A CLASS OF STRUCTURED LDPC CODES WITH LARGE GIRTH

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1 A CLASS OF STRUCTURED LDPC CODES WITH LARGE GIRTH Jin Lu, José M. F. Moura, and Urs Niesen Deartment of Electrical and Comuter Engineering Carnegie Mellon University, Pittsburgh, PA jinlu, ABSTRACT We introduce a class of structured LDPC codes turbostructured LDPC (TS-LDPC) codes comosed of two subtrees connected by an interleaver. TS-LDPC codes with good girth roerties are easy to design: careful design of the interleaver comonent revents short cycles in its Tanner grah. We resent a methodology to design TS-LDPC codes with arbitrary column weight j 2 and arbitrary girth. In addition, we describe a comlexity reduced decoding algorithm. Simulation results demonstrate the good erformance of TS-LDPC codes when comared to random LDPC codes of the similar size and rate. 1. INTRODUCTION Low-density arity-check (LDPC) codes [1], can be alied in numerous tasks, e.g., communication systems, magnetic recording channels. Their erformance is close to the Shannon limit using iterative decoding [2]. An LDPC code can be described by a biartite grah called Tanner grah [3]. The length of the shortest cycle in a Tanner grah is referred to as its girth g. Short cycles in Tanner grahs tax the comuting effort of the decoding algorithm and revent it from converging to the otimal decoding result. Furthermore, reference [3] derives a lower bound on the minimum distance d min. This lower bound increases exonentially with girth. Therefore, LDPC codes with good girth roerties are articularly desirable. Recently, cyclic and uasi-cyclic LDPC codes have drawn much attention in the sense that they facilitate low-comlexity encoder and decoder designs. However, they have limited girths. Tanner, [4], roved that the girth of such codes with column weight j 3 is less than or eual to 6. This revents the girth of (n, j, k) cyclic and uasi-cyclic LDPC codes of log n log[(j 1)(k 1)] growing as, [1], hence erforming oorly at very long code block length. We resent a new tye of structured LDPC codes with arbitrary girth that are stimulated from turbo designs, which This work was suorted by the Data Storage Systems Center (DSSC) at Carnegie Mellon University, and by NSF grant # ECS we refer to as turbo-structured LDPC codes (TS-LDPC). The Tanner grah of such codes is comosed of two subtrees that are connected in a turbo-like manner by an interleaver. This structure facilitates the systematic design of LDPC codes with large girth and flexible code rates. Our turbo-structured LDPC codes are distinct from the codes in [5] that are also insired by turbo codes in their design of concatenated tree codes. However, our TS-LDPC codes are turbo for the decoder side, while [5] is turbo for the encoder side. The resulting TS-LDPC codes and the concatenated tree codes are very different, see [6] for comments. 2. TS-LDPC CODES The Tanner grah of TS-LDPC codes is comosed of three comonents: two height-balanced trees, denoted as an uertree T U, and a lower-tree T L, and an interlaver that connect T U and T L. The leaf-nodes of T U are bit nodes whereas the leaf-nodes of T L are check nodes. Both trees T U and T L contain h tiers (or layers). The first tier of T U contains only one check node C the root, as shown in Figure 1. To match T U, we let the root of T L to be a bit node V and connect V to C. The two trees are couled in a turbolike manner such that many edges join the leaf-nodes of T U and T L together, see Figure 1. The structure formed by the edges connecting the leaf-nodes of T U and the leaf nodes of T L is named the interleaver I. Since we are interested in regular LDPC codes, we let all the bit nodes have uniform degree j and all the check nodes have uniform degree k. For examle, a TS-LDPC code with h =4, j =3, and k =4is deicted in Figure 1. Ignoring ossible deendency of a few rows in the arity-check matrix, the code rate is ρ =1 j k. We can easily choose the values of j and k to create a TS- LDPC code with the desired code rate. 3. INTERLEAVER DESIGN By construction, each leaf-node in T U is connected to = j 1 leaf-nodes in T L. This is a one-to- maing, while the standard interleaver is a one-to-one maing between elements of two sets with the same size. To get a standard IEEE Communications Society 425

