周期为N≡1（mod4）的平衡最优几乎二元序列对

doi:10.11959/j.issn.1000-436x.2021078

Abstract

Abstract:

Based on the combinatorial design theory， the constructions of balanced optimal almost binary sequence pairs of period N≡1（mod 4）were researched.The maximal cross-correlation values θ_c were obtained by different combinations of （almost） binary sequence pairs.Furthermore， three new bounds on the autocorrelation values under the precondition of the value of θ_c=1，2，3 were presented individually.Meanwhile，four types of balanced（almost）optimal almost binary sequence pairs were generated， which satisfied the cross-correlation values and autocorrelation theory bounds.Through the constructions， the range of the cross-correlation values is expanded and the cross-correlation value of the optimal binary sequence pairs is further reduced.More than odd， the value of sequence length parameter f can be any integer， which enriches the existence space of the optimal binary sequence pair.

Key words: almost binary sequence, balanced, autocorrelation, cross-correlation

CLC Number:

TN911.2

Xiuping PENG, Hongxiao LI, Shide WANG, Hongbin LIN. Balanced optimal almost binary sequence pairs of period N≡1（mod4）[J]. Journal on Communications, 2021, 42(12): 163-171.

Figures/Tables 5

θ_c	序列类型	互相关值	N_i
1	平衡最优互相关I型-几乎二元序列对	$R_{u^{'}, v^{'}} (τ) \in {0, 1, - 1}$	$N_{0} =1, N_{1} = \frac{N - 1}{2}, N_{- 1} = \frac{N - 1}{2}$
2	平衡最优互相关II型-几乎二元序列对	$R_{u, v^{'}} (τ) \in {0, 2, - 2}$	$N_{0} = \frac{N + 1}{2}, N_{2} = \frac{N - 1}{4}, N_{- 2} = \frac{N - 1}{4}$
3	平衡几乎最优互相关I型-几乎二元序列对	$R_{u^{'}, v^{'}} (τ) \in {0, 3, - 3}$	$N_{0} =1, N_{3} = \frac{N - 1}{2}, N_{- 3} = \frac{N - 1}{2}$

序列	特征集	互相关值	自相关值	x值	最优性
				x=±1	最优
$u^{'}, v^{'}$	$U^{'}, V^{'}$	$R_{u^{'}, v^{'}} (τ) = {\begin{cases} 0 ， 1 次 \\ x, \frac{N - 1}{2} 次 \\ - x, \frac{N - 1}{2} 次 \end{cases}$	$R_{u^{'} （ v^{'} ）} (τ) = {\begin{cases} N - 1 ， 1 次 \\ - 1 - 2 y, \frac{N - 1}{2} 次 \\ - 1 + 2 y, \frac{N - 1}{2} 次 \end{cases}$	x=±3	几乎最优
$u, v^{'} (u^{'}, v)$	$U, V^{'} (U^{'}, V)$	$R_{u^{'}, v （ u, v^{'})} (τ) = {\begin{cases} 0 ， \frac{N + 1}{2} 次 \\ x + 1, \frac{N - 1}{4} 次 \\ - x - 1, \frac{N - 1}{4} 次 \end{cases}$	$\begin{array}{l} R_{u^{'} （ v^{'} ）} (τ) = {\begin{cases} N - 1 ， 1 次 \\ - 1 - 2 y, \frac{N - 1}{2} 次 \\ - 1 + 2 y, \frac{N - 1}{2} 次 \end{cases} \\ R_{u （ v ）} (τ) = {\begin{cases} N ， 1 次 \\ - 1 - 2 y, \frac{N - 1}{2} 次 \\ - 1 + 2 y, \frac{N - 1}{2} 次 \end{cases} \end{array}$	x= 1	最优

