周期为4<i>v</i>(几乎)平衡理想二进制序列构造研究

彭秀平; 冀惠璞; 林洪彬; 刘刚

doi:10.11999/JEIT200829

周期为4v(几乎)平衡理想二进制序列构造研究

doi: 10.11999/JEIT200829

彭秀平^{1, 3, 4, ,},
冀惠璞¹,
林洪彬²,
刘刚²

1.
燕山大学信息科学与工程学院秦皇岛 066004
2.
燕山大学电气工程学院秦皇岛 066004
3.
河北省信息传输与信号处理重点实验室秦皇岛 066004
4.
通信网信息传输与分发技术重点实验室石家庄 050081

基金项目: 国家重点研发计划(2017YFB0306402)，河北省自然科学基金(F2021203040, E2020203188)，河北省高等学校科学技术研究基金(BJ2018018, ZD2019039, QN2019133)，弹性网络服务自主适应管控技术研究(6142104190109)

详细信息

作者简介:
彭秀平：女，1984年生，副教授，研究方向为编码理论、信号设计等

冀惠璞：女，1995年生，硕士生，研究方向为编码理论、信号设计等

林洪彬：男，1979年生，副教授，研究方向为RGBD图像/点云/网格曲面语义标注与分割、点云结构推理与纹理修复等

刘刚：男，1973年生，讲师，研究方向为无线网络通信等

通讯作者:
彭秀平　pengxp@ysu.edu.cn

中图分类号: TN911.2
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出版历程
- 收稿日期: 2020-09-23
- 修回日期: 2021-04-15
- 网络出版日期: 2021-07-13
- 刊出日期: 2022-01-10

Study on the Constructions of Balanced Optimal Binary Sequences with Period 4v (almost)

PENG Xiuping^{1, 3, 4
, ,},
JI Huipu¹,
LIN Hongbin²,
LIU Gang²

1.
School of Information Science & Engineering, Yanshan University, Qinhuangdao 066004, China
2.
School of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China
3.
Hebei Province Key Laboratory of Information Transmission and Signal Processing, Qinhuangdao 066004, China
4.
Key Laboratory of Communication Network Information Transmission and Distribution Technology, Shijiazhuang 050081, China

Funds: The National Key Research and Development Project (2017YFB0306402), The Natural Foundation of Hebei Province(F2021203040, E2020203188), Science and Technology Program of Universities and Colleges in Hebei Province (BJ2018018, ZD2019039, QN2019133), The Research on Self-Adaptive Control Technology of Elastic Network Service (6142104190109)

摘要

摘要: 具有理想自相关特性的序列在无线通信、雷达以及密码学中具有重要的作用。因此为了扩展更多可应用于通信系统的理想序列，该文基于2阶分圆类和中国剩余定理，提出3类新的周期为$T = 4v$(v是奇素数)平衡或几乎平衡理想二进制序列构造方法。构造所得序列的周期自相关函数满足：当$v \equiv 3{\text{ }}\left( {{\rm{mod}} 4} \right)$时，序列的周期自相关函数旁瓣值取值集合为$\left\{ {0, - 4} \right\}$或$\left\{ {0, 4, - 4} \right\}$；当$v \equiv 1{\text{ }}\left( {{\rm{mod}} 4} \right)$时，相应的取值集合为$\left\{ {0, 4, - 4} \right\}$。通过该文方法拓展了周期为4$v$平衡理想二进制序列的存在范围，从而可为工程应用提供更多性能优良的理想序列。
- 二进制序列 /
- 几乎差集 /
- 分圆 /
- 理想自相关性
Abstract: Sequences with optimal autocorrelation property have important roles in wireless communication, radar and cryptography. Therefore, in order to expand more ideal sequences that can be applied to communication systems, based on cyclotomy of order 2 and Chinese remainder theorem, three new constructions of balanced or almost balanced binary sequences of period $T = 4v$(v is odd prime) are presented in this paper. The periodic autocorrelation function of the constructed sequence satisfies: when $v \equiv 3{\text{ }}\left( {{\rm{mod}} 4} \right)$, the out-of-phase autocorrelation value set of the sequence is $\left\{ {0, - 4} \right\}$ or $\left\{ {0, 4, - 4} \right\}$; when $v \equiv 1{\text{ }}\left( {{\rm{mod}} 4} \right)$, the corresponding value set is $\left\{ {0, 4, - 4} \right\}$. The existing range of balanced optimal binary sequences with period of 4v is extended by this method, so that more optimal sequences with good property can be provided for engineering applications.
- Binary sequences /
- Almost difference set /
- Cyclotomy /
- Optimal autocorrelation property

