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Codebooks with small maximum cross-correlation amplitudes are used todistinguish the signals from different users in CDMA communication systems. Inthis paper, we first study the Jacobi sums over Galois rings of arbitrarycharacteristics and completely determine their absolute values, which extendsthe work in [34], where the Jacobi sums over Galois rings with characteristicsof a square of a prime number were discussed. Then, the deterministicconstruction of codebooks based on the Jacobi sums over Galois rings ofarbitrary characteristics is presented, which produces asymptotically optimalcodebooks with respect to the Welch bound. In addition, the parameters of thecodebooks provided in this paper are new.

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