On Generalized LDPC Codes and Their Decoders - Semantic Scholar

On Generalized LDPC Codes and Their Decoders Shadi Abu-Surra

Gianluigi Liva

William E. Ryan

University of Arizona Email: [email protected]

DLR (German Aerospace Center) Email: [email protected]

University of Arizona Email: [email protected]

Abstract— We first consider the design of generalized LPDC (G-LDPC) codes with recursive systematic convolutional (RSC) constraint nodes in place of the standard single parity-check constraint nodes. Because rate-1/2 RSC nodes lead to low-rate G-LDPC codes, we consider high-rate tail-biting RSC nodes for which Riedel’s APP-decoder based on the reciprocal-dual code trellis becomes necessary. We then consider higher-rate G-LDPC codes via the class of doubly generalized LDPC (DG-LDPC) codes. We show how the graph of a DG-LDPC code (called a DG-graph) may be transformed into a G-graph. This alternative representation of DG-LDPC codes leads to a modified-schedule G-graph decoder which is equivalent to the flooding-schedule DG-graph decoder.

I. INTRODUCTION A generalization of LDPC codes was suggested by Tanner [1] for which subsets of the set of code bits obey a more complex constraint than a single parity check (SPC) constraint. The generalized constraint nodes are called super constraint nodes (super-CNs). The Tanner graph of a generalized LDPC (G-LDPC) code with length n and mc constraints is depicted in Fig. 1(a). There are several advantages to employing superCNs. First, super-CNs tend to lead to larger minimum distances. Second, because a complex constraint node can encapsulate multiple SPC constraints, the resulting Tanner graph will contain fewer edges so that deleterious graphical properties are more easily avoided. Third, the belief propagation decoder tends to converge more quickly because the CN processors now correspond to stronger codes. The first two advantages lead to a lower error-rate floor. The third advantage leads to lower decoder complexity and/or higher decoding speed. Tanner’s G-LDPC codes were investigated by several researchers in recent years where Hamming codes, BCH codes, ReedSolomon codes, and recursive systematic convolutional (RSC) codes were considered for use in constraint nodes [2]–[11]. Because G-LDPC codes typically replace high-rate SPC nodes with lower-rate nodes (e.g, rate-4/7 Hamming nodes), the code rate of G-LDPC codes tends to be small. In this paper we consider two approaches to increasing the rate of G-LDPC codes. First, we employ high-rate RSC component codes in our design of G-LDPC codes. We employ rate-(κ−1)/κ tail-biting RSC component codes in our designs. Second, we explore a further generalization of the G-LDPC principle, called doubly generalized LDPC (DG-LDPC) codes, studied in [12], [13]. In DG-LDPC codes, the variable nodes, which represent (lowrate) repetition codes in the graphs of LDPC and G-LDPC This work was funded in part by University of Bologna, Progetto Pluriennale, and by NASA-JPL grant 1264726.

C0

Cmc −1

C1 ...

l V0

l V1

l V2

...

l Vn−2

l Vn−1

Fig. 1. (a) Tanner graph (G-graph) of G-LDPC code. (b) to (d) Transformation from DG-graph to G-graph.

codes, are replaced by super variable nodes (super-VNs) which represent more general codes (of higher rate). We will call the Tanner graph of a DG-LDPC code a DG-graph to distinguish it from the graph of a G-LDPC code, which we will call a Ggraph. We show how any DG-LDPC code can be represented by a G-graph. This leads to a decoder based on the G-graph which is an alternative to the one presented in [13] which is based on the DG-graph. II. G-LDPC CODES WITH HIGH-RATE RSC COMPONENTS A quasi-cyclic rate-1/2 (8160, 4080) G-LDPC code was constructed from the protograph in Fig. 2. In the decoder, we used the standard belief propagation algorithm. As indicated in the Introduction, the complexity of the BCJR decoder [14] for high-rate RSC nodes is prohibitive. Consequently, in this paper we adopted a variation of Riedel’s (tail-biting) decoder [15] (described in equation (20) in [15]) which uses the trellis of the reciprocal-dual code. III. A G-GRAPH DECODER FOR DG-LDPC CODES Fig. 1 summarizes the process of transforming the DG-graph to G-graph. The two equivalent graphical representations of a DG-LDPC code in Fig. 1(b) and Fig. 1(d) implies equivalence in the sense that they correspond to the same set of codewords. However, they do not imply equivalent iterative decoders. In fact, if the flooding schedule was employed in each of the iterative decoders corresponding to these graphs, the various

0

10

0

10

−1

10 −2

10

−2

−4

−3

10

FER

FER/BER

10 10

−4

10 −6

10

−5

10 −8

10

0.2

(8160, 4080)RSC, BER (8192, 4096)AR4JA, BER (8160, 4080)RSC, FER 4096 RCB 0.4

0.6

0.8 Eb/No [dB]

RCB FER Modified, 200it FER Flooding, 200it FER Modified, 20it FER Flooding, 20it