HTML全文

表 1 定理2中理想二进制序列的自相关函数值分布

$j$	$V \in $	$R\left( {{\tau _1},{\tau _2}} \right)$	$\left( {{\tau _1},{\tau _2}} \right) \in $
0	$\left\{ {\left\{ {\left( {i,0} \right),\left( {i + 1,0} \right)} \right\},} \right.$$\left. {\left\{ {\left( {i + 1,0} \right),\left( {i + 2,0} \right)} \right\}} \right\}$	–4	$\left\{ 1 \right\} \times {D_0} \cup \left\{ 3 \right\} \times {D_1} \cup \left\{ 0 \right\} \times Z_v^* \cup \left\{ {\left( {2,0} \right)} \right\}$,
1	$\left\{ {\left\{ {\left( {i,0} \right),\left( {i + 3,0} \right)} \right\},} \right.$$\left. {\left\{ {\left( {i + 2,0} \right),\left( {i + 3,0} \right)} \right\}} \right\}$	0	$\left\{ 2 \right\} \times Z_v^* \cup \left\{ {\left( {1,0} \right),\left( {3,0} \right)} \right\}$,
1		4	$\left\{ 1 \right\} \times {D_1} \cup \left\{ 3 \right\} \times {D_0}$
0	$\left\{ {\left\{ {\left( {i,0} \right),\left( {i + 2,0} \right)} \right\},} \right.$$\left. {\left\{ {\left( {i + 1,0} \right),\left( {i + 3,0} \right)} \right\}} \right\}$	–4	$\left\{ 0 \right\} \times Z_v^* \cup \left\{ {\left( {1,0} \right),\left( {3,0} \right)} \right\}$,
0		0	$\left\{ {1,2,3} \right\} \times Z_v^*$,
1		4	$\left\{ {\left( {2,0} \right)} \right\}$
0	$\left\{ {\left\{ {\left( {i + 3,0} \right),\left( {i + 2,0} \right)} \right\},} \right.$$\left. {\left\{ {\left( {i,0} \right),\left( {i + 3,0} \right)} \right\}} \right\}$	–4	$\left\{ 0 \right\} \times Z_v^* \cup \left\{ 1 \right\} \times {D_1} \cup \left\{ 3 \right\} \times {D_0} \cup \left\{ {\left( {2,0} \right)} \right\}$,
1	$\left\{ {\left\{ {\left( {i,0} \right),\left( {i + 1,0} \right)} \right\},} \right.$$\left. {\left\{ {\left( {i + 1,0} \right),\left( {i + 2,0} \right)} \right\}} \right\}$	0	$\left\{ 2 \right\} \times Z_v^* \cup \left\{ {\left( {3,0} \right),\left( {1,0} \right)} \right\}$,
1		4	$\left\{ 1 \right\} \times {D_0} \cup \left\{ 3 \right\} \times {D_1}$
注：${0 \le i \le 3}$