2 interleaver, we introduce auxiliary nodes (solid triangles) as shown in Figure 2 to facilitate the code design. For each leaf-node in T U, we add j 1 auxiliary nodes as its children. Similarly, each leaf-node in T L has k 1 auxiliary nodes as its descendants. After introducing auxiliary nodes, we notice that cycles resent in the codes must contain at least four auxiliary nodes two auxiliary nodes of T U and two auxiliary nodes of T L. We classify cycles into two disjoint categories: Tye- I cycles contain four and only four auxiliary nodes; tye-ii cycles contain more than four auxiliary nodes. We will disose of them searately. --alternate-decimal form is: X = x 4 x 3 x 2 x 1 In euation (1), x i, i =1 4, reresents the i th digit. The corresonding value of X in decimals is X =(x x 3 + x 2 + x 1 ) Tye-I Cycles: Digit-wise Reversal We consider first how to avoid short tye-i cycles. We start with a simle interleaver design digit-wise reversal. For an index X in --alternate-decimal form with h digits, its digit-wise reversal interchanges the i th digit and the (h + 1 i) th digit. We reresent the digit-wise reversal oerator by π d ( ). For the index X in euation (1), its digit-wise reversal is: (1) π d (X ) = x 1 x 2 x 3 x 4 = (x x 2 + x 3 + x 4 ) 10 We state the advantage of the digit-wise reversal interleaver in the following theorem and omit the roof. For a detailed roof, lease refer to [7]. Fig. 1. TS-LDPC code: Tanner grah, h =4, j =3, k =4. Theorem 1 Connecting the auxiliary nodes indexed by X in T U to the auxiliary nodes indexed by π d (X ) in T L guarantees the resulting tye-i cycles is at least of length 2h (h denotes the number of tiers in T U ). Following theorem 1, we can enlarge the length of tye-i cycles by simly increasing tiers of sub-trees Tye-II 4-Cycles Fig. 2. Auxiliary nodes in the uer tree T U For T U with h tiers, there are [(k 1)(j 1)] h/2 auxiliary nodes in T U. We address the interleaver design roblem algebraically by indexing all the auxiliary nodes of T U, from 0 to [(k 1)(j 1)] h/2 1 in a format we exlain next that we refer to as --alternate-decimal, where = k 1 and = j 1. We need h digits in the --alternatedecimal indexing and number these h digits from 1 to h, starting from the rightmost one. The odd digits take values 0 to 1 and the even digits take values 0 to 1. Similarly, we index all the auxiliary nodes of T L from 0 to [(k 1)(j 1)] h/2 1 and reresent all these indices in --alternate-decimal format. We rovide an examle. With reference to the index in Figure 2, its index X in Theorem 1 revents short tye-i cycles, however, it may lead to many short tye-ii cycles, even tye-ii 4-cycles. To avoid short tye-ii cycles, we roose a method called grouing and shifting. The shift S is defined to be a constant in --alternate-decimal format that is added to the original index π d (X ) to form a new index. For examle, let π d (X )=x 1 x 2 x 3 x 4,shiftS = s 1 s 2 s 3 s 4, and + reresent the digit-wise addition, then π d (X ) +S = y 1 y 2 y 3 y 4 (2) In euation (2), y i = x i +s i =mod(x i + s i, div i ). where div i = if i is even and div i = if i is odd. Similarly, we reresent the digit-wise subtraction by. Next, we divide all the auxiliary nodes of T U into k 1 grous, grouing together those nodes with the same leftmost digit in their --alternate-decimated indices. The auxiliary nodes of T L can, likewise, be classified into j 1 different grous based also on the values of the leftmost digits IEEE Communications Society 426

