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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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T. HayashiT. MaedaS. 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.   Google Scholar

[5]

T. HayashiT. Maeda and S. Okawa, A generalized construction of zero-correlation zone sequence set with sequence subsets, IEICE Trans. Fund., E94-A (2011), 1597-1602.   Google Scholar

[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.  Google Scholar

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S. MatsufujiN. KuroyanagiN. Suehiro and P. Z. Fan, Two types polyphase sequence sets for approximately synchronized CDMA systems, IEICE Trans. Fund., E86-A (2003), 229-234.   Google Scholar

[8]

X. H. TangP. 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.  Google Scholar

[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.  Google Scholar

[10]

X. H. TangP. 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.  Google Scholar

[11]

H. ToriiT. Matsumoto and M. Nakamura, A new method for constructing asymmetric {ZCZ} sequence sets, IEICE Trans. Fund., E95-A (2012), 1577-1586.   Google Scholar

[12]

H. ToriiT. Matsumoto and M. Nakamura, Extension of methods for constructing polyphase asymmetric ZCZ sequence sets, IEICE Trans. Fund., E96-A (2013), 2244-2252.   Google Scholar

[13]

H. Torii, M. Nakamurai and Makoto, Optimal polyphase asymmetric ZCZ sets including uncorrelated sequences, Journal of Signal Processing, 16 (2012), 487-494. Google Scholar

[14]

L. Y. WangX. 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.   Google Scholar

[15]

L. Y. WangX. 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.  Google Scholar

[16]

L. Y. WangX. 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.   Google Scholar

[17]

Z. C. ZhouZ. 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.   Google Scholar

[18]

Z. C. ZhouX. 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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. HayashiT. MaedaS. 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.   Google Scholar

[5]

T. HayashiT. Maeda and S. Okawa, A generalized construction of zero-correlation zone sequence set with sequence subsets, IEICE Trans. Fund., E94-A (2011), 1597-1602.   Google Scholar

[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.  Google Scholar

[7]

S. MatsufujiN. KuroyanagiN. Suehiro and P. Z. Fan, Two types polyphase sequence sets for approximately synchronized CDMA systems, IEICE Trans. Fund., E86-A (2003), 229-234.   Google Scholar

[8]

X. H. TangP. 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.  Google Scholar

[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.  Google Scholar

[10]

X. H. TangP. 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.  Google Scholar

[11]

H. ToriiT. Matsumoto and M. Nakamura, A new method for constructing asymmetric {ZCZ} sequence sets, IEICE Trans. Fund., E95-A (2012), 1577-1586.   Google Scholar

[12]

H. ToriiT. Matsumoto and M. Nakamura, Extension of methods for constructing polyphase asymmetric ZCZ sequence sets, IEICE Trans. Fund., E96-A (2013), 2244-2252.   Google Scholar

[13]

H. Torii, M. Nakamurai and Makoto, Optimal polyphase asymmetric ZCZ sets including uncorrelated sequences, Journal of Signal Processing, 16 (2012), 487-494. Google Scholar

[14]

L. Y. WangX. 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.   Google Scholar

[15]

L. Y. WangX. 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.  Google Scholar

[16]

L. Y. WangX. 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.   Google Scholar

[17]

Z. C. ZhouZ. 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.   Google Scholar

[18]

Z. C. ZhouX. 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.  Google Scholar

Figure 1.  Periodic correlation properties of U-ZCZ sequence set ${B}$.
Figure 2.  Comparison of Different Families of A-ZCZ Sequence Sets
Table 1.  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
Table 2.  Periodic correlation properties of UA-ZCZ sequence set $\mathcal{C}$.
Constructions Parameters Uncorrelated
or not
Flexible
ZCZ or not
Theorem$\ast$1 in [12] $\mathcal{Z}_A(LP, [L, N], [M-1, 2M-1])$ No No
Theorem2 in [12] $\mathcal{Z}_A(TL, \, [T, N], \, [M, TL])$ Yes No
Theorem2 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 Uncorrelated
or not
Flexible
ZCZ or not
Theorem$\ast$1 in [12] $\mathcal{Z}_A(LP, [L, N], [M-1, 2M-1])$ No No
Theorem2 in [12] $\mathcal{Z}_A(TL, \, [T, N], \, [M, TL])$ Yes No
Theorem2 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 $M=\lfloor\frac{P-1}{Z}\rfloor$.
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