# American Institute of Mathematical Sciences

August 2018, 12(3): 541-552. doi: 10.3934/amc.2018032

## An asymmetric ZCZ sequence set with inter-subset uncorrelated property and flexible ZCZ length

 1 The National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China 2 School of Electronic and Information Engineering, Henan University of Science and Technology, Luoyang, Henan 471022, China 3 School of Information Science and Technology, Tibet University, Lhasa, Tibet 850000, China

* Xiaoli Zeng is the corresponding author

Longye Wang is also affiliated with School of Engineering and Technology, Tibet University, Lhasa, Tibet 850000, China

Received  July 2017 Revised  January 2018 Published  July 2018

In this paper, we propose a novel method for constructing new uncorrelated asymmetric zero correlation zone (UA-ZCZ) sequence sets by interleaving perfect sequences. As a type of ZCZ sequence set, an A-ZCZ sequence set consists of multiple sequence subsets. Different subsets are correlated in conventional A-ZCZ sequence set but uncorrelated in our scheme. In other words, the cross-correlation function (CCF) between two arbitrary sequences which belong to different subsets has quite a large zero cross-correlation zone (ZCCZ). Analytical results demonstrate that the UA-ZCZ sequence set proposed herein is optimal with respect to the upper bound of ZCZ sequence set. Specifically, our scheme enables the flexible selection of ZCZ length, which makes it extremely valuable for designing spreading sequences for quasi-synchronous code-division multiple-access (QS-CDMA) systems.

