where q is a power of a prime This geometry consists of q 2 points and q(q+1) lines A point in EG(2,q) is simply a two-tuple a = (a 0,a 1 ) over GF(q) and the zero two-tuple (0,0) is called the origin A line in EG(2,q) is simply a one-dimensional subspace or its coset of the vector space of all the q 2 two-tuples over GF(q) A line contains q points If a point a is on a line L in EG(2,q), we say the line L passes through the point a Any two points in EG(m,q) are connected by a line Two points are connected by one and only one line in EG(2,q) For every point a in EG(2,q), there are (q + 1) lines that intersect at the point a The field GF(q 2 ), as an extension field of the ground field GF(q), is a realization of EG(2,q) Let α be a primitive element of GF(q 2 ) Then, the powers of α,α = 0,α 0 = 1,α,α 2,,α q2 2, give all the q 2 elements of GF(q 2 ) and they represent the q 2 points of EG(2,q) The 0-element represents the origin of EG(2,q) Let EG (2,q) be the subgeometry obtained from EG(2,q) by removing the origin and the q + 1 lines passing through the origin This subgeometry consists of q 2 1 non-origin points and q 2 1 lines not passing through the origin Each non-origin point in EG (2,q) is intersected by q lines Let L = {α j1,α j2,,α jq } be a line in EG*(m,q) For 0 i < q 2 1, let α i L = {α j1+i,α j2+i,,α jq+i } Then, α i L is also a line in EG (2,q) and α 0 L,αL,α q2 2 L give all the q 2 1 lines in EG (2,q) [2] This structure of lines is called cyclic structure Let L be a line EG (2,q) Based on L, we define the following (q 2 1)-tuple over GF(2), v L = (v 0,v 1,,v q2 2), whose components correspond to the q 2 1 non-origin points α 0,α,α 2,,α q2 2 of EG (2,q), where v j = 1 if α j is a point on L and v j = 0 otherwise It is clear the weight of v L is q This (q 2 1)-tuple v L is called the incidence vector of the line L Due to the cyclic structure of the lines in EG (2,q), the incidence vector v αl of the line αl is the cyclic-shift (one place to the right) of incidence vector v L of the line L Let n = q 2 1 Form an n n matrix H EG over GF(2) with the incidence vectors of the n lines, α 0 L,αL,,α n 1 L, of EG (2,q) as rows Then, H EG is an n n circulant with both column and row weights q H EG can be obtained by using the incidence vector v L of the line L (or any line in EG (2,q)) as the generator and cyclically shifting v L n 1 times Since the number of 1-entries in H EG is much smaller than the total entries in H EG, it is a sparse matrix Therefore, its null space gives a cyclic LDPC code C EG Since two lines in EG (2,q) have at most one point in common, their incidence vectors have at most one position where they both have 1-components Therefore, H EG has the following structure: no two rows (or two columns) have more than one place where they both have 1-components In almost all of the proposed constructions of LDPC codes, this structure is imposed on the rows and columns of the paritycheck matrix This structure is referred to as the row-column (RC) constraint Therefore, H EG satisfies the RC-constraint The RC-constraint ensures that the Tanner graph of the LDPC code given by the null space of a sparse parity-check H is free of cycles of 4 and hence has a girth of at least 6 and that the minimum weight of the code is at least γ min + 1, where γ min is the minimum column weight of H [1], [2] Since the column weight of H EG is constant and equals to q, therefore the cyclic EG-LDPC code C EG given by the null space of H EG has minimum weight at least q+1 Cyclic EG- LDPC codes can be effectively decoded with various messagepassing (MP) algorithms, one-step majority-logic decoding (OSMLGD), bit-flipping (BF) and various weighted BF (WBF) decoding algorithms [1], [2], [10] To find the generator polynomial g EG (X) of C EG, we express each row of