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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2018, Vol. 35 ›› Issue (3): 319-328.doi: 10.3969/j.issn.1005-3085.2018.03.007

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周期为$p^2$的完备高斯整数序列的新构造

柯品惠1,   胡电芬1,2,   常祖领3   

  1. 1- 福建师范大学数学与信息学院 福建省网络安全与密码技术重点实验室，福州  350117
    2- 福建北斗森林有限公司森林云安全联合实验室，福州  350000 
    3- 郑州大学数学与统计学院，郑州  450001
  • 收稿日期:2016-03-08 接受日期:2016-09-12 出版日期:2018-06-15 发布日期:2018-08-15
  • 基金资助:
    国家自然科学基金(61772292; 61772476)；福建省自然科学基金(2015J01237)；福建师范大学“网络与信息安全关键理论和技术”校创新团队基金(IRTL1207).

New Construction of Perfect Gaussian Integer Sequence with Period $p^2$

KE Pin-hui1,   HU Dian-fen1,2,   CHANG Zu-ling3   

  1. 1- School of Mathematics and Informatics, Fujian Normal University, Fujian Provincial Key Laboratory of Network Security and Cryptology, Fuzhou 350117
    2- Laboratory of Forest Cloud Security, Fujian Beidou Forest Company Limited, Fuzhou 350000 
    3- School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001
  • Received:2016-03-08 Accepted:2016-09-12 Online:2018-06-15 Published:2018-08-15
  • Supported by:
    The National Natural Science Foundation of China (61772292; 61772476); the Natural Science Foundation of Fujian Province (2015J01237); the Innovative Research Team of Fujian Normal University (IRTL1207).

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摘要： 由于具有良好的相关特性，完备高斯整数序列被广泛应用于现代通信系统，但迄今已知的完备高斯整数序列的构造方法比较有限．本文给出了周期为奇素数平方的完备高斯整数序列的新构造．基于模奇素数平方的2阶广义分圆，构造了一类新的周期为奇素数平方的高斯整数序列，并利用广义分圆数确定了该高斯整数序列的自相关函数值的分布．证明了该高斯整数序列成为完备序列等价于复数域上一类特殊形式的二次方程组的求解，并给出了一些特殊情形的解．

关键词: 高斯整数, 完备序列, 广义分圆数, 自相关函数

Abstract: Due to its good correlation property, perfect Gaussian integer sequence has been widely used in modern communication system. However, the known construction methods for perfect Gaussian integer sequence is limited. In this paper, we present a new construction method for perfect Gaussian integer sequences with their period being the square of an odd prime. Based on the generalized cyclotomy of order 2 over the ring of integers with modulo being an odd prime square, we construct Gaussian integer sequence with the period being an odd prime square and determine its autocorrelation function distributions. Furthermore, the construction of perfect Gaussian integer sequence is proved to be equivalent to the solution of an equation system of degree 2 over complex field. Special cases of above equation system are then considered.

Key words: Gaussian integer, perfect sequence, generalized cyclotomy, autocorrelation function

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