Gauss periods and codebooks from generalized cyclotomic sets of order four

Let p, q be distinct primes with gcd(p ? 1, q ? 1) = 4. Let D ₀, D ₁, D ₂, D ₃ be Whiteman’s generalized cyclotomic classes, satisfying the multiplicative group ${{\mathbb Z}^*_{pq}=D_0\cup D_1\cup D_2\cup D_3}$ . In this paper, we give formulas of Gauss periods: ${\sum_{i\in D_0\cup D_2}\zeta^i}$ and ${\sum_{i\in D_0}\zeta^i}$ , where ${\zeta}$ is a pqth primitive root of unity. As an application, we get the maximum cross-correlation amplitudes of three codebooks from generalized cyclotomic sets of order four and supply conditions on p and q such that they nearly meet the Welch bound.