Citation: Longye Wang, Gaoyuan Zhang, Hong Wen, Xiaoli Zeng. An asymmetric ZCZ sequence set with inter-subset uncorrelated property and flexible ZCZ length. Advances in Mathematics of Communications, 2018, 12 (3) : 541-552. doi: 10.3934/amc.2018032
##### References:
 [1] P. Z. Fan and L. Hao, Generalized orthogonal sequences and their applications in synchronous CDMA systems, IEICE Trans. Fund., E83-A (2000), 2054-2069. [2] G. Gong, New designs for signal sets with low cross correlation balance property and large linear span: GF(p) case, IEEE Trans. Inf. Theory, 48 (2002), 2847-2867. doi: 10.1109/TIT.2002.804044. [3] T. Hayashi, Zero-correlation zone sequence set constructed from a perfect sequence, IEICE Trans. Fund., E90-A (2007), 1-5. doi: 10.1109/CIT.2007.192. [4] 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. Fund., E94-A (2011), 2230-2235. [5] T. Hayashi, T. Maeda and S. Okawa, A generalized construction of zero-correlation zone sequence set with sequence subsets, IEICE Trans. Fund., E94-A (2011), 1597-1602. [6] P. H. Ke and Z. C. Zhou, A generic construction of Z-periodic complementary sequence sets with flexible flock size and zero correlation zone length, IEEE Signal Process. Lett., 22 (2015), 1462-1466. doi: 10.1109/LSP.2014.2369512. [7] S. Matsufuji, N. Kuroyanagi, N. Suehiro and P. Z. Fan, Two types polyphase sequence sets for approximately synchronized CDMA systems, IEICE Trans. Fund., E86-A (2003), 229-234. [8] X. H. Tang, P. Z. Fan and S. Matsufuji, Lower bounds on correlation of spreading sequence set with low or zero correlation zone, Electronics Letters, 36 (2000), 551-552. doi: 10.1049/el:20000462. [9] X. H. Tang and H. M. Wai, 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. [10] X. H. Tang, P. 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. [11] H. Torii, T. Matsumoto and M. Nakamura, A new method for constructing asymmetric {ZCZ} sequence sets, IEICE Trans. Fund., E95-A (2012), 1577-1586. [12] H. Torii, T. Matsumoto and M. Nakamura, Extension of methods for constructing polyphase asymmetric ZCZ sequence sets, IEICE Trans. Fund., E96-A (2013), 2244-2252. [13] H. Torii, M. Nakamurai and Makoto, Optimal polyphase asymmetric ZCZ sets including uncorrelated sequences, Journal of Signal Processing, 16 (2012), 487-494. [14] L. Y. Wang, X. L. Zeng and H. Wen, A novel construction of asymmetric ZCZ sequence sets from interleaving perfect sequence, IEICE Trans. Fund., E97-A (2014), 2556-2561. [15] L. Y. Wang, X. L. Zeng and H. Wen, Families of asymmetric sequence pair set with zero-correlation zone via interleaved technique, IET Commun., 10 (2016), 229-234. doi: 10.1049/iet-com.2015.0075. [16] L. Y. Wang, X. L. Zeng and H. Wen, Asymmetric ZCZ sequence sets with inter-subset uncorrelated sequences via interleaved technique, IEICE Trans. Fund., E100-A (2017), 751-756. [17] Z. C. Zhou, Z. Pan and X. H. Tang, New families of optimal zero correlation zone sequences based on interleaved technique and perfect, IEICE Trans. Fund., E91-A (2008), 3691-3697. [18] 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] P. Z. Fan and L. Hao, Generalized orthogonal sequences and their applications in synchronous CDMA systems, IEICE Trans. Fund., E83-A (2000), 2054-2069. [2] G. Gong, New designs for signal sets with low cross correlation balance property and large linear span: GF(p) case, IEEE Trans. Inf. Theory, 48 (2002), 2847-2867. doi: 