Two constructions for optimal Z-complementary sequence sets

Li Cui; Xiaoyu Chen

doi:10.3934/amc.2022090

Article Contents

2024, Volume 18, Issue 5: 1315-1333. Doi: 10.3934/amc.2022090

This issue Previous Article On the symmetric 2-adic complexity of periodic binary sequences Next Article Three constructions of asymptotically optimal codebooks via multiplicative characters of finite fields

Two constructions for optimal Z-complementary sequence sets

Li Cui^{1, 2, ,} and
Xiaoyu Chen^1,

1.
School of Information Science and Engineering, Yanshan University, Qinhuangdao, 066004, Hebei, China
2.
School of Mathematics and Information Science and Technology, Hebei Normal University of Science and Technology, Qinhuangdao, 066004, Hebei, China

^*Corresponding author: Li Cui
^*Corresponding author: Li Cui

Received: April 2022

Revised: October 2022

Early access: November 22, 2022

Published online: November 22, 2022

This work was supported by the National Natural Science Foundation of China (No.61671402), the Natural Science Foundation of Hebei Province (Nos.F2020203043, F2021203078), and the Science and Technology Project of Hebei Education Department (NO.ZD2022026).

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Abstract

Z-complementary sequence sets (ZCSSs) can be applied to multi-carrier code division multiple access (MC-CDMA) systems as user address codes to eliminate multi-path interference and multi-access interference. In this paper, two constructions of ZCSSs are investigated. First of all, flexible and diverse seed sequence sets are employed to generate a large number of optimal ZCSSs, which is a generalization of an existing construction. Next, note that most ZCSSs obtained by orthogonal-matrix-based constructions have lengths that are multiples of the ZCZ widths. Toward the challenge of designing ZCSSs of various lengths, we proposed a noval construction that can yield ZCSSs with arbitrary lengths based on orthogonal matrices. In this way, different transmission rates can be facility provided to cater for different application scenarios. In addition, the flock sizes and ZCZ widths can be chosen flexibly. It means that, in theory, the obtained ZCSSs can be applied to MC-CDMA systems of various sub-carriers. To the best of our knowledge, the construction of ZCSSs have not been reported in the literature.

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Mathematics Subject Classification: Primary: 94A05; Secondary: 60G35.

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Figure 1. Aperiodic auto-correlation complementarity of $ {{S}^{m}} $, $ 0\le m\le 15 $

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Figure 2. Aperiodic cross-correlation complementarity of $ {{S}^{0}} $ and $ {{S}^{m}} $, $ 1\le m\le 15 $

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Figure 3. Aperiodic auto-correlation complementarity of $ {{S}^{m}} $, $ 0\le m\le 13 $

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Figure 4. Aperiodic cross-correlation complementarity of $ {{S}^{0}} $ and $ {{S}^{m}} $, $ 1\le m\le 13 $

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Table 1. The parameters of the existing aperiodic ZCSSs