3 in their --alternate-decimated indices. We further let the shift to be the same when we connect the auxiliary nodes of T U in the same grou to the auxiliary nodes of T L in the same grou. Denote by S y,z the shift introduced when we connect the auxiliary nodes of T L in the y th grou to the auxiliary nodes of T U in the z th grou. For different y and z, the shifts S y,z may be the same or different from each other. In addition, since S y,z is not to affect the assigning of the grous, its leftmost and rightmost digits are always set to 0. The following theorem roved elsewhere relates tye-i cycles to shifts: Theorem 2 (TYPE-I 2h-CYCLES) Connecting the auxiliary node indexed by X in the z th grou in T U to the auxiliary node indexed by π d (X ) +S y,z (S y,z can be any ossible value) in the y th grou in T L guarantees that any tye-i cycle formed is at least of length 2h (h denotes the number of tiers in T U ). Hence, when following theorem 2, we have (j 1)(k 1) free arameters S y,z to adjust to revent short tye-ii cycles while in the mean time all short tye-i cycles are avoided. Theorem 3 rovides a sufficient condition on the shifts S y,z that can be used to revent tye-ii 4-cycles. Theorem 3 (NO TYPE-II 4-CYCLES) Consider the m th and the n th grous of nodes in the uer-tree T U and the i th and the j th grous in the lower-tree T L. If the associated shifts satisfy S i,m +S j,n S i,n +S j,m, then NO tye-ii 4- cycles are formed between these four grous. shifts MUST satisfy S i,m +S j,n = S i,n +S j,m. Figure 3.(a) shows a tye-ii 4-cycle containing four vertices A, B, C, and D. It can also be viewed as a cycle containing eight auxiliary nodes a 1, a 2, c 1, c 2, b 1, b 2, d 2, and d 1. With reference to the lot in Figure 3.(a), and assuming that the index for the auxiliary node a 1 is X a1, then according to the connecting rule resented in theorem 2, the index for the auxiliary node b 1 is X b1 = π d (X a1 ) +S i,m.letδ b denote the difference between X b1 and X b2, i.e., δ b = X b2 X b1. Since the auxiliary nodes b 1 and b 2 are connected to the same check node B, only the rightmost digit of the -alternate-decimal form of δ b is non-zero, all its other digits are zero. The index for the auxiliary node c 1 is X c1 = π d (X b2 S i,n ).Again,letδ c = X c2 X c1. Then the - alternate-decimal form of δ c has only one non-zero digit, its rightmost digit. The index for the auxiliary node d 1 is X d1 = π d (X c2 ) +S j,n. Let X d2 = X d1 + δ d ; the index for the auxiliary node a 2 in terms of X d2 is X a2 = π d (X d2 S j,m ). The relationshi between X a1 and X a2 is X a1 = X a2 +δ a. Iterating in the definition of X a2,wehave π d [π d (π d (π d (X a1 ) +S i,m +δ b S i,n ) +δ c ) +S j,n +δ d S j,m ]+δ a = X a1. (3) As π d (π d (X)) = X, this leads to π d (S i,m S i,n +S j,n S j,m ) +[π d (δ b +δ d ) +δ c +δ a ]=0 (4) In the --alternate-decimal form of a shift, the leftmost and the rightmost digits are always zero. On the other hand, excet for the leftmost and rightmost digits, all the digits in δ c, δ a, π d (δ b ), and π d (δ d ) are zero. Therefore, euation (4) can be slit into two sub-euations: Fig. 3. (a) Fig. 3. (b) π d (S i,m S i,n +S j,n S j,m ) = 0 (5) π d (δ b +δ d ) +δ c +δ a = 0. (6) It follows then from euation (5) S i,m +S j,n = S i,n +S j,m. (7) This comletes the roof Exclude Tye-II Cycles of Arbitrary Length Fig. 3. (c) Fig. 3. Fig. 3. (d) Proof : We rove theorem 3 by roving an euivalent roosition: If a tye-ii 4-cycle is formed, then the associated We observe that there are many different classes of tye-ii cycles. For examle, all tye-ii cycles of length six can be divided into three classes, as shown in Figure 3.