10.1109/TIT.2002.804044. [3] T. Hayashi, Zero-correlation zone sequence set constructed from a perfect sequence, IEICE Trans. Fund., E90-A (2007), 1-5. doi: 10.1109/CIT.2007.192. [4] 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. Fund., E94-A (2011), 2230-2235. [5] T. Hayashi, T. Maeda and S. Okawa, A generalized construction of zero-correlation zone sequence set with sequence subsets, IEICE Trans. Fund., E94-A (2011), 1597-1602. [6] P. H. Ke and Z. C. Zhou, A generic construction of Z-periodic complementary sequence sets with flexible flock size and zero correlation zone length, IEEE Signal Process. Lett., 22 (2015), 1462-1466. doi: 10.1109/LSP.2014.2369512. [7] S. Matsufuji, N. Kuroyanagi, N. Suehiro and P. Z. Fan, Two types polyphase sequence sets for approximately synchronized CDMA systems, IEICE Trans. Fund., E86-A (2003), 229-234. [8] X. H. Tang, P. Z. Fan and S. Matsufuji, Lower bounds on correlation of spreading sequence set with low or zero correlation zone, Electronics Letters, 36 (2000), 551-552. doi: 10.1049/el:20000462. [9] X. H. Tang and H. M. Wai, 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. [10] X. H. Tang, P. 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. [11] H. Torii, T. Matsumoto and M. Nakamura, A new method for constructing asymmetric {ZCZ} sequence sets, IEICE Trans. Fund., E95-A (2012), 1577-1586. [12] H. Torii, T. Matsumoto and M. Nakamura, Extension of methods for constructing polyphase asymmetric ZCZ sequence sets, IEICE Trans. Fund., E96-A (2013), 2244-2252. [13] H. Torii, M. Nakamurai and Makoto, Optimal polyphase asymmetric ZCZ sets including uncorrelated sequences, Journal of Signal Processing, 16 (2012), 487-494. [14] L. Y. Wang, X. L. Zeng and H. Wen, A novel construction of asymmetric ZCZ sequence sets from interleaving perfect sequence, IEICE Trans. Fund., E97-A (2014), 2556-2561. [15] L. Y. Wang, X. L. Zeng and H. Wen, Families of asymmetric sequence pair set with zero-correlation zone via interleaved technique, IET Commun., 10 (2016), 229-234. doi: 10.1049/iet-com.2015.0075. [16] L. Y. Wang, X. L. Zeng and H. Wen, Asymmetric ZCZ sequence sets with inter-subset uncorrelated sequences via interleaved technique, IEICE Trans. Fund., E100-A (2017), 751-756. [17] Z. C. Zhou, Z. Pan and X. H. Tang, New families of optimal zero correlation zone sequences based on interleaved technique and perfect, IEICE Trans. Fund., E91-A (2008), 3691-3697. [18] 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.
Periodic correlation properties of U-ZCZ sequence set ${B}$.
Comparison of Different Families of A-ZCZ Sequence Sets
The A-ZCZ Sequence Set $\mathcal{C} = \{{C}^{(0)}, \, {C}^{(1)}, \, {C}^{(2)}, \, {C}^{(3)}\}$
 ${C}^{(0)}$ $\mathit{\boldsymbol{c}}^{(0)}_0$ 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1 $\mathit{\boldsymbol{c}}^{(0)}_1$ 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1 $\mathit{\boldsymbol{c}}^{(0)}_2$ 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0 $\mathit{\boldsymbol{c}}^{(0)}_3$ 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0 ${C}^{(1)}$ $\mathit{\boldsymbol{c}}^{(1)}_0$ 1, 0, 0, -i, 0, 0, i, 0, 1, 1, i, {-i}, 0, 1, -i, 0, 0, -1, 0, 0, 1, i, -i, -i, 1, -1, i, -i, -1, i, i, i, i, i, -i, i, -1, -1, -i, i, i, -1, i, i, i, 1, i, i, -1, i, i, i, -1, 1, i, i $\mathit{\boldsymbol{c}}^{(1)}_1$ 1, 0, 0, i, 0, 0, i, 0, 1, -1, i, i, 0, -1, -i, 0, 0, 1, 0, 0, 1, 0, -i, i, 1, 1, 0, i, -1, 0, 0, -i, 0, 0, -i, 0, -1, 