In this paper, we propose a sequence construction framework that extends prime-length Björck sequences, a class of Constant Amplitude Zero Autocorrelation (CAZAC) sequences, to arbitrary lengths using Goldbach's conjecture for even and odd integers. The framework is generic and applies to any CAZAC family defined for prime lengths and supports extensions to both cyclically shifted sequences and sequences with different root indices. We analytically characterize the resulting correlation behavior and show that the construction preserves orthogonality among cyclic shifts while maintaining favorable zero-lag cross-correlation across different root-index sequences. We further investigate Björck sequences as candidates for reference signals in next-generation wireless systems. Using the proposed framework, we extend Björck sequences to arbitrary lengths and evaluate their time- and frequency-offset estimation performance in terrestrial (TNs) and non-terrestrial networks (NTNs). Results show performance comparable to Zadoff--Chu (ZC) sequences in low-Doppler TN environments and improved robustness in high-Doppler NTN scenarios due to superior ambiguity-function properties. We also identify an inherent Doppler-dependent behavior that can cause sequence misidentification under large Doppler shifts. To address this, we propose two mitigation strategies: (i) leveraging coarse Doppler estimates prior to detection, and (ii) selecting appropriately spaced subsets of orthogonal sequences. Ambiguity function-based analysis demonstrates the effectiveness of these approaches in improving estimation reliability. Overall, this work enables practical arbitrary-length CAZAC sequence design and establishes Björck sequences as a strong alternative for reference signal design in high-Doppler environments.

翻译：本文提出一种序列构造框架，利用Goldbach猜想将素数长度的Björck序列（一类恒幅零自相关序列）扩展至任意长度（包括偶数和奇数长度）。该框架具有通用性，适用于任何定义为素数长度的CAZAC族，并支持循环移位序列和不同根指数序列的扩展。我们解析地表征了由此产生的相关特性，证明该构造在保持循环移位间正交性的同时，不同根指数序列间的零延时互相关性能优良。进一步地，我们研究Björck序列作为下一代无线系统参考信号候选方案的可行性。利用所提框架将Björck序列扩展至任意长度后，评估其在陆地网络和非陆地网络中的时频偏移估计性能。结果表明：在低多普勒TN环境中性能与Zadoff-Chu序列相当，而在高多普勒NTN场景中，因具备更优的模糊函数特性而展现出更强的鲁棒性。同时，我们识别出一种固有依赖多普勒效应的现象，该现象可能导致大频偏下的序列误判。为此提出两种缓解策略：（i）在检测前利用粗多普勒估计；（ii）选取适当间隔的正交序列子集。基于模糊函数的分析验证了这些策略在提升估计可靠性方面的有效性。综上，本工作实现了实用化的任意长度CAZAC序列设计，并确立了Björck序列作为高多普勒环境参考信号设计的强有力替代方案。