Ref.	Based on	Parameters	Constraints	Optimality
[5], Th.1	MOCSS and OM	$ (MD, L){\rm{-ZCSS}}_{N}^{LD} $	gcd$ (L, D)=1 $, $ (M, N, L){\rm{-MOCSS}} $	Optimal when CCSS is Optimal
[5], Th.2	OM	$ (NK, Z){\rm{-ZCSS}}_{N}^{L} $	$ N=MZ $, $ L=KZ $	Optimal
[39], Th.1	GBF	$ (M, 2N){\rm{-ZCSS}}_{M}^{3N} $	$ M={{2}^{k+1}} $, $ N={{2}^{m}} $, $ m, k\in {{\mathbb{Z}}^{+}} $	Optimal
[39], Th.2	Orthogonal sequences and CCSS	$ ({ MK, N}){\rm{-ZCSS}}_{M}^{NL} $	$ M={{2}^{k}} $, $ N={{2}^{m}} $, $ m, k\in {{\mathbb{Z}}^{+}} $	Optimal when $ K=L $
[39], Th.3	ZCP and CCSS	$ (M, NZ){\rm{-ZCSS}}_{M}^{NL} $	$ M={{2}^{k+2}} $, $ N={{2}^{m}} $, $ m, k\in {{\mathbb{Z}}^{+}} $	Optimal when $ Z>L/2 $
[20], Th.1	Golay sequence and OM	$ (KL, Z){\rm{-ZCSS}}_{N}^{KZ} $	$ Z\vert{L} $, $ N\ge 3 $	Not Optimal
[1]	ZCP and $ {{2}^{n}}\times {{2}^{n}} $ Hadamard matrix	$ ({{2}^{n+1}}, Z){\rm{-ZCSS}}_{{{2}^{n+1}}}^{L} $	$ \left\lfloor {L}/{Z}\; \right\rfloor =1 $	Optimal
[18], Th.1	OM	$ (pNZ, Z){\rm{-ZCSS}}_{Q}^{NZ} $	$ p={Q}/{Z}\; $ and $ p $ is prime	Optimal
[18], Th.2	OM	$ ({{p}^{n}}NZ, Z){\rm{-ZCSS}}_{Q}^{NZ} $	$ {{p}^{n}}={Q}/{Z}\; $ and $ p $ is prime	Optimal
[19], Th.1	OM	$ (GN, Z){\rm{-ZCSS}}_{N}^{N} $	$ G={N}/{Z}\; $	Optimal
[12], Th.1	GBF	$ (\prod\nolimits_{i=1}^{l}{{{p}_{i}}{{2}^{n+1}}}, {{2}^{m}}){\rm{-ZCSS}}_{{{2}^{n+1}}}^{\prod\nolimits_{i=1}^{l}{{{p}_{i}}{{2}^{m}}}} $	$ l, m, n\in {{\mathbb{Z}}^{+}} $, $ {{p}_{i}} $ is prime	Optimal
[36], Th.3	GBF	$ ({{2}^{k+v}}, {{2}^{m-v}}){\rm{-ZCSS}}_{{{2}^{k}}}^{{{2}^{m}}} $	$ m, v, k\in {{\mathbb{Z}}^{+}} $, $ v\le M $, $ k\le m-v $	Optimal
[40], Th.1	CCSS and optimal ZCCS	$ ({{M}_{1}}{{M}_{2}}, {{Z}_{1}}{{L}_{2}}){\rm{-ZCSS}}_{{{N}_{1}}{{M}_{2}}}^{{{L}_{1}}{{L}_{2}}} $	$\begin{array}{c}({{M}_{2}}, {{L}_{2}})\text{-CC} \\{\rm{Optimal}} \ ({{M}_{1}}, {{Z}_{1}})\text{-ZCSS}_{{{N}_{1}}}^{{{L}_{1}}}\end{array}$	Optimal
[40], Th.2	CCSS and optimal ZCCS	$ ({{M}_{1}}, {{Z}_{1}}{{L}_{2}}{{N}_{1}}){\rm{-ZCSS}}_{{{N}_{1}}}^{{{L}_{1}}{{L}_{2}}{{N}_{1}}} $	$\begin{array}{c}({{N}_{2}}, {{L}_{2}})\text{-CC} \\{\rm{Optimal}} \ ({{M}_{1}}, {{Z}_{1}})\text{-ZCSS}_{{{N}_{1}}}^{{{L}_{1}}}\end{array}$	Optimal
[29], Th.1	PBF	$ (p{{2}^{k+1}}, {{2}^{m}}){\rm{-ZCSS}}_{{{2}^{k+1}}}^{p{{2}^{m}}} $	$ p $ is prime	Optimal
Th.1	OM	$ \left( HL, Z \right){\rm{-ZCSS}}_{HZ}^{ZN} $	$ L=ZN, Q=HZ $	Optimal
Th.2	OM	$ \left( HL, Z \right){\rm{-ZCSS}}_{Q}^{L} $,	$ H=\left\lfloor {Q}/{Z}\; \right\rfloor $	Optimal when $ \left\lfloor {L}/{Z}\; \right\rfloor \cdot \left( Q\bmod Z \right)=\left( L\bmod Z \right)\left\lfloor {Q}/{Z}\; \right\rfloor $
Note: OM denotes orthogonal matrix.

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