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, D, D, D 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. [ABSTRACT FROM AUTHOR]