H EG as a polynomial over GF(2) of degree n 1 or less with leftmost entry as the constant term and rightmost entry as the coefficient of X n 1 Let h(x) be greatest common divisor of the row polynomials of H EG The reciprocal h EG (X) of h EG (X) is the paritycheck polynomial Then, the generator polynomial g EG (X) = (X n 1)/h EG (X) The roots of g EG (X) in GF(q 2 ) were determined and can be found in [2], [13] - [15] For the special case with q = 2 s, the rank of H EG is 3 s 1 H EG has a very large row redundancy It has 4 s 3 s redundant rows (or dependent rows) Large row redundancy helps iterative decoding based on belief propagation to converge fast The cyclic EG-LDPC code C EG given by the null space of H EG is a (4 s 1,4 s 3 s ) code with minimum weight exactly 2 s +1 [1], [2], [10] Let c and l be two proper factors of n = q 2 1 such that n = c l Decompose the circulant H EG into a doubly cyclic c c array π(h EG ) = Ψ(v L ) of circulants of size l l of the form given by (4) through column and row permutation π defined by (3) Since H EG satisfies the RCconstraint, π(h EG ) also satisfies the RC-constraint The null spaces of the array π(h EG ) gives a QC-EG-LDPC code C EG,qc of length q 2 1 which is equivalent to the cyclic EG- LDPC code C EG given by the null space of H EG Based on the doubly cyclic array π(h EG ) of circulants, three types of cyclic descendant LDPC codes can be constructed Therefore, decomposition of H EG results in a family of new cyclic EG- LDPC codes This enlarges the repertoire of cyclic LDPC codes Theorems 2 and 3 can be used to determine the roots of the generator polynomials of type-1 and type-2 cyclic descendant LDPC codes of C EG

5 5 Cyclic LDPC(1365,765) SPA BER Cyclic LDPC(1365,765) SPA BLER Cyclic LDPC(1365,765) MS BER Cyclic LDPC(1365,765) MS BLER Sphere Packing Bound, BLER Uncoded BER SPA 50 BER MS 50 BER MS 10 BER MS 5 BER SRBI MLGD 50 BER Uncoded Shannon Limit E b /N 0 (db) Fig 1 Bit error performances of the binary (4095,3367) cyclic EG-LDPC code given in Example 2 decoded with the SPA and the scaled MSA Example 2 Let EG(2,2 6 ) be the code construction Euclidean geometry The field GF(2 12 ) is a realization of EG(2,2 6 ) Based on the incidence vectors of the = 4095 lines in the subgeometry EG (2,2 6 ), we can construct a RCconstrained circulant H EG with both column and row weights 64 The rank of H EG is = 728 The null space of H EG gives a (4095,3367) cyclic EG-LDPC code CEG with minimum distance 65 Its error performances decoded with the sum-product algorithm (SPA) and a scaled min-sum (MS) algorithm over the binary AWGN channel are shown in Figure 1 We see that decoding of the code with the MS algorithm converges very fast The performance curves with 5, 10 and 50 iterations of the scaled MS-algorithm almost overlap with each other Also included in Figure 1 is the error performance of the code decoded with a soft-reliability based iterative majoritylogic decoding (SRBI-MLGD) (a binary message-passing decoding algorithm) proposed in [16] We see that, at the biterror rate (BER) of, the SRBI-MLGD performs only 06 db from the scaled MS with 50 iterations The SRBI-MLGD requires only integer and binary logical operations with a computational complexity much less than that of the SPA and the MS-algorithm It offers more effective trade-off between error-performance and decoding complexity compared to the other reliability-based iterative decoding, such as the weighted bit-flipping (WBF) algorithms [10], [16] Example 3 Continue Example 3 Suppose we factor 4095 as the product of c = 3 and l = 1365 By column and row permutations, the 4095x4095 circulant H EG can be decomposed into a 3 3 doubly cyclic array π(h EG ) of circulants of size 1365x1365 in the form of (4) as shown below: π(h EG ) = Ψ 0 Ψ 1 Ψ 2 Ψ (1) 2 Ψ 0 