  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.