1, -i, -i, 0, 1, i, 0, 0, -1, 0, 0, -1, 0, i, -i, -1, -1, 0, -i $\mathit{\boldsymbol{c}}^{(1)}_2$ 1, 0, i, i, 0, 1, -i, i, 0, 0, 0, -i, 1, 0, -i, 0, 1, 1, 0, -i, -1, 1, 0, 0, 0, -1, -i, 0, -1, 0, -i, -i, 0, -1, i, -i, 0, 0, 0, i, -1, 0, i, 0, -1, -1, 0, i, 1, -1, 0, 0, 0, 1, i, 0 $\mathit{\boldsymbol{c}}^{(1)}_3$ 1, 0, i, -i, 0, -1, -i, -i, 0, 0, 0, i, 1, 0, -i, 0, 1, -1, 0, i, -1, -1, 0, 0, 0, 1, -i, 0, -1, 0, -i, i, 0, 1, i, i, 0, 0, 0, -i, -1, 0, i, 0, -1, 1, 0, -i, 1, 1, 0, 0, 0, -1, i, 0 ${C}^{(2)}$ $\mathit{\boldsymbol{c}}^{(2)}_0$ 1, 0, 0, i, 0, 0, -i, 0, 1, 1, -i, i, 0, 1, i, 0, 0, -1, 0, 0, 1, 0, i, i, 1, -1, 0, i, -1, 0, 0, -i, 0, 0, i, 0, -1, -1, i, -i, 0, -1, -i, 0, 0, 1, 0, 0, -1, 0, -i, -i, -1, 1, 0, -i $\mathit{\boldsymbol{c}}^{(2)}_1$ 1, 0, 0, -i, 0, 0, -i, 0, 1, -1, -i, -i, 0, -1, i, 0, 0, 1, 0, 0, 1, 0, i, -i, 1, 1, 0, -i, -1, 0, 0, i, 0, 0, i, 0, -1, 1, i, i, 0, 1, -i, 0, 0, -1, 0, 0, -1, 0, -i, i, -1, -1, 0, i $\mathit{\boldsymbol{c}}^{(2)}_2$ 1, 0, -i, -i, 0, 1, i, -i, 0, 0, 0, i, 1, 0, i, 0, 1, 1, 0, i, -1, 1, 0, 0, 0, -1, i, 0, -1, 0, i, i, 0, -1, -i, i, 0, 0, 0, -i, -1, 0, -i, 0, -1, -1, 0, -i, 1, -1, 0, 0, 0, 1, -i, 0 $\mathit{\boldsymbol{c}}^{(2)}_3$ 1, 0, -i, i, 0, -1, i, i, 0, 0, 0, -i, 1, 0, i, 0, 1, -1, 0, -i, -1, -1, 0, 0, 0, 1, i, 0, -1, 0, i, -i, 0, 1, -i, -i, 0, 0, 0, i, -1, 0, -i, 0, -1, 1, 0, i, 1, 1, 0, 0, 0, -1, -i, 0 ${C}^{(3)}$ $\mathit{\boldsymbol{c}}^{(3)}_0$ 1, 0, 0, -1, 0, 0, 1, 0, 1, -1, -1, -1, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1, 1, 0, 0, -1, 0, 0, 1, 0, 1, -1, -1, -1, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1 $\mathit{\boldsymbol{c}}^{(3)}_1$ 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, -1, 1, 0, -1, -1, 0, 0, -1, 0, 0, -1, 0, -1, -1, 1, -1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, -1, 1, 0, -1, -1, 0, 0, -1, 0, 0, -1, 0, -1, -1, 1, -1, 0, 1 $\mathit{\boldsymbol{c}}^{(3)}_2$ 1, 0, -1, 1, 0, 1, -1, -1, 0, 0, 0, -1, -1, 0, -1, 0, 1, -1, 0, -1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, -1, 1, 0, 1, -1, -1, 0, 0, 0, -1, -1, 0, -1, 0, 1, -1, 0, -1, 1, 1, 0, 0, 0, 1, 1, 0 $\mathit{\boldsymbol{c}}^{(3)}_3$ 1, 0, -1, -1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 0, -1, 1, 0, 1, 0, -1, -1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 0, -1, 1, 0
 ${C}^{(0)}$ $\mathit{\boldsymbol{c}}^{(0)}_0$ 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1 $\mathit{\boldsymbol{c}}^{(0)}_1$ 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1 $\mathit{\boldsymbol{c}}^{(0)}_2$ 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0 $\mathit{\boldsymbol{c}}^{(0)}_3$ 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0 ${C}^{(1)}$ $\mathit{\boldsymbol{c}}^{(1)}_0$ 1, 0, 0, -i, 0, 0, i, 0, 1, 1, i, {-i}, 0, 1, -i, 0, 0, -1, 0, 0, 1, i, -i, -i, 1, -1, i, -i, -1, i, i, i, i, i, -i, i, -1, -1, -i, i, i, -1, i, i, i, 1, i, i, -1, i, i, i, -1, 1, i, i $\mathit{\boldsymbol{c}}^{(1)}_1$ 1, 0, 0, i, 0, 0, i, 0, 1, -1, i, i, 0, -1, -i, 0, 0, 1, 0, 0, 1, 0, -i, i, 1, 1, 0, i, -1, 0, 0, -i, 0, 0, -i, 0, -1, 1, -i, -i, 0, 1, i, 0, 0, -1, 0, 0, -1, 0, i, -i, -1, -1, 0, -i $\mathit{\boldsymbol{c}}^{(1)}_2$ 1, 0, i, i, 0, 1, -i, i, 0, 0, 0, -i, 1, 0, -i, 0, 1, 1, 0, -i, -1, 1, 0, 0, 0, -1, -i, 0, -1, 0, -i, -i, 0, -1, i, -i, 0, 0, 0, i, -1, 0, i, 0, -1, -1, 0, i, 1, -1, 0, 0, 0, 1, i, 0 $\mathit{\boldsymbol{c}}^{(1)}_3$ 1, 0, i, -i, 0, -1, -i, -i, 0, 0, 0, i, 1, 0, -i, 0, 1, -1, 0, i, -1, -1, 