Abstract
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.
Highlights
F n l with minimum distance d, where l is a finite field with order l and n l denotes the n-dimensional linear space over F l
In order to search for more residue difference sets, Whiteman [12] introduced a generalized cyclotomy regarding pq
The generalized cyclotomies with order two over Zpq in [7, 16] are exactly the first type and the second type, respectively. By means of this general generalized cyclotomy, we constructed a class of the general two-prime GCSs of order two with period n over F l, where n pq and gcd(l, n) 1, and computed their minimal polynomials and linear complexities. e result shows that their linear complexities are high
Summary
It is widely known that such a positive integer L for any finite sequence always exists. E (generalized) cyclotomic numbers with order h are defined as (i, j) Wi + 1 ∩ Wj, 0 ≤ i, j ≤ h − 1. E second generalized cyclotomic classes D(i n)(0 ≤ i ≤ 1) with order two are defined as. The third generalized cyclotomic classes W(i n)(0 ≤ i ≤ 1) with order two are defined as. Let D0 be a fixed multiplicative subgroup of Z∗n with order φ(n)/2. In [7], the linear complexity and minimal polynomial of generalized cyclotomic sequence with period n over Fl based on the first generalized cyclotomy of order two have been determined. E generalized cyclotomic numbers with order two are defined by (i, j)(2n) D(i n) + 1 ∩ D(jn), 0 ≤ i, j ≤ 1.
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