# American Institute of Mathematical Sciences

February  2015, 9(1): 9-21. doi: 10.3934/amc.2015.9.9

## Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties

 1 College of Communication Engineering, Chongqing University, Chongqing 400044, China, China 2 College of Communication Engineering, Chongqing University, Chongqing 400044, China, and Chongqing Key Laboratory of Emergency Communication, Chongqing Communication Institute, Chongqing 400035 3 Chongqing Key Laboratory of Emergency Communication, Chongqing Communication Institute, Chongqing 400035

Received  August 2013 Published  February 2015

In this paper, a novel method for constructing complementary sequence set with zero correlation zone (ZCZ) is presented by interleaving and combining three orthogonal matrices. The constructed set can be divided into multiple sequence groups and each sequence group can be further divided into multiple sequence subgroups. In addition to ZCZ properties of sequences from the same sequence subgroup, sequences from different sequence groups are orthogonal to each other while sequences from different sequence subgroups within the same sequence group possess ideal cross-correlation properties, that is, the proposed ZCZ sequence set has inter-group orthogonal (IGO) and inter-subgroup complementary (ISC) properties. Compared with previous methods, the new construction can provide flexible choice for ZCZ width and set size, and the resultant sequences which are called IGO-ISC sequences in this paper can achieve the theoretical bound on the set size for the ZCZ width and sequence length.
Citation: Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9
##### References:
 [1] H. H. Chen, Y. C. Yeh, et al., Generalized pairwise complementary codes with set-wise uniform interference-free windows, IEEE J. Sel. Areas Commun., 24 (2006), 65-74. [2] P. Z. Fan, N. Suehiro, N. Kuroyanagi and X. M. Deng, A class of binary sequences with zero correlation zone, Electr. Lett., 35 (1999), 777-779. [3] P. Z. Fan, W. N. Yuan and Y. F. Tu, Z-complementary binary sequences, IEEE Signal Process. Lett., 14 (2007), 509-512. [4] L. F. Feng, P. Z. Fan, X. H. Tang and K.-K. Loo, Generalized pairwise Z-complementary codes, IEEE Signal Process. Lett., 15 (2008), 377-380. [5] L. F. Feng, X. W. Zhou and P. Z. Fan, A construction of inter-group complementary codes with flexible ZCZ length, J. Zhejiang Univ. Sci. C, 12 (2011), 846-854. [6] L. F. Feng, X. W. Zhou and X. Y. Li, A general construction of inter-group complementary codes based on Z-complementary codes and perfect periodic cross-correlation codes, Wireless Pers. Commun., 71 (2012), 695-706. [7] M. J. E. Golay, Complementary series, IRE. Trans. Inf. Theory, 7 (1961), 82-87. [8] T. Hayashi, Ternary sequence set having periodic and aperiodic zero-correlation zone, IEICE Trans. Fundamentals, E89-A (2006), 1825-1831. [9] T. Hayashi, T. Maeda and S. Matsufuji, A generalized construction scheme of a zero-correlation zone sequence set with a wide inter-subset zero-correlation zone, IEICE Trans. Fundamentals, E95-A (2012), 1931-1936. [10] T. Hayashi, T. Maeda, S. Matsufuji and S. Okawa, A ternary zero-correlation zone sequence set having wide inter-subset zero-correlation zone, IEICE Trans. Fundamentals, E94-A (2011), 2230-2235. [11] T. Hayashi, T. Maeda and S. Okawa, A generalized construction of zero-correlation zone sequence set with sequence subsets, IEICE Trans. Fundamentals, E94-A (2011), 1597-1602. [12] T. Hayashi and S. Matsufuji, A generalized construction of optimal zero-correlation zone sequence set from a perfect sequence pair, IEICE Trans. Fundamentals, E93-A (2010), 2337-2344. [13] H. G. Hu and G. Gong, New sets of zero or low correlation zone sequences via interleaving techniques, IEEE Trans. Inf. Theory, 56 (2010), 1702-1713. doi: 10.1109/TIT.2010.2040887. [14] J. W. Jang, Y. S. Kim and S. H. Kim, New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set, Adv. Math. Commun., 3 (2009), 115-124. doi: 10.3934/amc.2009.3.115. [15] J. W. Jang, Y. S. Kim, S. H. Kim and D. W. Lim, New construction methods of quaternary periodic complementary sequence sets, Adv. Math. Commun., 4 (2010), 61-68. doi: 10.3934/amc.2010.4.61. [16] J. Li, A. P. Huang, M. Guizani and H. H. Chen, Inter-group complementary codes for