Ψ 1 Ψ (1) 1 Ψ (1) 2 Ψ 0 The descendant circulants Ψ 0 and Ψ 2 both have column E /N (db) b 0 Fig 2 The error performances of the binary (1365,765) cyclic EG-LDPC code given in Example 3 decoded with 50 iterations of the SPA and the MSA and row weights 24 The descendant circulant Ψ 1 has both column and row weights 16 Consider the descendant circulant Ψ 1 Its rank is 600 The null space of Ψ 1 gives a (1365,765) type-1 cyclic descendant code C (1) EG of the (4095,3367) cyclic EG-LDPC code CEG with rate 056 and minimum distance at least 17 The generator polynomial g (1) EG (X) of this code has β = α3,β 2,,β 16 and their conjugates as roots By extensive computer search, we find that C (1) EG has no trapping set with size smaller than 17, however, we do find a (17,0) trapping set which gives a codeword of weight 17 Therefore, the minimum distance of C (1) EG is exactly 17 Therefore, the error-floor of this code is dominated by the minimum distance of the code The error performance of the code over the AWGN channel using BPSK signaling decoded with 50 iterations of SPA (or MS) is shown in Figure 2 At the block error rate (BLER) of, the code perform 16 db from the sphere packing bound Suppose we use H col,3 = [Ψ T 0Ψ1 T Ψ2 T ] T as the paritycheck matrix of a type-2 cyclic descendant C (2) EG of the (4095,3367) cyclic EG-LDPC code C EG This matrix has column weight 64 but two different row weights, 16 and 20, respectively It has rank 664 The null space of H col,3 gives a type-2 cyclic (1365,701) descendant LDPC code of the cyclic (4095,3367) EG-LDPC code C EG It has rate and minimum distance at least 65 The error performance of this code is shown in Figure 3 With OSMLGD, the code is capable of correcting 32 or fewer errors Suppose we replace Ψ 2 and its cyclic-shift Ψ (1) 2 by two zero matrices We obtain the following 3 3 masked array: π(h EG ) mask == Ψ 0 Ψ Ψ 0 Ψ 1 Ψ (1) 1 0 Ψ 0 The masked array π(h EG ) mask still has the doubly

6 Cyclic LDPC(1365,701), SRBI MLGD, BER Cyclic LDPC(1365,701), SRBI MLGD, BLER Cyclic LDPC(1365,701), MS, BER Cyclic LDPC(1365,701), MS, BLER Uncoded Sphere Packing Bound E /N (db) b 0 Fig 3 The error performances of the binary (1365,701) cyclic EG-LDPC code given in Example 3 decoded with the MSA and the SRBI-MLGDalgorithm another family of cyclic EG-LDPC codes For a pair of positive integers, (s,t) with 1 s,t c, let π(h EG )(s,t) be a s t subarray of π(h EG ) This subarray also satisfies the RC-constraint and its null space gives a QC descendant LDPC code C EG,qc of the cyclic EG-LDPC code C EG given by the null space of the n n circulant H EG Suppose q 1 can be factored as a product of two integers, b and l with 1 b,l < q Then n = q 2 1 can be factored as the following product: n = (q + 1)(q 1) = (q + 1)bl Let c = (q +1)b Then, the (q 2 1) (q 2 1) circulant H EG can be decomposed into a doubly cyclic c c array π(h EG ) cpm of l l circulants The following Theorem characterizes the structure of this doubly cyclic array The proof of this theorem can be found in [8] Theorem 4 Suppose q 1 can be factored as a product of two integers, b and l with 1 b,l < q Then, the circulant H EG can be decomposed into a (q + 1)b (q + 1)b doubly cyclic array π(h EG ) cpm of circulants of size l l Each circulant is either an l l circulant permutation matrix (CPM) or an l l zero matrix (ZM) Each row (or column) block of π(h EG ) cpm consists of q CPM and (q + 1)b q zero matrices Cyclic LDPC(4095,2703), SPA 3 BER Cyclic LDPC(4095,2703), SPA 3 BLER Cyclic LDPC(4095,2703), SPA 5 BER Cyclic LDPC(4095,2703), SPA 5 BLER Cyclic LDPC(4095,2703), SPA 50 BER Cyclic LDPC(4095,2703), SPA 50 BLER Sphere Packing Bound For b = 1, it follows from Theorem 4 that π(h EG ) cpm is a (q + 1) (q + 1) doubly cyclic array of circulants of size (q 1) (q 1) Each row (or