(b), 3.(c), and 3.(d). For convenience, we divide all tye-ii cycles u to length 2L into L 1 euivalence sets based on the number of edges N I contained in the interleaver comonent. For examle, all tye-ii cycles of length six belong to two euivalence sets: Cycles shown in Figure 3.(b) and 3.(c), with four edges in the interleaver, are in the same euivalence set. Another euivalence set contains the cycles shown in IEEE Communications Society 427

4 Figure 3.(d) that have six edges in the interleaver. The following theorem eliminates the cycles in each euivalence set. Each cycle in the euivalence set with N I =2k is associated with 2k shifts S y1,z 1 S y2,z 2... S y2k,z 2k whose subindices satisfy the following two conditions: (i) y 2t 1 = y 2t, t = 1, 2, 3,...,k and z 2t = z 2t+1, t =1, 2, 3,...,k 1 (ii) y 2t y 2t+1, t =1, 2, 3,...,k 1 and z 2t 1 z 2t, t =1, 2, 3,...,k Define the shift matrix S =[S y,z ] to be a matrix that collects all the shifts. From the oint of view of grah theory, S y1,z 1, S y2,z 2,...,S y2k,z 2k are adjacent vertices of a closed ath in the S matrix. For examle, as shown in Figure 4, S 1,1, S 1,2, S 2,2, and S 2,1 are vertices of a closed ath of length four (dashed line) in S and S 2,5, S 2,6, S 4,6, S 4,4, S 3,4, and S 3,5 are vertices of a closed ath of length six (solid line) in S. Theorem 4 For the euivalence set with N I =2k,let S = k u=1 S y 2u,z k 2u v=1 S y 2v 1,z 2v 1. Each S having h digits is free of any cycle having less than N T =2ledges in the trees (both T U and T L ) if it contains at most d = h l 2 consecutive digits 0 in its --alternate-decimal exansion. The roof is rather lengthy. We rove it elsewhere. By choosing suitable shifts S y,z according to theorems 2-4, we can avoid all short tye-i and tye-ii cycles u to the desired length g 2. Fig. 5. Parity-check matrix of a (6666, 3, 6) TS-LDPC code with girth PERFORMANCE EVALUATION We comare by simulation the bit error rate (BER) of the TS-LDPC codes with the BER of randomly constructed LDPC codes that are free of 4-cycles [2] in additive white Gauss noise (AWGN) channels. The codes are decoded with the sum-roduct algorithm [9], and we adot the signal to noise ratio (SNR) defined in [2]: SNR = 10 log 10 [ Eb / ( 2rσ 2)] where r denotes the code rate. Bit Error Rate (BER) TS LDPC code with g=10 Random LDPC code(4 cycles free) Fig. 6. BER erformance comarison between a column weight j =3TS-LDPC code with girth 10 and a randomly constructed LDPC code of the same size free of 4-cycles. Fig. 4. Two closed aths in the shift matrix S As an illustration, we alied the above methods to construct a (6666, 3, 6) regular LDPC code, rate.5, with girth g =10. Its structure is given by the matrix H shown in Figure 5. We can clearly identify T U, T L, and the interleaver comonent I from the constructed matrix, as labelled in Figure 5. In this matrix, along the solid lines, there is a single 1 in each row, while along the dashed thicker diagonals there are five 1 s in each row, so that er row there are six 1 s. The lot in Figure 6 shows the BER erformance for a column weight j =3TS-LDPC code with girth 10 (solid line). For comarison, we also show the BER erformance of a randomly constructed LDPC code (no 4-cycles) with column weight j =3(dashed line). Both codes have the same block length 6666 and the same code rate 1/2. From Figure 6, it can be seen that the BER erformance of the TS-LDPC code is 0.12dB better than that of the random LDPC code at BER= 10 5 while at low SNR both codes have similar error-correcting erformance. According to [3], the (6666, 3, 6) TS-LDPC code has minimum distance d min 10. Since the lower bound of d min derived in [3] is not tight, the actual d min of the (6666, 3, 6) TS-LDPC code may be much larger than 10. In the high SNR region, d min is a dominant factor in determining the code BER erformance. This exlains why the TS-LDPC IEEE Communications Society 428

