Jacobi sums over Galois rings of arbitrary characters and their applications in constructing asymptotically optimal codebooks

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.