each column) block of π(h EG ) cpm consists q CPMs and one ZM The array π(h EG ) cpm allows us to construct a large class of QC-EG-LDPC codes For any pair of integers, (γ, ρ) with 1 γ,ρ q + 1, let π(h EG (γ,ρ)) cpm be a γ ρ subarray of π(h EG ) cpm The null space of π(h EG (γ,ρ)) cpm gives a descendant QC-LDPC code C EG,qc of the cyclic EG-LDPC code C EG, whose Tanner graph has a girth of at least 6 The above decomposition and construction give a large class of QC-EG-LDPC codes with various lengths, rates and minimum distances The special case with b = 1 was proposed earlier in [17] E b /N 0 (db) Fig 4 The bit and block error performances of the binary (4095,2703) cyclic EG-LDPC code given in Example 3 cyclic structure Applying the inverse permutation π 1 to π(h EG ) mask, we obtain a masked circulant H EG,mask with both column and row weights 40 The null space of H EG,mask gives a (4095,2703) type-3 cyclic descendant code of the (4095,3367) cyclic EG-LDPC code constructed based on EG(2,2 6 ) over GF(2 6 ) The code has rate 066 and minimum distance at least 41 The error performances of this cyclic descendant LDPC code decoded with 3, 5, and 50 iterations of the SPA is shown in Figure 4 We can factor 4096 as the product of c = 15 and l = 273 Then the circulant H EG constructed in Example 2 can be decomposed into a doubly cyclic array π(h EG ) of circulants Based on this array, we can construct Example 4 Consider the circulant H EG over GF(2) constructed based the two-dimensional Euclidean geometry EG(2,2 6 ) given in Example 3 We factor = 4095 as the product of = 65 and = 63 Let c = 65 and l = 63 Decompose the circulant H EG into a array π(h EG ) cpm of CPMs and ZMs of size Suppose H EG is constructed by choosing a line L not passing through the origin of EG(2,2 6 ) such that, after decomposition of H EG, the 65 ZMs of π(h EG ) cpm lie on its main diagonal The null space of π(h EG ) cpm gives a (4095,3367) QC-EG-LDPC code which is combinatorially equivalent to the (4095,3367) cyclic EG-LDPC code given in Example 2 Suppose we choose the first 6 rows of π(h EG ) cpm to from a 6 65 subarray π(h EG (6,65)) cpm of π(h EG ) cpm It is a matrix over GF(2) with constant row weight 64 and two column weights, 5 and 6 The null space of this matrix gives a (4095,3771) type-2 descendant QC-EG-LDPC code with rate 0921 Its error performance with 50 iterations of the SPA is shown in Figure 5 At the BLER of 10 4, the code performs 075 db from the sphere packing bound At the BER of, it performs 13 db from the Shannon limit

7 7 QC LDPC(4095,3771) BER QC LDPC(4096,3771) BLER Sphere Packing Bound Shannon Limit Uncoded QC LDPC(32768,31747), SPA, BER QC LDPC(32768,31747), SPA, BLER Shannon Limit E b /N 0 (db) Fig 5 The bit and block error performances of the binary (4095,3771) QC-LDPC code given in Example 4 Example 5 In this example, we construct a long highrate code and show how close the code performs to the Shannon limit Let the two-dimensional Euclidean geometry EG(2,257) over the prime field GF(257) be the code construction geometry Based on the incidence vectors of the lines in EG(2,257) not passing through the origin of the geometry, we construct a circulant H EG with both column and row weights 257 The null space of H EG gives a cyclic-eg-ldpc code of length of with minimum distance at least 258 Set c = q + 1 = = 258 and l = q 1 = = 257 Decompose H EG into a array π(h EG ) cpm of CPMs and ZMs of size In this CPM-decomposition, every row and every column consists of 257 CPMs and a single ZM Suppose H EG is constructed by choosing a line not passing through the origin of EG(2,257) such that the 258 ZMs lie on the main diagonal of the array π(h EG ) cpm Let γ = 4 and ρ = 128 Take a subarray π(h EG )(4,128) cpm from π(h EG ) cpm, avoiding the ZMs on the main diagonal