Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in multiple access communication systems in code division. Firstly, this paper studies the Jacobi sums over Galois rings of arbitrary characteristics and completely determines their absolute values. This extends the work by Li et al. (Sci China 56(7):1457–1465, 2013), where the Jacobi sums over some Galois rings with characteristics of a prime square were discussed. It is worth mentioning that the generalization is not trivial, as the Galois rings of arbitrary characteristics have a more complicated structure. Our deterministic construction of codebooks is based on Jacobi sums over Galois rings of arbitrary characteristics, producing asymptotically optimal codebooks for the Welch bound. Finally, compared to the literature, this article proposes for the first time design of codebooks based on Jacobi sums over Galois rings. In addition, the parameters of the presented codebooks are new.

最大互相关幅度较小的码本用于在码分多址通信系统中区分来自不同用户的信号。首先，研究了任意特征伽罗瓦环上的雅可比和，并完全确定了它们的绝对值。这扩展了 Li 等人的工作。(Sci China 56(7):1457–1465, 2013)，其中讨论了一些具有素数平方特征的伽罗瓦环的雅可比求和。值得一提的是，推广并不简单，因为任意特征的伽罗瓦环具有更复杂的结构。我们对码本的确定性构造基于任意特征的伽罗瓦环上的雅可比和，产生韦尔奇界的渐近最优码本。最后，与文献相比，本文首次提出了基于伽罗瓦环上雅可比和的码书设计。此外，所提供的码本的参数是新的。