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

通信学报 ›› 2021, Vol. 42 ›› Issue (12): 163-171.doi: 10.11959/j.issn.1000-436x.2021078

周期为N≡1（mod4）的平衡最优几乎二元序列对

彭秀平¹^,², 李红晓¹^,², 王仕德¹^,², 林洪彬³

¹ 燕山大学信息科学与工程学院，河北秦皇岛 066004
² 河北省信息传输与信号处理重点实验室，河北秦皇岛 066004
³ 燕山大学电气工程学院，河北秦皇岛 066004

修回日期:2021-10-04 出版日期:2021-12-01 发布日期:2021-12-01
作者简介:彭秀平（1984- ），女，安徽安庆人，博士，燕山大学副教授，主要研究方向为编码理论、信号设计等
李红晓（1995- ），女，河北邢台人，燕山大学硕士生，主要研究方向为编码理论、信号设计等
王仕德（1997- ），男，河北石家庄人，燕山大学硕士生，主要研究方向为编码理论、信号设计等
林洪彬（1979- ），男，河北秦皇岛人，燕山大学副教授，主要研究方向为智能信息处理、动态复杂场景认知
基金资助:
国家自然科学基金资助项目(61601401);河北省自然科学基金资助项目(F2021203040);河北省高等学校科学技术研究基金资助项目(BJ2018018);河北省高等学校科学技术研究基金资助项目(ZD2019039);河北省高等学校科学技术研究基金资助项目(QN2021144)

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

Xiuping PENG¹^,², Hongxiao LI¹^,², Shide WANG¹^,², Hongbin LIN³

¹ School of Information Science ＆Engineering， Yanshan University， Qinhuangdao 066004， China
² Hebei Province Key Laboratory of Information Transmission and Signal Processing， Qinhuangdao 066004， China
³ School of Electrical Engineering， Yanshan University， Qinhuangdao 066004， China

Revised:2021-10-04 Online:2021-12-01 Published:2021-12-01
Supported by:
The National Natural Science Foundation of China(61601401);The Natural Foundation of Hebei Province(F2021203040);Young Talent Program of Colleges in Hebei Province under Grant(BJ2018018);Young Talent Program of Colleges in Hebei Province under Grant(ZD2019039);Young Talent Program of Colleges in Hebei Province under Grant(QN2021144)

摘要/Abstract

摘要：

基于组合设计理论，对周期为N≡1（mod 4）的平衡最优几乎二元序列对的构造方法进行了研究。根据（几乎）二元序列对的不同组合，得出了最大互相关值分别为θ_c =1，2，3，以此θ_c值为前提，推导得出 3 种情况的自相关理论界，并生成了4类满足互相关值和自相关理论界下界值的平衡（几乎）最优几乎二元序列对。所提构造方法扩展了互相关值的取值范围并进一步降低了最优二元序列对的互相关值，且序列长度参数 f 可选择任意整数，丰富了最优二元序列对的存在空间。

关键词: 几乎二元序列, 平衡性, 自相关性, 互相关性

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

中图分类号:

TN911.2

彭秀平, 李红晓, 王仕德, 林洪彬. 周期为N≡1（mod4）的平衡最优几乎二元序列对[J]. 通信学报, 2021, 42(12): 163-171.

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.

图/表 5

表1

表2

表3

几乎二元序列对的互相关值分布情况"

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	$R_{u^{'}, v^{'}} (τ) \in {0, 3, - 3}$	$N_{0} =1, N_{3} = \frac{N - 1}{2}, N_{- 3} = \frac{N - 1}{2}$

表3

表4

满足条件的平衡（几乎）最优几乎二元序列对"

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

表4

表5

现有的平衡（几乎）最优几乎二元序列对的比较"

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

表5

参考文献 23

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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

周期为N≡1（mod4）的平衡最优几乎二元序列对

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

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