在擁有相同的碼率及相同的碼長之線性區塊編碼當中，里德所羅門碼能夠達到最大可能的最小距離，並且在眾多的應用之中，可能會變成最廣受大家使用的錯誤更正碼之一。雖然代數的硬式解碼演算法可以提供實際的效能，但是里德所羅門碼之錯誤更正能力的淺力，在還沒有導入軟資訊的情況之下，仍然沒有被完全地展現出來。因此，有相當多的研究已經專注於在里德所羅門碼的軟式解碼演算法上。在本次成果當中，我們提出了里德所羅門碼之軟式疊代解碼演算法。我們提出的解碼演算法結合了變換矩陣、消息動態排程以及位元翻轉解碼等觀念。奇偶檢驗矩陣會在每次疊代時重新排列，其中，系統化部分對應到最低信息位元，以此來降低他們對其他位元的影響。使用動態排程，相對來講較為重要的解碼訊息會被用來更新這些最低位元信息；也就是說，大部分擁有低信息度的錯誤位元都可以被更正回來。最後，位元翻轉解碼被應用在最高信息位元，以此解決那些剩餘的且擁有高信息度的位元。當我們提出的解碼演算法做一個整合，並且將之應用到(255,239)的里德所羅門碼上時，其錯誤率的效能和最大似然較低界線的差距可以縮小到0.25dB；並且，和先前其他文獻所發表的里德所羅門碼之軟式解碼相比之下，我們會有超過0.4dB的效能增益。

This paper, proposes an iterative soft-decision decoding algorithm for Reed-Solomon (RS) codes. The proposed decoding algorithm combines the concepts of adapting the parity-check matrix, informed dynamic scheduling, and bit-flipping decoding. The parity-check matrix is re-arranged before each iteration, where the systematic part is mapped to the least reliable bits, consequently reducing their influence on the other bits. Using dynamic scheduling, the more important decoding messages are updated to these least reliable bits, meaning that the majority of the error bits with low reliability can be corrected. Finally, bit-flipping decoding is applied to the most reliable bits, thereby solving the remaining error bits that have a high reliability. When the proposed integrated decoding is applied to a (255, 239) RS code, the difference between its frame error rate performance (FER) and the maximum-likelihood (ML) bound can be reduced to 0.25 dB, and a gain of more than 0.4 dB is achieved compared to all the previously recorded soft-decision decoding for RS codes.

Abstract I
中文摘要 I
1 Introduction 1
2 Preliminaries 5
2.1 A Brief Introduction of Reed-Solomon Codes . . . . . . . . . . . 5
2.2 Binary Image of the Parity-Check Matrix . . . . . . . . . . . . . 6
2.3 Adaptive Belief Propagation Algorithm . . . . . . . . . . . . . . 10
2.4 Layered Residual Belief Propagation . . . . . . . . . . . . . . . 13
2.5 Order Statistic Decoding . . . . . . . . . . . . . . . . . . . . . . 14
3 Proposed Double-polling Residual Belief Propagation 17
3.1 Flooding algorithm with Adapting Parity-Check Matrix . . . . . 17
3.2 Global Search Residual Belief Propagation . . . . . . . . . . . . 18
3.3 Global Search RBP with restricted variable nodes . . . . . . . . 20
III
3.4 Global Search RBP with restricted check nodes . . . . . . . . . 23
3.5 Global Search RBP with restricted edges . . . . . . . . . . . . . 25
3.6 Double-polling Residual Belief Propagation with channel value . 27
3.7 Double-polling Residual Belief Propagation . . . . . . . . . . . . 32
3.8 Bit-Flipping on MRVNs . . . . . . . . . . . . . . . . . . . . . . 36
4 Numerical results 43
4.1 Probability of occurrence of wrong check per codeword . . . . . 43
4.2 Convergence rate . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Calculation of Cycle-4 . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Conclusion 51
Bibliography 52