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表 2 定理3中理想二进制序列的自相关函数值分布

$j$	$V \in $	$R\left( {{\tau _1},{\tau _2}} \right)$	$\left( {{\tau _1},{\tau _2}} \right) \in $
0	$\left\{ {\left\{ {\left( {i + 1,0} \right)} \right\},\left\{ {\left( {i + 3,0} \right)} \right\}} \right\}$	–4	$\left\{ 0 \right\} \times Z_v^* \cup \left\{ {1,3} \right\} \times {D_1}$,
1	$\left\{ {\left\{ {\left( {i,0} \right),\left( {i + 1,0} \right),\left( {i + 2,0} \right)} \right\},} \right.$	0	$\left\{ 2 \right\} \times Z_v^* \cup \left\{ {\left( {1,0} \right),\left( {2,0} \right),\left( {3,0} \right)} \right\}$,
1	$\left. {\left\{ {\left( {i,0} \right),\left( {i + 2,0} \right),\left( {i + 3,0} \right)} \right\}} \right\}$	4	$\left\{ {1,3} \right\} \times {D_0}$
0	$\left\{ {\left\{ {\left( {i,0} \right),\left( {i + 1,0} \right),\left( {i + 2,0} \right)} \right\},} \right.$	–4	$\left\{ 0 \right\} \times Z_v^* \cup \left\{ {1,3} \right\} \times {D_0}$,
0	$\left. {\left\{ {\left( {i,0} \right),\left( {i + 2,0} \right),\left( {i + 3,0} \right)} \right\}} \right\}$	0	$\left\{ 2 \right\} \times Z_v^* \cup \left\{ {\left( {1,0} \right),\left( {2,0} \right),\left( {3,0} \right)} \right\}$,
1	$\left\{ {\left\{ {\left( {i + 1,0} \right)} \right\},\left\{ {\left( {i + 3,0} \right)} \right\}} \right\}$	4	$\left\{ {1,3} \right\} \times {D_1}$
注：${0 \le i \le 3}$

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表 3 已知周期为$T \equiv 0\;\left( {{\rm{mod 4}}} \right)$的具有理想自相关值/幅度的二进制序列总结

周期 $T$	$R\left( {\tau \ne 0} \right)$	平衡性	构造方法
$T = 4v$, $v \equiv 3(\bmod 4)$	$\left\{ {0, - 4} \right\}$	几乎平衡	交织法^[10]
$T = 4v$, $v = {2^{2k}} - 1$	$\left\{ {0, \pm 4} \right\}$	几乎平衡	交织法^[11]
$T = 4v$, $v = p\left( {p + 2} \right)$, $p$和$p + 2$为素数	$\left\{ {0, \pm 4} \right\}$	不平衡	交织法^[11]
$T = 4v$, $v = {2^m} - 1$, $m$为整数	$\left\{ {0, \pm 4} \right\}$	几乎平衡	交织法^[12]
$T = 4v$, $v \equiv 3(\bmod 4)$, $v$为整数	$\left\{ {0, - 4} \right\}$	几乎平衡	中国剩余定理^[16]
$T = 4v$, $v \equiv 3(\bmod 4)$, $v$为素数	$\left\{ {0, - 4} \right\}$	几乎平衡	分圆类^[15]
$T = {p^m} - 1$, $\dfrac{ { {p^m} - 1} }{2}$为偶数	$\left\{ {0, - 4} \right\}$	平衡	基于多项式$z\left( {1 - z} \right)$^[8]
$T = {p^m} - 1$, $p$为奇素数	$\left\{ {0, - 4} \right\}$	平衡或几乎平衡	基于多项式${\left( {z + 1} \right)^d} + a{z^d} + b$^[9]
$T = 4v$, $v \equiv 3(\bmod 4)$	$\left\{ {0, \pm 4} \right\}$	几乎平衡或不平衡	一般化交织法^[7]
$T = 4v$, $v \equiv 2(\bmod 4)$	$\left\{ {0, \pm 4} \right\}$	几乎平衡	交织法^[13]
$T = 4v$, $v \equiv 1(\bmod 4)$, $v$为素数	$\left\{ {0, \pm 4} \right\}$	平衡或不平衡	交织法^[14]
$T = 4v$, $v \equiv 3(\bmod 4)$, $v$为素数	$\left\{ {0, - 4} \right\}$	几乎平衡	广义分圆, 定理1
$T = 4v$, $v \equiv 3(\bmod 4)$, $v$为素数	$\left\{ {0, \pm 4} \right\}$	平衡	广义分圆, 定理2
$T = 4v$, $v \equiv 1(\bmod 4)$, $v$为素数	$\left\{ {0, \pm 4} \right\}$	几乎平衡	广义分圆, 定理3

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