Constructions of optimal Z-periodic complementary sequence sets with large zero correlation zone

Tao Yu; Yang Yang; Chunming Tang; Meng Yang

doi:10.3934/amc.2023049

Article Contents

2025, Volume 19, Issue 1: 202-218. Doi: 10.3934/amc.2023049

This issue Previous Article The error-correcting pair for direct sum codes Next Article On MSR subspace families of lines

Constructions of optimal Z-periodic complementary sequence sets with large zero correlation zone

1.
School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, China
2.
School of Information Science and Technology, Southwest Jiaotong University, Chengdu, 611756, China

^*Corresponding author: Yang Yang
^*Corresponding author: Yang Yang

Received: September 2022

Revised: July 2023

Early access: November 16, 2023

Published online: November 16, 2023

This work was supported in part by the National Natural Science Foundation of China under 12201523, in part by the Sichuan Provincial Fund for Distinguished Young Scholars under Grant 2023NSFSC1912, and in part by the Fundamental Research Funds for the Central Universities of China under Grant 2682023CG007, 2682023ZTPY021, 2682023GF013, and 2682023CX079.

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Abstract

Z-periodic complementary sequence sets (ZPCSSs) have potential applications in multi-carrier code-division multiple access (MC-CDMA) communication systems and MIMO channel estimation due to their favourable correlation properties. In this paper, we explore several methods to design optimal ZPCSSs. More specifically, combining complete complementary code (CCC) and optimal ZPCSS, we design optimal ZPCSSs with flexible parameters, using the Kronecker product. Besides, using product sequences, we obtain the optimal ZPCSS with a large zero correlation zone (ZCZ) and optimal ZPCSSs with flexible flock sizes, respectively. Finally, we can obtain more optimal ZPCSSs using the left shift operator. In particular, some lengths of the resultant sequence sets have not been reported before.

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

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Table 1. Summary of known optimal ZPCSSs

Ref.	Based on	Parameters	Conditions
[33, Th. 1]	Optimal ZCZ sequence sets and orthogonal matrix	$ \left(MU, U, L, Z\right) $	Optimal periodic $ (M,L,Z) $-ZCZ sequence set and $ U\times U $ orthogonal matrix
[33, Th. 2]	Optimal ZCZ sequence sets	$ \left(2^nM, 2^n, 2^nL, 2^nZ\right) $	Optimal periodic $ (M,L,Z) $-ZCZ sequence set, $ n\in \mathbb{Z}^+ $
[23, Section 3]	PCCC	$ \left(MG,M,L,Z\right) $	$ G=\lfloor \frac{L}{Z}\rfloor $, $ 0<Z\leq L $, $ (M,M,L) $-PCCC
[6, Th. 1]	PCCC and shift sequence	$ (2MG,M,2L,Z) $	$ (M,M,L) $-PCCC, $ 1<Z< L $, when $ Z $ is even, $ G=\lfloor \frac{L-2}{Z}\rfloor $; when $ Z $ is odd, $ G=\lfloor \frac{L-1}{Z}\rfloor $
[7, Section 3]	PCCC, orthogonal matrix and almost optimal shift sequence	$ (UM,M,UL,Z) $	$ (M,M,L) $-PCCC, $ U\times U $ orthogonal matrix, length-$ U $ almost optimal shift sequence, $ L-2< Z\leq L $
[32, Th. 10]	Binary optimal ZPCSS and Gray mapping	$ \left(M, N, L, Z\right) $	Binary optimal $ (M,N,L,Z) $-ZPCSS
[31, Th. 3.5]	Binary optimal ZPCSS and Gray mapping	$ \left(M, N, 2L,2Z+1\right) $	Binary optimal $ (M,N,L,Z) $-ZPCSS, $ L $ is odd
[11, Th. 1]	Perfect sequence and orthogonal matrix	$ (M,M_2,M_1L,L) $	Length-$ L $ perfect sequence and $ M\times M $ orthogonal matrix, $ M=M_1M_2 $, $ \gcd{(M_1,L)}=1 $
[11, Th. 2]	Perfect sequence and orthogonal matrix	$ (M,M_2,L_1L,L-1) $	Length-$ L $ perfect sequence and $ M\times M $ orthogonal matrix, $ M=L_1M_2 $, $ L=L_1L_2 $, $ L_1< L-1 $
[10, Th. 2]	Perfect sequence, shift sequence and orthogonal matrix	$ (M, M_2, L_1L,L_1\cdot \lfloor\frac{L}{L_1}\rfloor-1) $	Length-$ L $ perfect sequence and $ M\times M $ orthogonal matrix, $ M=L_1M_2 $, length-$ L_1 $ shift sequence, $ L=qL_1+r $, $ 0\leq r< L_1 $, $ r+1\leq q $
[10, Th. 3]	Perfect sequence, shift sequence and orthogonal matrix	$ (MK, M_2, 2L,L_1) $	Length-$ L $ perfect sequence and $ M\times M $ orthogonal matrix, $ M=2M_2 $, $ K $ length-$ 2 $ shift sequence, $ L=qL_1+r $, $ 0\leq r< L_1 $, $ L_1 $ is even, $ 1\leq r\leq \frac{L_1}{2} $; $ L_1 $ is odd, $ 0\leq r\leq \frac{L_1}{2} $.
[27, Th. 1]	GCPs and orthogonal matrix	$ (2M,2,LM,L-1) $	Length-$ L $ GCPs and $ M\times M $ orthogonal matrix, $ M<L-1 $
Th. 3.1	Optimal ZPCSS and CCC	$ (M_1M_2,N_1M_2,L_1L_2,Z_1L_2) $	$ (M_2,M_2,L_2) $-CCC and $ M_2 $ Optimal $ (M_1,N_1,L_1,Z_1) $-ZPCSS
Th. 3.3	Optimal APCSS and PCCC	$ (M_1M_2,M_1M_2/2,L_1L_2,L_1L_2/2) $	Optimal $ (M_1,M_1/2,L_1,L_1/2) $-APCSS and $ (M_2,M_2,L_2) $-PCCC, $ \gcd{(L_1,L_2)}=1 $
Th. 3.4	Orthogonal matrix and PCCC	$ (MN,M_2N,M_1L,L) $	$ M\times M $ orthogonal matrix and $ (N,N,L) $-PCCC $ M=M_1M_2 $, $ \gcd{(M_1,L)}=1 $
Th. 3.5	Optimal ZPCSS and CCC	$ (M_1,N_1,L_1L_2N_1,Z_1L_2N_1) $	Optimal $ (M_1,N_1,L_1,Z_1) $-ZPCSS and $ (N_1,N_1,L_2) $-CCC
Remark 3.7	Optimal ZPCSS	$ (M_1M,N,L,Z_1) $	Optimal $ (M,N,L,Z) $-ZPCSS, $ M_1=\lfloor\frac{Z}{Z_1}\rfloor $, $ 0<Z_1\leq Z $, $ \lfloor\frac{L}{Z_1}\rfloor=\lfloor\frac{L}{Z}\rfloor\cdot \lfloor\frac{Z}{Z_1}\rfloor $

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