A Unified Approach to Whiteman´s and Ding-Helleseth´s Generalized Cyclotomy Over Residue Class Rings

The theory of cyclotomy dates back to Gauss and has a number of applications in combinatorics, coding theory, and cryptography. Cyclotomy over a residue class ring BBZv can be divided into classical cyclotomy or generalized cyclotomy, depending on v prime or composite. In this paper, we introduce a generalized cyclotomy of order d over BBZp1^e1p2^e2,..., pn^en, which includes Whiteman´s and Ding-Helleseth´s generalized cyclotomy as special cases. Here, p1,p2,...,pn are pairwise distinct odd primes satisfying d|(pi-1) for all 1 ≤ i ≤ n and e1,e2,...,en are positive integers. We derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers. As applications, we completely solve an open problem and a conjecture on Whiteman´s generalized cyclotomy of order four over BBZp1p2. Besides, we also construct an infinite series of near-optimal codebooks over BBZp1p2, as well as some infinite series of asymptotically optimal difference systems of sets over BBZp1^e1p2^e2,...,pn^en.