interference-resistant CDMA wireless communications, IEEE Trans. Wireless Commun., 7 (2008), 166-174. [17] X. D. Li, P. Z. Fan, X. H. Tang and L. Hao, Quadriphase Z-complementary sequences, IEICE Trans. Fundamentals, E93-A (2010), 2251-2257. [18] Y. B. Li, C. Q. Xu and K. Liu, Construction of mutually orthogonal zero correlation zone polyphase sequence sets, IEICE Trans. Fundamentals, E94-A (2011), 1159-1164. [19] S. Matsufuji, T. Matsumoto, T. Hayashida, T. Hayashi, N. Kuroyanagi and P. Z. FAN, On a ZCZ code including a sequence used for a synchronization symbol, IEICE Trans. Fundamentals, E93-A (2010), 2286-2290. [20] K. Omata, H. Torii and T. Matsumoto, Zero-cross-correlation properties of asymmetric ZCZ sequence sets, IEICE Trans. Fundamentals, E95-A (2012), 1926-1930. [21] A. Rathinakumar and A. K. Chaturvedi, Mutually orthogonal sets of ZCZ sequences, Electron. Lett., 40 (2004), 1133-1134. [22] A. Rathinakumar and A. K. Chaturvedi, A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences, IEEE Trans. Inf. Theory, 52 (2006), 3817-3826. doi: 10.1109/TIT.2006.878171. [23] A. Rathinakumar and A. K. Chaturvedi, Complete mutually orthogonal Golay complementary sets from Reed-Muller codes, IEEE Trans. Inf. Theory, 54 (2008), 1339-1346. doi: 10.1109/TIT.2007.915980. [24] X. H. Tang, P. Z. Fan and J. Lindner, Multiple binary ZCZ sequence sets with good cross-correlation property based on complementary sequence sets, IEEE Trans. Inf. Theory, 56 (2010), 4038-4045. doi: 10.1109/TIT.2010.2050796. [25] X. H. Tang and W. H. Mow, Design of spreading codes for quasi-synchronous CDMA with intercell interference, IEEE J. Sel. Areas Commun., 24 (2006), 84-93. [26] X. H. Tang and W. H. Mow, A new systematic construction of zero correlation zone sequences based on interleaved perfect sequences, IEEE Trans. Inf. Theory, 54 (2008), 5729-5734. doi: 10.1109/TIT.2008.2006574. [27] H. Torii, T. Matsumoto and M. Nakamura, A new method for constructing asymmetric ZCZ sequences sets, IEICE Trans. Fundamentals, E95-A (2012), 1577-1586. [28] H. Torii, M. Nakamura and N. Suehiro, A new class of zero-correlation zone sequences, IEEE Trans. Inf. Theory, 50 (2004), 559-565. doi: 10.1109/TIT.2004.825399. [29] H. Torii, M. Satoh, T. Matsumoto and M. Nakamura, Generalized mutually orthogonal ZCZ sequence sets based on perfect sequences and orthogonal codes, in Proc. 15th Int. Conf. Adv. Commun. Techn., 2013, 894-899. [30] H. Torii, M. Satoh, T. Matsumoto and M. Nakamura, Quasi-optimal and optimal generalized mutually orthogonal ZCZ sequence sets based on an interleaving technique, Int. J. Commun., 7 (2013), 18-25. [31] Y. F. Tu, P. Z. Fan, L. Hao and X. Y. Li, Construction of binary array set with zero correlation zone based on interleaving technique, IEICE Trans. Fundamentals, E94-A (2011), 766-772. doi: 10.1587/transfun.E94.A.766. [32] Y. F. Tu, P. Z. Fan, L. Hao and X. H. Tang, A simple method for generating optimal Z-periodic complementary sequence set based on phase shift, IEEE Signal Process. Lett., 17 (2010), 891-893. [33] F. X. Zeng, New perfect ployphase sequences and mutually orthogonal ZCZ polyphase sequence sets, IEICE trans. Fundamentals, E92-A (2009), 1731-1736. [34] F. X. Zeng, X. P. Zeng, Z. Y. Zhang and G. X. Xuan, Quaternary periodic complementary/Z-complementary sequence sets based upon interleaving technique and Gray mapping, Adv. Math. Commun., 6 (2012), 237-247. doi: 10.3934/amc.2012.6.237. [35] C. Zhang, X. M. Tao, S. Yamada and M. Hatori, Sequence set with three zero correlation zone and its application in MC-CDMA system, IEICE Trans. Fundamentals, E89-A (2006), 2275-2282. doi: 10.1093/ietfec/e89-a.9.2275. [36] Z. Y. Zhang, W. Chen, F. X. Zeng, H. Wu and Y. H. Zhong, Z-complementary sets based on sequences with periodic and aperiodic zero correlation zone, EURASIP J. Wireless Comm. Networking, 2009 (2009), 1-8. doi: 10.1155/2009/418026. [37] Z. Y. Zhang, F. X. Zeng and G. X. Xuan, A class of complementary sequences with multi-width zero cross-correlation zone, IEICE Trans. Fundamentals, E93-A (2010), 1508-1517. doi: 10.1587/transfun.E93.A.1508. [38] Z. C. Zhou, X. H. Tang and G. Gong, A new class of sequences with zero or low correlation zone based on interleaving technique, IEEE Trans. Inf. Theory, 54 (2008), 4267-4273. doi: 10.1109/TIT.2008.928256.

