严格达到Welch界的最优三元序列集  

Optimal ternary sequence sets rigorously achieving the Welch bound

作　　者：叶智钒 周正春 张胜元[2] 唐小虎[3] Zhifan Ye;Zhengchun Zhou;Shengyuan Zhang;Xiaohu Tang

机构地区：[1]西南交通大学数学学院,成都611756 [2]福建师范大学数学与统计学院,福州350117 [3]西南交通大学信息科学与技术学院,成都611756

出　　处：《中国科学：数学》2023年第2期395-406,共12页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:62071397和62131016)资助项目。

摘　　要：序列是复数域上的有限维离散信号,因其抗干扰、稳定和易实现等特点,被广泛应用于通信、雷达、声呐和信息安全中,用于实现同步、多址随机接入、信道估计、测距、抗干扰和数字水印等需求.序列与循环Hadamard矩阵、差集(族)、紧框架、无偏基和循环码等数学对象之间存在密切的联系.构造相关性达到或逼近理论界的序列一直是序列编码领域关注的核心问题.本文基于semi-bent函数和差集,构造一类具有最优相关性的三元序列集(序列元素取值为0和±1).这是自1974年Welch界提出以来,第一类严格达到Welch界的最优序列集.相比二元序列集(序列元素取值为±1),新序列集更适用于超宽带通信、数字水印和频谱受限等应用场景;相比已有基于bent函数的三元序列集(相关性渐近达到2倍Welch界),新序列集具有更好的相关性.Sequences are finite-dimensional discrete signals in the complex domain, which are widely used in communication, radar, sonar, and information security to complete the requirements for synchronization, multiple random access, channel estimation, ranging, anti-interference, and digital watermarking due to their antiinterference, stability, and easy implementation. There are close relations between sequences and mathematical objects such as cyclic Hadamard matrices, difference sets(families), tight frames, unbiased bases, and cyclic codes.Constructing sequences reaching or approximating the theoretical bounds has always been a core topic in sequence design and coding theory. In this paper, based on the semi-bent function and the difference set, we propose a class of optimal ternary sequence sets(the elements of sequences belong to {0, ±1}), which is the first class of optimal sequence set strictly achieving the Welch bound since this bound was established in 1974. Compared with the binary sequence sets(the elements of sequences take values ±1), the new sequence set is more suitable for the ultra-wideband communication, the digital watermarking, and the spectrum constrained scenarios. While compared with the known ternary sequence sets which asymptotically approach two times the Welch bound, the proposed ternary sequence sets have better correlation properties.

关 键 词：最优三元序列集 Welch界 semi-bent函数 超宽带通信 数字水印 频谱受限 

分 类 号：O157.4[理学—数学]