Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two

Abstract

Suppose that p and q are two distinct odd prime numbers with n=pq. In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤn was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over Fl was presented and their minimal polynomials and linear complexities were derived, where l=rs with a prime number r and gcdl,n=1. The results have indicated that the linear complexities of these sequences are high without any special requirements on the prime numbers. Furthermore, we employed these sequences to obtain a few cyclic codes over Fl with length n and developed the lower bounds of the minimum distances of many cyclic codes. It is important to stress that some cyclic codes in this paper are optimal.