−6

10

−7

1

1.2

1.4

Fig. 2. Performance of the (8160, 4080) G-LDPC code compared with that of the (8192, 4096) AR4JA code. The G-LDPC code is constructed as follows. First, we use tail-biting RSC component codes with memory υ = 4, blocklength 3T = 60, and generator polynomials (32, 36, 31)8 . The protograph, shown in the figure, has 40 information bits and 40 parity bits, and q = 102 replicas of the protograph were made to design the code. RCB = Gallager’s random coding bound.

decoder metrics would differ somewhat after each iteration, although the decoder output decisions would almost always agree after a sufficient number of iterations. However, it is possible to adjust to decoding schedule of the G-graph decoder so that it is equivalent to the flooding-schedule DGgraph decoder (under the assumption that the corresponding nodal decoders for the two graphs are identical). The modified schedule G-graph decoder effectively accelerates the messagepassing between node C and CNs W, X, Y, and Z so that these messages are passed in one half-iteration as in the DGgraph case. That is, the two edges between CN C and CN W are treated as one edge, and similarly for the two edges between CN C and each of the CNs X, Y, and Z. For the case of nodes W and X, channel LLRs are added to the messages coming from node C (see Fig. 1(d)), whereas the messages are unaltered for the cases of nodes Y and Z. Fig. 3 compares the performance of the two decoders. R EFERENCES [1] R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. on Inform. Theory, vol. 27, pp. 533–547, September 1981. [2] J. Boutros, O. Pothier, and G. Zemor, “Generalized low density (Tanner) codes,” in IEEE Int. Conf. on Commun., ICC ’99, pp. 441–445, June 1999. [3] M. Lentmaier and K. S. Zigangirov, “Iterative decoding of generalized low-density parity-check codes,” in IEEE Int. Symp. on Inform. Theory, p. 149, August 1998. [4] N. Miladinovic and M. Fossorier, “Generalized LDPC codes with ReedSolomon and BCH codes as component codes for binary channels,” in IEEE Global Telecommunications Conf., GLOBECOM ’05, November 2005. [5] S. Vialle and J. Boutros, “A Gallager-Tanner construction based on convolutional codes,” in Proceedings of Int. Workshop on Coding and Cryptography, WCC’99, pp. 393–404, January 1999.

10

1

1.5

2

2.5

3

Eb/No [dB] Fig. 3. Performance of the (1800,960) DG-LDPC code using the floodingschedule and modified-schedule decoders, both based on the code’s Ggraph representation. The DG-graph has 450 super-VNs based on the (7, 4) Hamming code and 210 super-CNs based on the (15, 11) Hamming code. The DG-LDPC code was obtained from 30 copies of a protograph consisting of 15 Hamming (7, 4) super-VNs and 7 Hamming (15, 11) super-CNs. The protograph expansion was performed by progressive edge-growth (PEG) [16] using cyclic edge permutations.

[6] R. M. Tanner, “A hybrid coding scheme for the Gillbert-Elliot channel,” in Proc. of the 42th Annual Allerton Conf. on Commun., Control, and Computing, Illinois, September 2004. [7] G. Liva and W. E. Ryan, “Short low-error-floor Tanner codes with Hamming nodes,” in IEEE Military Commun. Conf., MILCOM ’05, 2005. [8] G. Liva, W. E. Ryan, and M. Chiani, “Design of quasi-cyclic Tanner codes with low error floors,” in 4th Int. Symp. on Turbo Codes, ISTC2006, April 2006. [9] J. Thorpe, “Low-density parity-check (LDPC) codes constructed from protographs,” Tech. Rep. 42-154, IPN Progress Report, August 2003. [10] S. Abu-Surra, G. Liva, and W. E. Ryan, “Design and performance of selected classes of Tanner codes,” in UCSD Workshop on Information Theory and Its Applications, February 2006. http://ita.ucsd.edu/workshop/06/talks/papers/129.pdf. [11] S. Abu-Surra, G. Liva, and W. E. Ryan, “Low-floor Tanner codes via Hamming-node or RSCC-node doping,” in Lecture Notes in Computer Science, vol. 3857, pp. 245–254, January 2006. [12] A. Ashikhmin, G. Kramer, and S. ten Brink, “Extrinsic information transfer functions: Model and erasure channel properties,” IEEE Trans. on Inform. Theory, pp. 2657–2673, November 2004. [13] Y. Wang and M. Fossorier, “Doubly generalized LDPC codes,” in IEEE Int. Symp. on Inform. Theory, 2006. to appear. [14] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. on Inform. Theory, vol. 20, pp. 284–287, March 1974. [15] S. Riedel, “Symbol-by-symbol MAP decoding algorithm for high-rate convolutional codes that use reciprocal dual codes,” IEEE J. on Select. Areas in Commun., vol. 16, pp. 175–185, February 1998. [16] X. Y. Hu, E. Eleftheriou, and D. M. Arnold, “Progressive edge-growth Tanner graphs,” in IEEE Global Telecommunications Conf., GLOBECOM ’01, pp. 995–1001, November 2001.