在此篇論文中，我們提出了直接利用布林函數建構零相關區間互補序列集合的新方法。此方法可以直接建構出所需要的序列集合，而不同於現有的文獻必須由特殊的序列來組合出零相關區間互補序列集合。在我們的方法中，序列集合的長度、數量、集合大小、字集大小和零相關區間的寬度都可以很有彈性地調整。與目前的文獻中建構方法相比，我們是第一個提出具有系統，且可達到集合大小理論值上限的建構方法。對於建構出的最佳零相關區間互補序列集合，我們也分析了關於其數量大小、漢明距離以及尖峰平均功率比之特性。除此之外，我們也提出了長度非二冪次方的零相關區間互補序列集合的建構法。更進一步將建構拓展到完全互補碼。這也是第一個使用布林函數來建構長度為非二冪次方的完全互補碼之代數建構法。對於我們提出的建構方法，產生的序列集合大小和序列長度更具有彈性，也更加提高了在實際系統上應用的可能性。

In this thesis, new direct constructions of Z-complementary sequence (ZCS) sets are proposed based on generalized Boolean functions. Different from previous results in the literature, our methods is a direct construction without the aid of other special sequences. The sequence length, the flock size, the set size, the constellation size, and the width of the zero correlation zone (ZCZ) are all very flexible. Compared with previous methods, this thesis is the first work to propose direct constructions of optimal ZCS sets with respect to the theoretical bound on the set size. The properties of the constructed ZCS set including the enumeration, Hamming distance, and peak-to-average power ratio (PAPR) property are also discussed. To the best of our knowledge, the PAPR property of ZCS sets have not been studied in the literature. Furthermore, we propose constructions of ZCS sets of non-power-of two length based on truncated Boolean functions. And we, then, extend the construction to complete complementary codes (CCCs). It is the first construction of CCCs of non-power-of two length based on generalized Boolean functions. The flexible sequence lengths and set size of our constructed CCCs will increase more possible applications in practical systems.

[1] M. J. E. Golay, “Complementary series,” IRE Trans. Inf. Theory, vol. IT-7, pp. 82–87, Apr. 1961.
[2] C.-C. Tseng and C. L. Liu, “Complementary sets of sequences,” IEEE Trans. Inf. Theory, vol. IT-18, pp. 644–652, Sep. 1972.
[3] G. Welti, “Quaternary codes for pulsed radar,” IRE Trans. Inf. Theory, vol. IT-6, no. 3, pp. 400–408, Jun. 1960.
[4] P. Spasojevic and C. N. Georghiades, “Complementary sequences for ISI channel estimation,” IEEE Trans. Inf. Theory, vol. 47, no. 3, pp. 1145–1152, Mar. 2001.
[5] J. M. Groenewald and B. T. Maharaj, “MIMO channel synchronization using Golay complementary pairs,” in Proc. AFRICON 2007, Windhoek, South Africa, Sep. 2007, pp. 1–5.
[6] S. Boyd, “Multitone signals with low crest factor,” IEEE Trans. Circuits Syst., vol. CAS-33, no. 10, pp. 1018–1022, Oct. 1986.
[7] B. M. Popovi´c, “Synthesis of power efficient multitone signals with flat amplitude spectrum,” IEEE Trans. Commun., vol. 39, no. 7, pp. 1031–1033, Jul. 1991.
[8] R. van Nee, “OFDM codes for peak-to-average power reduction and error correction,” in Proc. IEEE Global Telecommun. Conf., London, U.K., Nov. 1996, pp. 740–744.
[9] K. G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” IEEE Trans. Inf. Theory, vol. 46, no. 1, pp. 104–120, Jan. 2000.
[10] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2397–2417, Nov. 1999.
[11] K.-U. Schmidt, “Complementary sets, generalized Reed-Muller codes, and power control for OFDM,” IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 808–814, Feb. 2007.
[12] M. G. Parker and C. Tellambura, “A construction for binary sequence sets with low peak-to-average power ratio,” Department of Informatics, University of Bergen, Norway, 2003. [Online]. Available: http://www.ii.uib.no/ matthew/ConstructReport.pdf.
[13] W. Chen and C. Tellambura, “Identifying a class of multiple shift complementary sequences in the second order cosets of the first order Reed-Muller codes,” in Proc. IEEE
Int. Conf. Commun., Seoul, Korea, May 2005, pp. 618–621.
[14] C.-Y. Chen, C.-H.Wang, and C.-C. Chao, “Complementary sets and Reed-Muller codes for peak-to-average power ratio reduction in OFDM,” in Proc. 16th Int. Symp. AAECC, LNCS 3857, Las Vegas, NV, Feb. 2006, pp. 317–327.
