The 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 CDMA communication systems. In this paper, we first study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in [34], where the Jacobi sums over Galois rings with characteristics of a square of a prime number were discussed. Then, the deterministic construction of codebooks based on the Jacobi sums over Galois rings of arbitrary characteristics is presented, which produces asymptotically optimal codebooks with respect to the Welch bound. In addition, the parameters of the codebooks provided in this paper are new.