show all references

##### References:
 [1] H. H. Chen, Y. C. Yeh, et al., Generalized pairwise complementary codes with set-wise uniform interference-free windows, IEEE J. Sel. Areas Commun., 24 (2006), 65-74. [2] P. Z. Fan, N. Suehiro, N. Kuroyanagi and X. M. Deng, A class of binary sequences with zero correlation zone, Electr. Lett., 35 (1999), 777-779. [3] P. Z. Fan, W. N. Yuan and Y. F. Tu, Z-complementary binary sequences, IEEE Signal Process. Lett., 14 (2007), 509-512. [4] L. F. Feng, P. Z. Fan, X. H. Tang and K.-K. Loo, Generalized pairwise Z-complementary codes, IEEE Signal Process. Lett., 15 (2008), 377-380. [5] L. F. Feng, X. W. Zhou and P. Z. Fan, A construction of inter-group complementary codes with flexible ZCZ length, J. Zhejiang Univ. Sci. C, 12 (2011), 846-854. [6] L. F. Feng, X. W. Zhou and X. Y. Li, A general construction of inter-group complementary codes based on Z-complementary codes and perfect periodic cross-correlation codes, Wireless Pers. Commun., 71 (2012), 695-706. [7] M. J. E. Golay, Complementary series, IRE. Trans. Inf. Theory, 7 (1961), 82-87. [8] T. Hayashi, Ternary sequence set having periodic and aperiodic zero-correlation zone, IEICE Trans. Fundamentals, E89-A (2006), 1825-1831. [9] T. Hayashi, T. Maeda and S. Matsufuji, A generalized construction scheme of a zero-correlation zone sequence set with a wide inter-subset zero-correlation zone, IEICE Trans. Fundamentals, E95-A (2012), 1931-1936. [10] T. Hayashi, T. Maeda, S. Matsufuji and S. Okawa, A ternary zero-correlation zone sequence set having wide inter-subset zero-correlation zone, IEICE Trans. Fundamentals, E94-A (2011), 2230-2235. [11] T. Hayashi, T. Maeda and S. Okawa, A generalized construction of zero-correlation zone sequence set with sequence subsets, IEICE Trans. Fundamentals, E94-A (2011), 1597-1602. [12] T. Hayashi and S. Matsufuji, A generalized construction of optimal zero-correlation zone sequence set from a perfect sequence pair, IEICE Trans. Fundamentals, E93-A (2010), 2337-2344. [13] H. G. Hu and G. Gong, New sets of zero or low correlation zone sequences via interleaving techniques, IEEE Trans. Inf. Theory, 56 (2010), 1702-1713. doi: 10.1109/TIT.2010.2040887. [14] J. W. Jang, Y. S. Kim and S. H. Kim, New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set, Adv. Math. Commun., 3 (2009), 115-124. doi: 10.3934/amc.2009.3.115. [15] J. W. Jang, Y. S. Kim, S. H. Kim and D. W. Lim, New construction methods of quaternary periodic complementary sequence sets, Adv. Math. Commun., 4 (2010), 61-68. doi: 10.3934/amc.2010.4.61. [16] J. Li, A. P. Huang, M. Guizani and H. H. Chen, Inter-group complementary codes for interference-resistant CDMA wireless communications, IEEE Trans. Wireless Commun., 7 (2008), 166-174. [17] X. D. Li, P. Z. Fan, X. H. Tang and L. Hao, Quadriphase Z-complementary sequences, IEICE Trans. Fundamentals, E93-A (2010), 2251-2257. [18] Y. B. Li, C. Q. Xu and K. Liu, Construction of mutually orthogonal zero correlation zone polyphase sequence sets, IEICE Trans. Fundamentals, E94-A (2011), 1159-1164. [19] S. Matsufuji, T. Matsumoto, T. Hayashida, T. Hayashi, N. Kuroyanagi and P. Z. FAN, On a ZCZ code including a sequence used for a synchronization symbol, IEICE Trans. Fundamentals, E93-A (2010), 2286-2290. [20] K. Omata, H. Torii and T. Matsumoto, Zero-cross-correlation properties of asymmetric ZCZ sequence sets, IEICE Trans. Fundamentals, E95-A (2012), 1926-1930. [21] A. Rathinakumar and A. K. Chaturvedi, Mutually orthogonal sets of ZCZ sequences, Electron. Lett., 40 (2004), 1133-1134. [22] A. Rathinakumar and A. K. Chaturvedi, A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences, IEEE Trans. Inf. Theory, 52 (2006), 3817-3826. doi: 10.1109/TIT.2006.878171. [23] A. Rathinakumar and A. K. Chaturvedi, Complete mutually orthogonal Golay complementary sets from Reed-Muller codes, IEEE Trans. Inf. Theory, 54 (2008), 1339-1346. doi: 10.1109/TIT.2007.915980. [24] X. H. Tang, P. Z. Fan and J. Lindner, Multiple binary ZCZ sequence sets with good cross-correlation property based on complementary sequence sets, IEEE Trans. Inf. Theory, 56 (2010), 4038-4045. doi: 10.1109/TIT.2010.2050796. [25] X. H. Tang and W. H. Mow, Design of spreading codes for quasi-synchronous CDMA with intercell interference, IEEE J. Sel. Areas Commun., 24 (2006), 84-93. [26] X. H. Tang and W. H. Mow, A new systematic construction of zero correlation zone sequences based on interleaved perfect sequences, IEEE Trans. Inf. Theory, 54 (2008), 5729-5734. doi: 10.1109/TIT.2008.2006574. [27] H. Torii, T. Matsumoto and M. Nakamura, A new method for constructing asymmetric ZCZ sequences sets, IEICE Trans. Fundamentals, E95-A (2012), 1577-1586. [28] H. Torii, M. Nakamura and N. Suehiro, A new class of zero-correlation zone sequences, IEEE Trans. Inf. Theory, 50 (2004), 559-565. doi: 10.1109/TIT.2004.825399. [29] H. Torii, M. Satoh, T. Matsumoto and M. Nakamura, Generalized mutually orthogonal ZCZ sequence sets based on perfect sequences and orthogonal codes, in Proc. 15th Int. Conf. Adv. Commun. Techn., 2013, 894-899. [30] H. Torii, M. Satoh, T. Matsumoto and M. Nakamura, Quasi-optimal and optimal generalized mutually orthogonal ZCZ sequence sets based on an interleaving technique, Int. J. Commun., 7 (2013), 18-25. [31] Y. F. Tu, P. Z. Fan, L. Hao and X. Y. Li, Construction of binary array set with zero correlation zone based on interleaving technique, IEICE Trans. Fundamentals, E94-A (2011), 766-772. doi: 10.1587/transfun.E94.A.766. [32] Y. F. Tu, P. Z. Fan, L. Hao and X. H. Tang, A simple method for generating optimal Z-periodic complementary sequence set based on phase shift, IEEE Signal Process. Lett., 17 (2010), 891-893. [33] F. X. Zeng, New perfect ployphase sequences and mutually orthogonal ZCZ polyphase sequence sets, IEICE trans. Fundamentals, E92-A (2009), 1731-1736. [34] F. X. Zeng, X. P. Zeng, Z. Y. Zhang and G. X. Xuan, Quaternary periodic complementary/Z-complementary sequence sets based upon interleaving technique and Gray mapping, Adv. Math. Commun., 6 (2012), 237-247. doi: 10.3934/amc.2012.6.237. [35] C. Zhang, X. M. Tao, S. Yamada and M. Hatori, Sequence set with three zero correlation zone and its application in MC-CDMA system, IEICE Trans. Fundamentals, E89-A (2006), 2275-2282. doi: 10.1093/ietfec/e89-a.9.2275. [36] Z. Y. Zhang, W. Chen, F. X. Zeng, H. Wu and Y. H. Zhong, Z-complementary sets based on sequences with periodic and aperiodic zero correlation zone, EURASIP J. Wireless Comm. Networking, 2009 (2009), 1-8. doi: 10.1155/2009/418026. [37] Z. Y. Zhang, F. X. Zeng and G. X. Xuan, A class of complementary sequences with multi-width zero cross-correlation zone, IEICE Trans. Fundamentals, E93-A (2010), 1508-1517. doi: 10.1587/transfun.E93.A.1508. [38] Z. C. Zhou, X. H. Tang and G. Gong, A new class of sequences with zero or low correlation zone based on interleaving technique, IEEE Trans. Inf. Theory, 54 (2008), 4267-4273. doi: 10.1109/TIT.2008.928256.
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