數位通訊已經成為人類生活中不可或缺的一部分，其中錯誤更正編碼技術是很重要的一環，而極化碼(polar codes)是這幾年最熱門的錯誤更正編碼。它在碼長很長時，它的錯誤率可以降得很低；但如果屬於中段碼長時，它的效果就沒有表現的很好。這時我們提出結合第二層的錯誤更正碼，希望能有效改善極化碼在中段碼長的效果。
本篇論文的方向是研究極化碼去結合低密度同位檢測碼(LDPC codes)以及渦輪碼(turbo codes)。在極化碼結合低密度同位檢測碼的部分，我們會在極化碼編碼端做系統性以及非系統性的搭配，之後分別測試在LDPC生成矩陣不同大小的情況下的錯誤率，然後我們再將不同情況下最佳的數據，拿去跟系統性及非系統性極化碼去做比較。
在極化碼結合渦輪碼的部分，我們一樣會在極化碼編碼端做系統性及非系統性的搭配，在不同碼率下，我們去觀察有使用刺穿機制跟沒有使用刺穿機制的極化-渦輪碼錯誤率，然後調整不同的刺穿碼率，在各種狀況下，找出錯誤率最低的數據。我們再把最佳的數據拿去跟系統性極化碼以及非系統性極化碼做比較，最後再去探討是否有必要去結合第二層編碼。

Digital communication has already become a part of daily life. Error correction coding is a very important part of digital communication, and polar codes are on of the most popular error correction codes in recent years. When the length of codewords of polar ocdes is long, the bit error rate(BER) is very low. However, when the codelength is medium, the performance of polar codes is not good. In this work, therefore, we try to investigate whether it is feasible to improve the BER by cascading polar codes to some other error correction codes.
We try to combine polar codes with LDPC codes and turbo codes. Regarding the combination of polar codes with LDPC codes, we use systematic polar encoder and nonsystematic polar encoder. First, we analyze the bit error rates with respect to different sizes of LDPC generator matrix. Then we try to find the best results and compare them with the performance of polar codes.
Regarding to use of turbo codes, we also match them with systematic polar encoder and nonsystematic polar encoder. With respect to different code rates of polar codes, we observe the bit error rates of polar-turbo codes, with puncturing or not. We do test on several rates. We find the best results in each situation, and then we compare them with the performance of polar codes. In the end, we will discuss whether it is better to combine polar codes with LDPC codes and turbo codes

[1].Arikan, E., Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels. IEEE Transactions on Information Theory, 2009. 55(7): p. 3051-3073.
[2].Sae-Young, C., T.J. Richardson, and R.L. Urbanke, Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation. IEEE Transactions on Information Theory, 2001. 47(2): p. 657-670.
[3].Du, W., S. Zhang, and F. Ding. Exploiting the UEP property of polar codes to reduce image distortions induced by transmission errors. in 2015 IEEE/CIC International Conference on Communications in China (ICCC). 2015.
[4].Balatsoukas-Stimming, A., M.B. Parizi, and A. Burg, LLR-Based Successive Cancellation List Decoding of Polar Codes. IEEE Transactions on Signal Processing, 2015. 63(19): p. 5165-5179.
[5].Tal, I. and A. Vardy, List Decoding of Polar Codes. IEEE Transactions on Information Theory, 2015. 61(5): p. 2213-2226.
[6].William, H.T., et al., Near Optimum Error Correcting Coding and Decoding: TurboCodes, in The Best of the Best:Fifty Years of Communications and Networking Research. Wiley-IEEE Press ,2007. p. 692.
[7].Bahl, L., et al., Optimal decoding of linear codes for minimizing symbol error rate (Corresp.). IEEE Transactions on Information Theory, 1974. 20(2): p. 284-287.
[8].Franz, V. and J.B. Anderson, Concatenated decoding with a reduced-search BCJR algorithm. IEEE Journal on Selected Areas in Communications, 1998. 16(2): p. 186-195.
[9].Gallager, R., Low-density parity-check codes. IRE Transactions on Information Theory, 1962. 8(1): p. 21-28.
[10].Vangala, H., Y. Hong, and E. Viterbo, Efficient Algorithms for Systematic Polar Encoding. IEEE Communications Letters, 2016. 20(1): p. 17-20.