of π(h EG ) cpm This subarray π(h EG )(4,128) cpm is a matrix with column and row weights 4 and 128, respectively The null space of π(h EG )(4,128) cpm gives a (4,128)-regular (32768,31747) QC-EG-LDPC code with rate 0969 The error performance of this code over the AWGN channel decoded with 50 iterations of the SPA is shown in Figure 6 At the BER of, the code performs 06 db from the Shannon limit If we select a set of CPMs and their cyclic-shifts in π(h EG ) cpm and replace them by zero matrices of size (q 1) (q 1), we obtain an array π(h EG,mask ) cpm of CPMs and ZMs which has the form of (4) with doubly cyclic structure Applying inverse permutation π 1 to the rows and columns of π(h EG,mask ) cpm, we obtain a (q 2 1) (q 2 1) masked circulant H EG,mask over GF(2) The null space H EG,mask gives a cyclic-eg-ldpc code of length q 2 1 For a pair of two positive integers, (γ,ρ) with 1 γ,ρ q + 1, let π(h EG )(γ,ρ) cpm = [B i,j ] be a γ ρ subarray of π(h EG ) cpm, where B i,j is either a CPM or a ZM with E b /N 0 (db) Fig 6 The bit and block error performances of the binary (32768,31747) QC-LDPC code given in Example 5 0 i < γ and 0 j < ρ A set of CPMs in π(h EG )(γ,ρ) cpm can be replaced by a set of ZMs This replacement is referred to as masking [2], [18] - [20] Masking results in a sparser matrix whose associated Tanner graph has fewer edges and hence fewer short cycles and probably a larger girth than that of the associated Tanner graph of the original γ ρ subarray of π(h EG ) cpm To carry out masking, we first design a low density γ ρ matrix Z(γ,ρ) = [z i,j ] over GF(2) Then take the following matrix product: π(m EG )(γ,ρ) cpm = Z(γ,ρ) π(h EG ) cpm = [z i,j B i,j ], where z i,j B i,j = B i,j for z i,j = 1 and z i,j B i,j = O (a (q 1) (q 1) zero matrix) for z i,j = 0 We call Z(γ,ρ) the masking matrix, π(h EG ) cpm the base array and π(m EG )(γ,ρ) cpm the masked array Since the base array π(h EG ) cpm satisfies the RC-constraint, the masked array π(m EG )(γ,ρ) cpm also satisfies the RC-constraint, regardless of the masking matrix Hence, the associated Tanner graph of the masked array π(m EG )(γ,ρ) cpm has a girth at least 6 The null space of the masked array π(m EG )(γ,ρ) cpm gives a new QC-EG-LDPC code If both the masking matrix and the base array are regular, the masked array is also regular and its null space gives a regular QC-LDPC code However, if the masking matrix is irregular and base array is regular, the masked array is irregular and its null space gives an irregular code A well designed masking matrix results in a good LDPC code Design and construction of masking matrices for constructing binary LDPC codes are discussed in [2], [18]- [20] Example 6 In this example, we construct a long irregular QC-EG-LDPC code using the masking technique presented above Consider the array π(h EG ) cpm of CPMs and ZMs of size constructed in Example 5 Take a subarray π(h EG )(128,256) cpm from π(h EG ) cpm We use this subarray as a base array for masking to construct an irregular code of rate 1/2 Next we construct a masking matrix Z(128, 286) (by computer search) with column and row weight distributions close to the follow-

8 8 BER Irregular QC LDPC code, 05 DVB LDPC code, 05 Shannon Limit E /N (db) b 0 Fig 7 The error performances of the binary (65536,32768) QC-LDPC code and the DVB S-2 standard code given in Example 6 ing variable-node and check-node degree distributions (node perspective) of a Tanner graph optimally designed for an irregular code of rate 1/2 and infinite length (using density evolution [21], [10]): λ(x) = 04410X X X X X X X X X 29, and ρ(x) = X X X 9 where the coefficient of X i represents the percentage of nodes with degree i+1 The column and row weight distributions of the constructed masking matrix Z(128,256) are given below: v(x) = 