[15] N. Suehiro and M. Hatori, “N-shift cross-orthogonal sequences,” IEEE Trans. Inf. Theory, vol. 34, pp. 143–146, Jan. 1988.
[16] S.-M. Tseng and M. R. Bell, “Asynchronous multicarrier DS-CDMA using mutually orthogonal complementary sets of sequences,” IEEE Trans. Commun., vol. 48, pp. 53–59, Jan. 2000.
[17] H.-H. Chen, J.-F. Yeh, and N. Suehiro, “A multicarrier CDMA architecture based on orthogonal complete complementary codes for new generations of wideband wireless communications,” IEEE Commun. Mag., vol. 39, pp. 126–134, Oct. 2001.
[18] C.-Y. Chen, Y.-J. Min, K.-Y. Lu, and C.-C. Chao, “Cell search for cell-based OFDM systems using quasi complete complementary codes,” in Proc. IEEE Int. Conf. Commun., Beijing, China, May 2008, pp. 4840–4844.
[19] P. Fan, W. Yuan, and Y. Tu, “Z-complementary binary sequences,” IEEE Signal Process.
Lett., vol. 14, no. 8, pp. 509–512, Aug. 2007.
[20] L. Feng, P. Fan, X. Tang, and K. K. Loo, “Generalized pairwise Z-complementary codes,” IEEE Signal Process. Lett., vol. 15, pp. 377–380, Mar. 2008.
[21] W. Yuan, Y. Tu, and P. Fan, “Optimal training sequences for cyclic-prefix-based singlecarrier multi-antenna systems with space-time block-coding,” IEEE Trans. Commun., vol. 7, no. 11, pp. 4047–4050, Nov. 2008.
[22] Y. Li and W. B. Chu, “More Golay sequences,” IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 1141–1145, Mar. 2005.
[23] Z. Liu, U. Parampalli, and Y. L. Guan, “On even-period binary Z-complementary pairs with large ZCZs,” IEEE Signal Process. Lett., vol. 21, no. 3, pp. 284–287, Mar. 2014.
[24] ——, “Optimal odd-length binary Z-complementary pairs,” IEEE Trans. Inf. Theory, vol. 60, no. 9, pp. 5768–5781, Sep. 2014.
[25] F. Fiedler, J. Jedwab, and M. G. Parker, “A framework for the construction of Golay sequences,” IEEE Trans. Inf. Theory, vol. 54, no. 7, pp. 3114–3129, Jul. 2008.
[26] P. Fan and M. Darnell, Sequence Design for Communications Applications. NY:Wiley, 1996.
[27] J. Jedwab and M. G. Parker, “Golay complementary array pairs,” Designs, Codes and Cryptography, vol. 44, pp. 209–216, Sep. 2007.
[28] X. Li, P. Fan, X. Tang, and Y. Tu, “Existence of binary Z-complementary pairs,” IEEE Signal Process. Lett., vol. 18, no. 1, pp. 63–66, Jan. 2011.
[29] X. Li,W. H. Mow, and X. Niu, “New construction of Z-complementary pairs,” Electron. Lett., vol. 52, no. 8, pp. 609–611, Apr. 2016.
[30] A. R. Adhikary, S. Majhi, Z. Liu, and Y. L. Guan, “New optimal binary Z-complementary pairs of odd lengths,” in Proc. Int. Eighth Int. Workshop on Signal Design and Its Appl. in Commun. (IWSDA), Sapporo, Japan, 2017, pp. 14–18.
[31] C.-Y. Chen, “A novel construction of Z-complementary pairs based on generalized Boolean functions,” IEEE Signal Process. Lett., vol. 24, no. 7, pp. 987–990, Jul 2017.
[32] A. Rathinakumar and A. K. Chaturvedi, “Complete mutually orthogonal Golay complementary sets from Reed-Muller codes,” IEEE Trans. Inf. Theory, vol. 54, pp. 1339–1346, Mar. 2008.
[33] C.-Y. Chen, C.-H. Wang, and C.-C. Chao, “Complete complementary codes and generalized Reed-Muller codes,” IEEE Commun. Lett., vol. 12, pp. 849–851, Nov. 2008.
[34] X. Li, P. Fan, X. Tang, and L. Hao, “Constructions of quadriphase Z-complementary sequences,” in Proc. Int.Workshop Signal Design and its Applicat. in Commun., Fukuoka, Japan, Oct. 2009, pp. 36–39.
[35] Y. Tu, P. Fan, L. Hao, and X. Tang, “A simple method for generating optimal Z-periodic complementary sequence set based on phase shift,” IEEE Signal Process. Lett., vol. 17, no. 10, pp. 891–893, Oct. 2010.