周期N	序列对	互相关	自相关	理论界f	参考文献
$\begin{array}{l} N = x^{2} + 4 \\ x \equiv 1 (\mod 4) \end{array}$	平衡二元序列对	{1，x，-x±2}	{1，-3}	$\begin{array}{l} θ_{a} = 3 \\ θ_{c} \geq ⌈ \sqrt{N} ⌉ \end{array}$ f 为奇数	文献[18]
$\begin{array}{l} N =1 + 4 y^{2} \\ y \equiv 1 (\mod 2) \end{array}$	平衡二元序列对	{1，-3}	{-1± 2y}	$\begin{array}{l} θ_{c} = 3 \\ θ_{a} \geq ⌈ \sqrt{N} ⌉ \end{array}$ f 为奇数	文献[20]
$\begin{array}{l} N =1 + 4 y^{2} \\ y > 0 \end{array}$	平衡最优几乎二元序列对	{0，±1}	{-1± 2y}	$\begin{array}{l} θ_{c} = 1 \\ θ_{a} \geq ⌈ \sqrt{N} ⌉ \end{array}$ f 为奇数f 为偶数	定理4
$\begin{array}{l} N =1 + 4 y^{2} \\ y > 0 \end{array}$	平衡最优几乎二元序列对	{0，±2}	{-1± 2y}	$\begin{array}{l} θ_{c} = 2 \\ θ_{a} \geq ⌈ \sqrt{N} ⌉ \end{array}$ f 为奇数f 为偶数	定理4
$\begin{array}{l} N = 4 y^{2} + (- 3)^{2} \\ y < 0 \end{array}$	平衡几乎最优几乎二元序列对	{0，±3}	{-1± 2y}	$θ_{c} = 3$ f 为奇数	定理4
$\begin{array}{l} N = 4 y^{2} + (- 3)^{2} \\ y > 0 \end{array}$	平衡几乎最优几乎二元序列对	{0，±3}	{-1± 2y}	$θ_{a} \geq ⌈ \sqrt{N - 8} ⌉$ f 为偶数	定理4

References 23

[1]	DIMITROV M , BAITCHEVA T , NIKOLOV N . Efficient generation of low autocorrelation binary sequences[J]. IEEE Signal Processing Letters, 2020,27: 341-345.
[2]	康家方, 王红星, 赵志勇 ,等. 新的扩频通信调制方法[J]. 通信学报, 2013,34(5): 79-87.
	KANG J F , WANG H X , ZHAO Z Y ,et al. Novel method of spread spectrum communication[J]. Journal on Communications, 2013,34(5): 79-87.
[3]	LUO J Q . Binary sequences with three-valued cross correlations of different lengths[J]. IEEE Transactions on Information Theory, 2016,62(12): 7532-7537.
[4]	WELCH L . Lower bounds on the maximum cross correlation of signals[J]. IEEE Transactions on Information Theory, 1974,20(3): 397-399.
[5]	LIU Z L , GUAN Y L , PARAMPALLI U ,et al. Spectrally-constrained sequences:bounds and constructions[J]. IEEE Transactions on Information Theory, 2018,64(4): 2571-2582.
[6]	SHEDD D , SARWATE D . Construction of sequences with good correlation properties[J]. IEEE Transactions on Information Theory, 1979,25(1): 94-97.
[7]	LUO L F , MA W P . Balanced quaternary sequences of even period with optimal autocorrelation[J]. IET Communications, 2019,13(12): 1808-1812.
[8]	TANG X H , GONG G . New constructions of binary sequences with optimal autocorrelation value/magnitude[J]. IEEE Transactions on Information Theory, 2010,56(3): 1278-1286.
[9]	PENG X P , LIN H B , LIU Y M ,et al. New families of almost binary sequences with optimal autocorrelation property[J]. IEICE Transactions on Fundamentals of Electronics,Communications and Computer Sciences, 2019,E102.A(2): 467-470.
[10]	SU W , YANG Y , ZHOU Z C ,et al. New quaternary sequences of even length with optimal auto-correlation[J]. Science China Information Sciences, 2018,61(2): 022308.
[11]	彭秀平, 冀惠璞, 郑德亮 ,等. 周期为2q理想几乎四进制序列构造研究[J]. 通信学报, 2019,40(12): 105-113.
	PENG X P , JI H P , ZHENG D L ,et al. Study on the constructions of optimal almost quaternary sequences with period 2q[J]. Journal on Communications, 2019,40(12): 105-113.
[12]	SONG M K , SONG H Y . Perfect polyphase sequences from cubic polynomials[C]// Proceedings of 2017 IEEE International Symposium on Information Theory (ISIT). Piscataway:IEEE Press, 2017: 1292-1295.
[13]	ZHANG D , HELLESETH T . New optimal sets of perfect polyphase sequences based on circular Florentine arrays[C]// Proceedings of 2020 IEEE International Symposium on Information Theory (ISIT). Piscataway:IEEE Press, 2020: 2921-2925.
[14]	PARK K H , SONG H Y , KIM D S ,et al. Optimal families of perfect polyphase sequences from the array structure of Fermat-quotient sequences[J]. IEEE Transactions on Information Theory, 2016,62(2): 1076-1086.
[15]	CHONG J F , ZHUO Z P . Cross-correlation properties of some perfect binary sequences[C]// Proceedings of 2011 IEEE International Conference on Computer Science and Automation Engineering. Piscataway:IEEE Press, 2011: 654-657.
[16]	ZHANG T , LI S X , FENG T ,et al. Some new results on the cross correlation of m-sequences[J]. IEEE Transactions on Information Theory, 2014,60(5): 3062-3068.
[17]	WU Y S , YUE Q , SHI X Y ,et al. Binary and ternary sequences with a few cross correlations[J]. Cryptography and Communications, 2020,12(3): 511-525.
[18]	DING C S , TANG X H . The cross-correlation of binary sequences with optimal autocorrelation[J]. IEEE Transactions on Information Theory, 2010,56(4): 1694-1701.
[19]	YANG Y , TANG X H . Balanced quaternary sequences pairs of odd period with (almost) optimal autocorrelation and cross-correlation[J]. IEEE Communications Letters, 2014,18(8): 1327-1330.
[20]	YANG Y , TANG X H , ZHOU Z C . The autocorrelation magnitude of balanced binary sequence pairs of prime period N ≡1(mod 4)with optimal cross-correlation[J]. IEEE Communications Letters, 2015,19(4): 585-588.
[21]	沈秀敏 . 基于分圆类的理想序列偶设计[D]. 秦皇岛:燕山大学, 2017.
	SHEN X M . The design of optimal sequence pairs based on cyclotomy[D]. Qinhuangdao:Yanshan University, 2017.
[22]	DING C S , YIN J X . Sets of optimal frequency-hopping sequences[J]. IEEE Transactions on Information Theory, 2008,54(8): 3741-3745.
[23]	FAN P Z , DARNELL M . Sequence design for communications applications[M]. Chichester: John Wiley ＆ Sons, 1996.

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（i，j）	Z_N x≡1（mod 4）	x≡3（mod 4）
16（0，0）	N- 7+2x	N- 7-2x
16（0，1）	N+ 1+2x-8y	N+ 1-2x-8y
16（0，2）	N+ 1-6x	N+ 1+6x
16（0，3）	N+ 1+2x+8y	N+ 1-2x+8y
16（1，0）	N- 3-2x	N- 3+2x

（i，j）	x≡1（mod 4）	x≡3（mod 4）
16（0，0）	N-11-6x	N-11+6x
16（0，1）	N- 3+2x+8y	N- 3-2x+8y
16（0，2）	N- 3+2x	N- 3-2x
16（0，3）	N- 3+2x-8y	N- 3-2x-8y
16（1，0）	N+ 1-2x	N+ 1+2x

Balanced optimal almost binary sequence pairs of period N≡1（mod4）

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