[1] E. R. Berlekamp, R. E. peile, and S. P. Pope, “The application of error
control to communications,” IEEE Commun. Mag, vol.25, pp. 44-57, 1987.
[2] W. W. Wu, D. Haccoun, R. E. peile, and Y. Hirata, “Coding for satellite
communication,” IEEE J. Select. Areas Commun, vol. SAC-5, pp. 724
785, 1987.
[3] Consultative Committee for Space Data Systems, “Recommendations for
Space Data System Standards: Telemetry Channel Coding, Blue Book,
1984.
[4] E. R. Berlekamp, J. Shifman, and W. Toms, “An application of Reed
Solomon codes to a satellite TDMA system, MILCOM’86, Monterey, CA.
[5] B. C. Mortimer, M. J. Moore, and M. Sablatash, “The design of a high
performance error-correcting coding scheme for the Canadian broadcast
telidon system based on Reed-Solomon codes,” IEEE Trans. Commun.,
vol. COM-35, pp. 1113-1138, 1987.
[6] M. B. Pursley and W. E. Stark, “Performance of Reed-Solomon coded
frequency-hop spread-spectrum communication in partial-band interfer
ence,” IEEE Trans. Commun., vol. COM-33, pp. 767-774, 1985.
[7] M. P. C. Fossorier and A. Valembois, “Reliability-based decoding of Reed
Solomon codes using their binary image,” IEEE Commun. Lett., vol. 7,
pp. 452-454, Jul. 2004.
[8] J. Jiang, K. R. Narayanan, “Iterative soft-input soft-output decoding of
Reed-Solomon codes by adapting the parity-check matrix,” IEEE Trasn.
Inf. Theory, vol. 52, no. 8, 2006.
[9] W.J.Gross,F.R.Kschischang R.Koetter, and P. G. Gulak, “Applications
of algebraic soft-decision decoding of Reed-Solomon codes,” IEEE Trans.
Communi., vol. 54, no. 7, pp. 1224-1234, Jul. 2006.
[10] H. Tang, Y. Liu, M. P. C. Fossorier, and S. Lin, “Combining Chase-2 and
GMD decoding algorithms for nonbinary block codes,” IEEE Commun.
Lett., vol. 5, no. 5, pp. 209-211, May 2000.
[11] A. V. Casado, M. Griot, and R. Wesel, “LDPC decoders with informed
dynamic scheduling,” IEEE Trans. on Commun., vol. 58, no. 12, pp.
3470-3479, Dec. 2010
[12] H.-C. Lee, Y.-L. Ueng, S.-M. Yeh, and W.-Y. Weng, “Two informed dy
namic scheduling strategies for iterative LDPC decoder,” IEEE Trans. on
Commun., vol. 61, no. 3, pp. 886-896, Mar. 2013.
[13] H.-C. Lee and Y.-L. Ueng, “Informed dynamic schedules for LDPC decod
ing using belief propagation,” in Proc. 24th IEEE Int. Symp. on Personal
Indoor and Mobile Radio Communications (PIMRC 2013), London, UK,
8-11 Sept., 2013.
[14] H.-C. Lee and Y.-L. Ueng, “LDPC decoding scheduling for faster conver
gence and lower error floor,” IEEE Trans. on Commun., vol. 62, no. 9,
pp. 3104-3113, Sept. 2014.
[15] V. Guruswami and M. Sudan, “Improved decoding of ReedSolomon and
algebraic-geometric codes,” IEEE Trans. Inf. Theory, vol. 45, pp. 1757
1767, Sept. 1999.
[16] R. Koetter and A. Vardy, “Algebraic soft-decision decoding of Reed
Solomon codes,” IEEE Trans. Inf. Theory, 49: 2809-2825, November
2003.
[17] A. Kothiyal and O. Y. Takeshita, “A comparison of adaptive belief prop
agation and the best graph algorithm for the decoding of block codes,” in
Proc. IEEE Int. Symp. on Inform. Theory 2005, pp. 724-728.
[18] D. Agrawal and A. Vardy, “Generalized-minimum-distance decoding in
euclidean space: Performance analysis,” in IEEE Trans. Inform. Theory,
vol. 46, pp. 60-83, Jan. 2000.
[19] M. Fossorier and S. Lin, “Error performance analysis for reliability-based
decoding algorithms,” in IEEE Trans. Inform. Theory, vol. 48, pp. 287
293, Jan. 2002.