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

Suppose that p and q are two distinct odd prime numbers with n = p q . In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤ n was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over F l was presented and their minimal polynomials and linear complexities were derived, where l = r s with a prime number r and gcd l , 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 F l 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.

Linear Complexity of a New Class of Quaternary Generalized Cyclotomic Sequence with Period 2pm

Sequences with high linear complexity property are of importance in applications. In this paper, based on the theory of generalized cyclotomy, new classes of quaternary generalized cyclotomic sequences with order 4 and period 2pm are constructed. In addition, we determine their linear complexities over finite field F4 and over ℤ4, respectively.