106X + 105X 2 +35X 8 +10X 29, and c(x) = 10X X 8, where the coefficient X i gives the number of columns (or rows) of Z(128,256) with weight i + 1 Masking the subarray π(h EG )(128,256) cpm with Z(128,256), we obtain a masked array π(m EG )(128,256) cpm = Z(128,256) π(h EG )(128,256) cpm of CPMs and ZMs It is a matrix over GF(2) with average column and row weights 3875 and 775, respectively The null space of π(m EG )(128,256) cpm gives an irregular (65536,32768) QC-EG-LDPC code The error performance of this code with 50 iterations of the SPA is shown in Figure 7 We see that at a BER of 10 9, the code performs 06 db from the Shannon limit Also included in Figure 7 is the performance of the DVB S-2 standard (64800,32400) LDPC code [22] with a BCH outer code The DVB S-2 LDPC code is an IRA (irregular repeat-accumulated) code [10], [23] The BCH code is a (32400,32208) shortened BCH code with errorcorrection capability 12 The BCH outer code is used to push down the error-floor of the DVB S-2 code We see that the (65536,32768) QC-EG-LDPC code outperforms DVB S-2 code with a BCH outer code VI DECOMPOSITION BASED ON MULTI-DIMENSIONAL EUCLIDEAN GEOMETRIES Cyclic and QC descendant EG-LDPC codes can be constructed from Euclidean geometries with dimension higher than two Consider the m-dimensional Euclidean geometry EG(m,q) over GF(q) with m 3 This geometry consists of q m points and J = q m 1 (q m 1)/(q 1) lines Each point is an m-tuple over GF(q) Each line consists of q points The field GF(q m ) as an extension field of the ground field GF(q) is a realization of the geometry EG(m,q) [2], [11], [12] Let α be a primitive element of GF(q m ) Then, the powers of α, α = 0,α 0 = 1,α,,α qm 2, represent q m points of EG(m,q) The element α = 0 represents the origin of the geometry Let EG (m,q) be the subgeometry obtained by removing the origin and the line passing through the origin from EG(m,q) This subgeometry consists of q m 1 non-origin points and and J 0 = (q m 1 1)(q m 1)/(q 1) lines not passing through the origin of EG(m,q) Based on the lines of EG (m,q), we can form K = (q m 1 1)/(q 1) circulants of size (q m 1) (q m 1), denoted H EG,0,H EG,2,,H EG,K 1 [1], [12] For 0 i < K, the columns of each circulant H EG,i are incidence vectors of a group of q m 1 lines Suppose q m 1 can be factored as product of two positive integers c and l, ie, q m 1 = cl Then each circulant H EG,i can be decomposed into a doubly cyclic c c array π(h EG,i ) of circulants of size l l This results in K doubly cyclic arrays, π(h EG,0 ),π(h EG,2 ),,π(h EG,K 1 ) Based on these arrays, we can construct large classes of cyclic and QC descendant EG-LDPC codes A special decomposition for constructing QC-EG-LDPC codes is given in the rest of this section Factor q m 1 as the following product: q m 1 = (q m 1 +q m 2 ++q+1)(q 1) Let c = (q m 1 + q m q + 1) and l = q 1 For 0 i < K, each (q m 1) (q m 1) circulant H EG,i can be decomposed into a doubly cyclic c c array π(h EG,i ) cpm of CPMs and MZs of size (q 1) (q 1) by applying the π-permutation to both the columns and rows of H EG,i Each row (or column) block of π(h EG,i ) cpm consists of q CPMs and c q ZMs Form a c ck array of CPMs and ZMs of size (q 1) (q 1) over GF(2) as follows: π(h EG ) cpm = [π(h EG,0 ) cpm π(h EG,1 ) cpm π(h EG,K 1 ) cpm ] It is a (q m 1) K(q m 1) matrix over GF(2) with column and row weights q and Kq, respectively For 1 γ c and 1 ρ ck, take a γ ρ subarray π(h EG (γ,ρ)) cpm from π(h EG ) cpm This subarray is a γ(q 1) ρ(q 1) matrix over GF(2) The null space of π(h EG (γ,ρ)) cpm gives a QC-EG-LDPC code of length ρ(q 1) The above construction gives a large family of structurally compatible QC-EG-LDPC codes from a single geometry The null space of the entire array π(h EG ) cpm gives the longest code with length K(q m 1) with