極化碼是一種可被證明達到向農容量的編碼方式，但對於碼長為短和中的極化碼，其解碼性能一直被認為無法勝過低密度奇偶檢查碼和渦輪碼。本篇論文採用一種名為A*演算法的最大似然法則演算法，探討短極化碼能達到的錯誤率性能，跟現有常見的解碼演算法所能達到的錯誤率之間的差距。除此之外，本篇論文不僅提出兩種修改過的啟發式，用以降低A*演算法的複雜度，更提出兩種修改過的A*演算法，此兩種修改過的A*演算法具有不須將接收訊號依照可靠度重新排序的特性，可以在解碼過程中不需對生成矩陣做高斯消去，因此更適合用於硬體實作。修改過的A*演算法充分搭配極化碼結構的特性，有效地利用極化碼的結構來實現將訊號依照可靠度重新排序的概念，得以幫助降低解碼的複雜度。此外，修改過的A*演算法利用極化碼生成矩陣為一個下三角矩陣的特殊結構，非常適合用來搭配特定的啟發式來有效地降低解碼複雜度。模擬結果顯示了A*演算法和其它常見的解碼演算法(如: 在第五代移動通信系統規格中的循環冗餘校驗輔助的列表名單串聯連續消除解碼演算法、排序統計解碼演算法)之間存在錯誤率性能的差距。最後，利用統計搜索樹的枝幹數量來量化解碼複雜度，統計結果顯示修改過的A*演算法的解碼複雜度搭配合適的啟發式之後可以有效地貼近A*演算法的解碼複雜度，並且保留了極化碼特殊的結構性，其它相關的演算法解碼複雜度也被呈現在本篇論文的模擬結果當中。

It is known that polar codes can not perform as well as LDPC and turbo codes for short and moderate code length. We focus on investigating the error rate capability for short polar codes by using an efficient ML decoder called A* algorithm. In addition, we not only propose two modified heuristic functions that are used to reduce the decoding complexity of A* algorithm but also propose two modified A* algorithms that are sorting-free and are fully optimized for polar codes to be more hardware friendly. The modified A* algorithms are suitable for exploiting the polar code structure to help reducing the decoding complexity. Simulation results show the error rate performance gap between A* algorithm and other common decoding algorithms, such as CA-LSC decoder in 5G standard. The related decoding complexity estimated by search tree edges are simulated as well.

[1] C. E. Shannon, “A mathematical theory of communication,” in The Bell
System Technical Journal, vol. 27, pp. 379-423, 623-656, July 1948.
[2] R. Gallager, “Low-density parity-check codes,” in IRE Transactions on
Information Theory, vol. 8, no. 1, pp. 21-28, January 1962.
[3] C. Berrou, A. Glavieux and P. Thitimajshima, “Near Shannon limit error-
correcting coding and decoding: Turbo-codes. 1,” Communications, 1993.
ICC ’93 Geneva. Technical Program, Conference Record, IEEE Interna-
tional Conference on, Geneva, 1993, pp. 1064-1070 vol.2.
[4] E. Arikan, “Channel Polarization: A Method for Constructing Capacity-
Achieving Codes for Symmetric Binary-Input Memoryless Channels,” in
IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 3051-3073,
July 2009.
[5] B. Yuan and K. K. Parhi, “Architecture optimizations for BP polar de-
coders,” 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, BC, 2013, pp. 2654-2658.
[6] U. U. Fayyaz and J. R. Barry, “Low-Complexity Soft-Output Decoding
of Polar Codes,” in IEEE Journal on Selected Areas in Communications,
vol. 32, no. 5, pp. 958-966, May 2014.
[7] S. Kahraman and M. E. elebi, “Code based efficient maximum-likelihood
decoding of short polar codes,” 2012 IEEE International Symposium on
Information Theory Proceedings, Cambridge, MA, 2012, pp. 1967-1971.
[8] B. Li, H. Shen and D. Tse, “An Adaptive Successive Cancellation List
Decoder for Polar Codes with Cyclic Redundancy Check,” in IEEE Com-
munications Letters, vol. 16, no. 12, pp. 2044-2047, December 2012.
[9] N. Hussami, S. B. Korada and R. Urbanke, “Performance of polar codes
for channel and source coding,” 2009 IEEE International Symposium on
Information Theory, Seoul, 2009, pp. 1488-1492.
[10] Z. Liu, K. Chen, K. Niu and Z. He, “Distance spectrum analysis of polar
codes,” 2014 IEEE Wireless Communications and Networking Conference
(WCNC), Istanbul, 2014, pp. 490-495.
[11] A. Hadi and E. Alsusa, “On enhancing the minimum Hamming distance of
polar codes,” 2016 IEEE 17th International Workshop on Signal Process-
ing Advances in Wireless Communications (SPAWC), Edinburgh, 2016,
pp. 1-5.
[12] L. Ekroot and S. Dolinar, “A* decoding of block codes,” in IEEE Trans-
actions on Communications, vol. 44, no. 9, pp. 1052-1056, Sep 1996.
[13] R. Mori and T. Tanaka, “Performance and construction of polar codes on
symmetric binary-input memoryless channels,” IEEE International Sym-
posium on Information Theory, pp. 1496-1500, June 2009.
[14] T. Richardson and R. Urbanke, “Modern coding theory,” in CAM-
BRIDGE UNIVERSITY PRESS, 2008, p. 202.
[15] E. Arikan, “Systematic Polar Coding,” in IEEE Communications Letters,
vol. 15, no. 8, pp. 860-862, August 2011.
[16] Yunghsiang S. Han, C. R. P. Hartmann and Chih-Chieh Chen, “Efficient
Maximum-Likelihood Soft-Decision Decoding of Linear Block Codes Us-
ing Algorithm A*,” Proceedings. IEEE International Symposium on In-
formation Theory, 1993, pp. 27-27.
[17] Y. S. Han, C. R. P. Hartmann and Chih-Chieh Chen, “Efficient priority-
first search maximum-likelihood soft-decision decoding of linear block
codes,” in IEEE Transactions on Information Theory, vol. 39, no. 5, pp.
1514-1523, Sep 1993.
[18] Y. S. Han, C. R. P. Hartmann and K. G. Mehrotra, “Decoding linear block
codes using a priority-first search: performance analysis and suboptimal version,” in IEEE Transactions on Information Theory, vol. 44, no. 3, pp.
1233-1246, May 1998.
[19] C. F. Chang, T. Y. Lin, C. L. Tai and M. C. Lin, “Advanced Information
of Parity Bits for Decoding Short Linear Block Codes Using the A* Al-
gorithm,” in IEEE Transactions on Communications, vol. 61, no. 4, pp.
1201-1211, April 2013.
[20] L. A. Dunning and W. E. Robbins, “Optimal encodings of linear block
codes for unequal error protection,” Inform. Contr., vol. 37, pp. 150-177,
1978.
[21] D. Wu, Y. Li, X. Guo and Y. Sun, “Ordered Statistic Decoding for Short
Polar Codes,” in IEEE Communications Letters, vol. 20, no. 6, pp. 1064-
1067, June 2016.
[22] G. Liva, L. Gaudio, T. Ninacs, and T. Jerkovits, “Code design for
short blocks: A survey,” arXiv preprint, 2016. [Online]. Available:
http://arxiv.org/abs/1610.00873
[23] K. Chen, K. Niu and J. R. Lin, “List successive cancellation decoding of
polar codes,” in Electronics Letters, vol. 48, no. 9, pp. 500-501, April 26
2012.