A Unified Approach to Whiteman's and Ding-Helleseth's Generalized Cyclotomy Over Residue Class Rings
Cuiling Fan, Gennian Ge · IEEE Transactions on Information Theory · 2014
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${\BBZ}_{v}$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${\BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},\ldots, p_{n}^{e_{n}}}$, which includes Whiteman's and Ding-Helleseth's generalized cyclotomy as special cases. Here,$p_{1},p_{2},\ldots,p_{n}$are pairwise distinct odd primes satisfying$d\vert (p_{i}-1)$for all$1\leq i\leq n$and$e_{1},e_{2},\ldots,e_{n}$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${\BBZ}_{p_{1}p_{2}}$. Besides, we also construct an infinite series of near-optimal codebooks over${\BBZ}_{p_{1}p_{2}}$, as well as some infinite series of asymptotically optimal difference systems of sets over${\BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},\ldots,p_{n}^{e_{n}}}$.