minimum distance at least q + 1 Consider the c c array π(h EG,i ) cpm of CPMs and ZMs As pointed out above, each column (or row block) consists of q CPMs and c q ZMs Suppose q can be factored as a product e and f, ie, q = ef We can split each column block of π(h EG,i ) cpm into e column blocks of the same length with the q CPMs distributed evenly into the new e column blocks, each with f CPMs This column splitting operation is referred to column block splitting In distributing the CPMs into e new column blocks, their relative positions are not changed This column block splitting results in a c ce array M col,i (e) of CPMs and ZMs of size (q 1) (q 1), each column block consisting of f CPMs and each row block consisting of q CPMs Next, we split each row block of

9 9 M col,i (e) into e new row blocks of the same length with the q CPMs evenly distributed among the e new row blocks, each with f CPMs This row splitting operation is referred to as the row block splitting This row block splitting of M col,i (e) results in a ce ce array M col,row,i (e,e) of CPMs and ZMs of size (q 1) (q 1) The array M col,row,i (e,e) is called the e-fold expansion of π(h EG,i ) cpm Each column block and each row block of M col,row,i (e,e) consists of f CPMs If we replace each c c subarray π(h EG,i ) cpm in π(h EG ) cpm with its e-fold expansion M col,row,i (e,e), we obtain the following ce cek array: M EG = [M col,row,0 (e,e)m col,row,1 (e,e),,m col,row,k 1 (e,e)] M EG has a much smaller density of CPMs than that of the array π(h EG ) cpm The null space of any subarray of M EG gives a QC-EG-LDPC code 0 QC (8176,7156) LDPC code, SPA 50, BER QC (8176,7156) LDPC code, SPA 50, BLER QC (8176,7156) LDPC code, MS 15, BER Shannon Limit Example 7 Consider the 3-dimensional Euclidean geometry EG(3,2 3 ) over GF(2 3 ) This geometry has 511 non-origin points and 4599 lines not passing through the origin of the geometry Based on the incidence vectors of these 4599 lines, we can form 9 circulants, H EG,0,H EG,1,,H EG,8, of size Factor 511 as product of c = 73 and l = q 1 = 7 Each circulant H EG,i can be factored as a array π(h EG,i ) cpm of CPMs and ZMs of size 7 7 Each column (row) block consists of 8 CPMs and 65 ZMs Form the following array of CPMs and ZMs of size 7 7: π(h EG ) cpm = [π(h EG,0 ) cpm π(h EG,1 ) cpm π(h EG,8 ) cpm ] This array is a matrix with column and row weights 8 and 72, respectively Factor q = 8 as product of e = 2 and f = 4 Using column and row block splitting, each array π(h EG,i ) cpm can be expanded into a array M col,row,i (2,2) of CPMs and ZMs of size 7 7, each row and column block consisting of 4 CPMs and 142 ZMs Suppose we take first 8 of these arrays to form the following array of CPMs and ZMs of size 7 7 : M EG (8) = [M col,row,0 (2,2)M col,row,1 (2,2)M col,row,7 (2,2)] It is a matrix over GF(2) with column and row weight 4 and 32, respectively The null space of this matrix gives a (4,32)-regular (8176,7156) QC-EG-LDPC code with rate The error performance of this code decoded with 50 iterations of the SPA is shown in Figure 8 We see that there is no visible error-floor down to the BER of 4 The estimated error-floor of this code is below the BER of 5 At the BER of 4, it performs only 16 db from the Shannon limit A hardware decoder for this code has been built VII CONCLUSION AND REMARKS In this paper, we have shown that cyclic and QC descendant codes can be derived from a given cyclic code through decomposition of its parity-check matrix in circulant form using specific column and row permutations Applying this decomposition technique to cyclic EG-LDPC codes, we can construct a large class of cyclic EG-LDPC codes and a large class of QC-EG-LDPC codes Some example codes demonstrate very low error-floor 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