[36] F. Zeng, X. Zeng, Z. Zhang, X. Zeng, G. Xuan, and L. Xiao, “New construction method for quaternary aperiodic, periodic, and Z-complementary sequence sets,” Journal of Wireless Commun. and Netw., vol. 14, no. 3, pp. 230–236, Jun. 2012.
[37] Y. Li, C. Xu, N. Jing, and K. Liu, “Constructions of Z-periodic complementary sequence set with flexible flock size,” IEEE Commun. Lett., vol. 18, no. 2, pp. 201–204, Feb. 2014.
[38] P. Ke and Z. Zhou, “A generic construction of Z-periodic complementary sequence sets with flexible flock size and zero correlation zone length,” IEEE Signal Process. Lett., vol. 22, no. 9, pp. 1462–1466, Sep. 2015.
[39] L. Feng, X. Zhou, and X. Li, “A new construction of Z-complementary codes,” in Proc. Int. Conf. on Parallel and Distributed Computing, Applicat. and Technol., Beijing, China, Dec. 2012, pp. 574–577.
[40] X. Long, F. Zeng, and W. Zhang, “New construction of Z-complementary sequences,” in Proc. Int. Conf. on Inform. Science and Eng., Nanjing, China, Dec. 2009, pp. 608–611.
[41] Z. Zhang, W. Chen, F. Zeng, H. Wu, and Y. Zhong, “Z-complementary sets based on sequences with periodic and aperiodic zero correlation zone,” EURASIP J. Wireless Commun. and Netw., vol. 2009, no. 1, pp. 1–9, Mar. 2009.
[42] G. Xuan, Y. Zhao, S. Lu, H. Pu, F. Zeng, and Z. Zhang, “Construction of mutually orthogonal Z-periodic complementary sequences,” in Proc. Int. Conf. on Wireless, Mobile and Multimedia Netw., Beijing, China, Nov. 2013, pp. 110–114.
[43] Y. Li and C. Xu, “ZCZ aperiodic complementary sequence sets with low column sequence PMEPR,” IEEE Commun. Lett., vol. 19, no. 8, pp. 1303–1306, Aug. 2015.
[44] L. Feng, P. Fan, and X. Zhou, “Lower bounds on correlation of Z-complementary code sets,” Wireless Pers. Commun., vol. 72, no. 2, pp. 1475–1488, Sep 2013.
[45] C. Han, N. Suehiro, and T. Hashimoto, “N-shift cross-orthogonal sequences and complete complementary codes,” in Proc. IEEE Int. Symp. Inf. Theory, Nice, France, Jun. 2007, pp. 2611–2615.
[46] ——, “A systematic framework for the construction of optimal complete complementary codes,” IEEE Trans. Inf. Theory, vol. 57, no. 9, pp. 6033–6042, 2011.
[47] P. Z. Fan, N. Suehiro, N. Kuroyanagi, and X. M. Deng, “Class of binary sequences with zero correlation zone,” Electron. Lett., vol. 35, no. 10, pp. 777–779, May 1999.
[48] X. H. Tang, P. Z. Fan, and S. Matsufuji, “Lower bounds on correlation of spreading sequence set with low or zero correlation zone,” Electron. Lett., vol. 36, no. 13, pp. 551–552, Mar. 2000.
[49] H. Torii, M. Nakamura, and N. Suehiro, “A new class of zero-correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 50, pp. 559–565, Mar. 2004.
[50] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.
[51] C.-Y. Chen, “Complementary sets of non-power-of-two length for peak-to-average power ratio reduction in OFDM,” IEEE Trans. Inf. Theory, vol. 62, no. 12, pp. 7538–7545, Dec. 2016.
[52] Y.-S. Tang, C.-Y. Chen, and C.-C. Chao, “A novel construction of zero correlation zone sequences based on boolean functions,” in Proc. IEEE Int. Symp. on Spread Spectrum Tech. and Applicat., Taichung, Taiwan, Oct. 2010, pp. 198–203.
[53] P. B. Borwein and R. A. Ferguson, “A complete description of Golay pairs for lengths up to 100,” Math. Comput., vol. 73, no. 246, pp. 967–985, Jul. 2003.
[54] Z. Zhou, X. Tang, and G. Gong, “A new class of sequences with zero or low correlation zone based on interleaving technique,” IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4267–4273, Sep. 2008.
[55] C. Y. Chen, “A new construction of Golay complementary sets of non-power-of-two length based on boolean functions,” in Proc. IEEE Wireless Commun. and Netw. Conf. (WCNC), San Francisco, California, Mar. 2017, pp. 1–6.