在本論文中，將提出一種里德所羅門碼的軟疊代解碼。
在中提出的變換矩陣概念用來跟依消息動
態排程的解碼方法合併。
在每一次的解碼疊代之前，奇偶檢驗矩陣會根據碼字在上次疊代完的對數概數比
重新排列, 如此可以讓最低信度的位元節點在解碼過程中可以
降低它的影響力。加上依消息動態排程的幫助下，重要的解碼訊息將會優先排在第一
順位更新使得信度可以被增強。模擬結果中顯示出所提出的解
碼衍算法在錯誤率方面有重大的表現。跟傳統的自適應置信傳
播方法相比，可以有0.5dB的增益。
加上另一個以信度為基準的解碼演算法-排列統計解碼演算法，可以讓效能表現更提升。

並且以排序統計解碼演算法將效能表現再加強，在階層式信度置信傳播衍算法疊代完之後，
將可靠度高的位元加以挑出，並以可靠度高的這些位元為新位元，重新編碼即可得到新的可能
解碼。並且可以假設錯誤發生在可靠度高的這些位元較少，可以一一翻過位元來找出所有可能
的解碼。如此一來只要可靠度推估的夠準確，效能表現則可以進一步再提升。

本論文提出之演算法為第一個提出將里德所羅門碼以動態排程來解碼的衍算法。
在效能上也有不錯的表現，研究上可以推廣至非二進位低密度奇偶檢查碼或者其他
非二進位為單位的檢查碼，以期能得到好的結果。

In this thesis, an iterative soft-decision decoding algorithm is proposed for Reed-Solomon (RS)
codes. The concept of adapting the parity-check matrix in is combined with informed dynamic
scheduling in this proposed decod- ing algorithm. Prior to each decoding iteration, the
parity-check matrix is re-arranged according to the log-likelihood-ratio (LLR) of the codeword bits
on the last decoding iteration, on purpose to lower the influence of the least reliable variable
nodes (LRVNs) on the decoding process. With the help of informed dynamic scheduling, the
important decoding messages can be scheduled to be updated first, and the reliability of the
LRVNs can be en- hanced. The simulation results show that the proposed decoding algorithm can
provide significant improvement in the error-rate performance. By us- ing the proposed algorithm, a
gain of 0.5 dB can be achieved compared to the conventional adapting belief propagation algorithm.
Another reliability- based decoding algorithm, ordered statistic decoding (OSD) algorithm, can be
added to improve performance further.

[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] D. Divsalar, R. M. Gagliardi, and J. H. Yuen, “PPM performance for Reed-Solomon decoding over
an optical-RF relay link,” IEEE Trans, Commun., vol. COM-32, pp. 302-305, 2984.

[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] M. P. C. Fossorier, S. Lin, “Soft-decision decoding of linear block codes based on ordered
statistics,” IEEE Trasn. Inf. Theory, vol. 41, pp. 1379-
1396, Sep. 1995.

[10] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor Graphs and the Sum-Product
Algorithm,” IEEE Trasn. Inf. Theory, vol. 47, no. 2, Feb. 2001.

[11] R. G. Gallager, “Low-density parity-check codes,” Cambridge, MA:MIT Press, 1963.

[12] W. J. Gross, F. R. Kschischang R. Koetter, and P. G. Gulak, “Applica- tions of algebraic
soft-decision decoding of Reed-Solomon codes,” IEEE Trans. Communi., vol. 54, no. 7, pp. 1224-1234,
Jul. 2006.

[13] 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.

[14] 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.

[15] A. V. Casado, M. Griot, and R. Wesel, “LDPC decoders with in- formed dynamic
scheduling,” IEEE Trans. on Commun., vol. 58, no. 12, pp. 3470-3479, Dec. 2010.

[16] H.-C. Lee, Y.-L. Ueng, S.-M. Yeh, and W.-Y. Weng, “Two informed dy- namic scheduling
strategies for iterative LDPC decoders,” IEEE Trans. on Commun., vol. 61, no. 3, pp. 886-896, Mar.
2013.

[17] H.-C. Lee, and Y.-L. Ueng, “Informed dynamic schedules for LDPC de- coding using belief
propagation,” in Proc. 24th IEEE Int. Symp. on Per- sonal Indoor and Mobile Radio Communications
(PIMRC 2013), Lon- don, UK, 8-11 Sept., 2013

[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.