5 code with girth 10 has good BER erformance in the high SNR region. 5. FAST DECODING The TS-LDPC codes can be effectively decoded by what we refer to as the turbo like decoding algorithm (TLDA) [8]. The TLDA is as follows: Ste 1: Initialization. Ste 2: Decode T U, using the udated information rovided by I, and, after decoding, transmit the new robabilistic information to T L through I. Ste 3: Decode T L, using the udated information from I and then transmit the new robabilistic information back to T U through I. Ste 4: Comute the temorary decoding oututs. If decoding success is achieved, go to ste 5. Otherwise, go back to ste 2. Ste 5: End. The message udatings of the TLDA are the same as those for the sum-roduct decoding algorithm [1]. We comment briefly on the advantages of TLDA. It is well known, [9], that with a cycle-free Tanner grah, the sum-roduct algorithm terminates in a finite number of stes and yields minimum symbol error robability. Therefore, in isolation, the local decoding for each cycle-free comonent is otimal. The TLDA is still iterative: each comonent transmits its a osteriori robability (APP) information to the others through the interleaver and, in turn, these comonents use these APPs as a riori information to start their own decoding rocess. We resent a simulation study by Monte Carlo simulations comaring the erformance of the code using TLDA and the standard sum-roduct decoding. This is a (6666, 3, 6) TS-LDPC code with uniform column weight j =3and code rate 0.5. Figure 7.(a) shows the BER erformance of the code. The lots in Figure 7.(b) show that TLDA converges faster by about 50% (smaller number of iteration stes) than the sum-roduct decoding algorithm. 6. CONCLUSION In this aer, we roose a new class of well-structured LDPC codes TS-LDPC codes. We resented designs of flexible code rate with arbitray column weight and arbitrary girth. TS-LDPC codes can be decoded efficiently, characteristics that make them attractive in alications, e.g., communication systems and data storage systems. 7. REFERENCES [1] R. G. Gallager, Low-Density Parity Check Codes, MIT Press, Cambridge, MA, [2] D. J. C. Mackay, Good error-correcting codes based on very sarse matrices, IEEE Trans. on Information Theory, vol. 45, no. 2, March 1999, [3] R. Michael Tanner, A recursive aroach to low comlexity codes, IEEE Trans. on Information Theory, vol. IT-27, no. 5, Se. 1981, [4] R. Michael Tanner, Sectral grahs for uasi-cyclic LDPC codes, in Proceedings of International Symosium on Information Theory, Washington, DC, June [5] Li Ping and Keying Y. Wu, Concatenated tree codes: A low-comlexity, high-erformance aroac, in IEEE Trans. Inform. Theory, vol. 47, no.2, Feb. 2001, [6] Jos M. F. Moura, J. Lu, H. Zhang, Structured LDPC codes with large girth,, in IEEE singal rocessing magazine, vol. 21, no.1, Jan. 2004, [7] J. Lu and José M. F. Moura, Turbo design for LDPC codes with large girth, in IEEE Signal Processing and Wireless Communications (SPAWC) Worksho, Rome, Italy, June Bit Error Rate (BER) 10 1 Sum roduct decoding TLDA decoding Fig. 7. (a) The average number of iteration stes Sum roudct decoding TLDA decoding Fig. 7. (b) [8] Jin Lu and José M. F. Moura, Turbo Like Decoding of LDPC Codes, in IEEE INTERMAG 2003, Ar. 2003, Boston, MA. [9] F. R. Kschischang, B. J. Frey, H. A. Loeliger, Factor grahs and the sum-roduct algorithm, IEEE Transactions on Information Theory, vol. 47, no.2, , Feb Fig. 7. IEEE Communications Society 429