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

TL;DR

This paper investigates Jacobi sums over Galois rings of any characteristic, determines their absolute values, and uses them to construct asymptotically optimal codebooks for CDMA systems, extending previous work and introducing new parameters.

Contribution

It provides a complete analysis of Jacobi sums over Galois rings of arbitrary characteristics and introduces a deterministic method to construct asymptotically optimal codebooks.

Findings

Determined the absolute values of Jacobi sums over Galois rings of arbitrary characteristics.

Constructed new asymptotically optimal codebooks with parameters surpassing previous designs.

Extended the theoretical framework of Jacobi sums from prime square characteristics to arbitrary characteristics.

Abstract

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.