0, 0, 0, 1, -i, 0, -1, 0, -i, i, 0, 1, i, i, 0, 0, 0, -i, -1, 0, i, 0, -1, 1, 0, -i, 1, 1, 0, 0, 0, -1, i, 0 ${C}^{(2)}$ $\mathit{\boldsymbol{c}}^{(2)}_0$ 1, 0, 0, i, 0, 0, -i, 0, 1, 1, -i, i, 0, 1, i, 0, 0, -1, 0, 0, 1, 0, i, i, 1, -1, 0, i, -1, 0, 0, -i, 0, 0, i, 0, -1, -1, i, -i, 0, -1, -i, 0, 0, 1, 0, 0, -1, 0, -i, -i, -1, 1, 0, -i $\mathit{\boldsymbol{c}}^{(2)}_1$ 1, 0, 0, -i, 0, 0, -i, 0, 1, -1, -i, -i, 0, -1, i, 0, 0, 1, 0, 0, 1, 0, i, -i, 1, 1, 0, -i, -1, 0, 0, i, 0, 0, i, 0, -1, 1, i, i, 0, 1, -i, 0, 0, -1, 0, 0, -1, 0, -i, i, -1, -1, 0, i $\mathit{\boldsymbol{c}}^{(2)}_2$ 1, 0, -i, -i, 0, 1, i, -i, 0, 0, 0, i, 1, 0, i, 0, 1, 1, 0, i, -1, 1, 0, 0, 0, -1, i, 0, -1, 0, i, i, 0, -1, -i, i, 0, 0, 0, -i, -1, 0, -i, 0, -1, -1, 0, -i, 1, -1, 0, 0, 0, 1, -i, 0 $\mathit{\boldsymbol{c}}^{(2)}_3$ 1, 0, -i, i, 0, -1, i, i, 0, 0, 0, -i, 1, 0, i, 0, 1, -1, 0, -i, -1, -1, 0, 0, 0, 1, i, 0, -1, 0, i, -i, 0, 1, -i, -i, 0, 0, 0, i, -1, 0, -i, 0, -1, 1, 0, i, 1, 1, 0, 0, 0, -1, -i, 0 ${C}^{(3)}$ $\mathit{\boldsymbol{c}}^{(3)}_0$ 1, 0, 0, -1, 0, 0, 1, 0, 1, -1, -1, -1, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1, 1, 0, 0, -1, 0, 0, 1, 0, 1, -1, -1, -1, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1 $\mathit{\boldsymbol{c}}^{(3)}_1$ 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, -1, 1, 0, -1, -1, 0, 0, -1, 0, 0, -1, 0, -1, -1, 1, -1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, -1, 1, 0, -1, -1, 0, 0, -1, 0, 0, -1, 0, -1, -1, 1, -1, 0, 1 $\mathit{\boldsymbol{c}}^{(3)}_2$ 1, 0, -1, 1, 0, 1, -1, -1, 0, 0, 0, -1, -1, 0, -1, 0, 1, -1, 0, -1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, -1, 1, 0, 1, -1, -1, 0, 0, 0, -1, -1, 0, -1, 0, 1, -1, 0, -1, 1, 1, 0, 0, 0, 1, 1, 0 $\mathit{\boldsymbol{c}}^{(3)}_3$ 1, 0, -1, -1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 0, -1, 1, 0, 1, 0, -1, -1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 0, -1, 1, 0
Periodic correlation properties of UA-ZCZ sequence set $\mathcal{C}$.
 Constructions Parameters Uncorrelatedor not FlexibleZCZ or not Theorem$\ast$1 in [12] $\mathcal{Z}_A(LP, [L, N], [M-1, 2M-1])$ No No Theorem†2 in [12] $\mathcal{Z}_A(TL, \, [T, N], \, [M, TL])$ Yes No Theorem‡2 in [16] $\mathcal{Z}_A(TLP, \, [L, T], \, [P, TLP])$ or $\mathcal{Z}_A(TLP, \, [L, T], \, [P-1, TLP])$ Yes No Theorem$\sharp$3.4 $\mathcal{Z}_A(2TP, \, [2M, T], \, [Z, 2TP])$ Yes Yes $\ast$ $L$ is the order of orthogonal matrix $O_L$, $P$ is length of perfect sequence, and $L=KM$, $N=\lfloor\frac{T}{M}\rfloor>1$, $K>1$, $M>1$. † $T$ is the order of DFT matrix $H_T$, $L$ is the order of orthogonal matrix $O_L$, and $L=KM$, $N=\lfloor\frac{T}{M}\rfloor>1$, $K>1$, $M>1$. ‡ $T$ is the order of DFT matrix $H_T$, $L$ is the order of orthogonal matrix $O_L$, $P$ is length of perfect sequence, and $\gcd{(T, P)}=1$, $\gcd{(L, P)}=1$ (or $L|P$ or $P|L$). $\sharp$ $T$ is the order of DFT matrix $H_T$, $P$ is length of perfect sequence, and $Z\leq2$, $M=\lfloor\frac{P-2}{Z}\rfloor$ or $M=\lfloor\frac{P-1}{Z}\rfloor$.
 Constructions Parameters Uncorrelatedor not FlexibleZCZ or not Theorem$\ast$1 in [12] $\mathcal{Z}_A(LP, [L, N], [M-1, 2M-1])$ No No Theorem†2 in [12] $\mathcal{Z}_A(TL, \, [T, N], \, [M, TL])$ Yes No Theorem‡2 in [16] $\mathcal{Z}_A(TLP, \, [L, T], \, [P, TLP])$ or $\mathcal{Z}_A(TLP, \, [L, T], \, [P-1, TLP])$ Yes No Theorem$\sharp$3.4 $\mathcal{Z}_A(2TP, \, [2M, T], \, [Z, 2TP])$ Yes Yes $\ast$ $L$ is the order of orthogonal matrix $O_L$, $P$ is length of perfect sequence, and $L=KM$, $N=\lfloor\frac{T}{M}\rfloor>1$, $K>1$, $M>1$. † $T$ is the order of DFT matrix $H_T$, $L$ is the order of orthogonal matrix $O_L$, and $L=KM$, $N=\lfloor\frac{T}{M}\rfloor>1$, $K>1$, $M>1$. ‡ $T$ is the order of DFT matrix $H_T$, $L$ is the order of orthogonal matrix $O_L$, $P$ is length of perfect sequence, and $\gcd{(T, P)}=1$, $\gcd{(L, P)}=1$ (or $L|P$ or $P|L$). $\sharp$ $T$ is the order of DFT matrix $H_T$, $P$ is length of perfect sequence, and $Z\leq2$, $M=\lfloor\frac{P-2}{Z}\rfloor$ or $M=\lfloor\frac{P-1}{Z}\rfloor$.
 [1] 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 [2] Wei-Wen Hu. Integer-valued Alexis sequences with large zero correlation zone. Advances in Mathematics of Communications, 2017, 11 (3) : 445-452. doi: 10.3934/amc.2017037 [3] Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004 [4] Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018 [5] Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475 [6] Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55 [7] Hua Liang, Wenbing Chen, Jinquan Luo, Yuansheng Tang. A new nonbinary sequence family with low correlation and large size. Advances in Mathematics of Communications, 2017, 11 (4) : 671-691. doi: 10.3934/amc.2017049 [8] Karol Mikula, Mariana Remešíková, Peter Novysedlák. Truss structure design using a length-oriented surface remeshing technique. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 933-951. doi: 10.3934/dcdss.2015.8.933 [9] Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375 [10] Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115 [11] Fanxin Zeng, Xiaoping Zeng, Zhenyu Zhang, Guixin Xuan. Quaternary periodic complementary/Z-complementary sequence sets based on interleaving technique and Gray mapping. Advances in Mathematics of Communications, 2012, 6 (2) : 237-247. doi: 10.3934/amc.2012.6.237 [12] Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117 [13] Yuhua Sun, Zilong Wang, Hui Li, Tongjiang Yan. The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$. Advances in Mathematics of Communications, 2013, 7 (4) : 409-424. doi: 10.3934/amc.2013.7.409 [14] Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 [15] Bin Li, Hai Huyen Dam, Antonio Cantoni. A low-complexity zero-forcing Beamformer design for multiuser MIMO systems via a dual gradient method. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 297-304. doi: 10.3934/naco.2016012 [16] Bin Li, Hai Huyen Dam, Antonio Cantoni. A global optimal zero-forcing Beamformer design with signed power-of-two coefficients. Journal of Industrial & Management Optimization, 2016, 12 (2) : 595-607. doi: 10.3934/jimo.2016.12.595 [17] Hua Liang, Jinquan Luo, Yuansheng Tang. On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$. Advances in Mathematics of Communications, 2017, 11 (4) : 693-703. doi: 10.3934/amc.2017050 [18] Lassi Roininen, Markku S. Lehtinen, Sari Lasanen, Mikko Orispää, Markku Markkanen. Correlation priors. Inverse Problems & Imaging, 2011, 5 (1) : 167-184. doi: 10.3934/ipi.2011.5.167 [19] Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015 [20] Ming Su, Arne Winterhof. Hamming correlation of higher order. Advances in Mathematics of Communications, 2018, 12 (3) : 505-513. doi: 10.3934/amc.2018029

2017 Impact Factor: 0.564

## Metrics

• PDF downloads (80)
• HTML views (187)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]