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 � pq . 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


Introduction
roughout this paper, assume that l is a power of a prime number r. A linear [n,k,d] code C over F l is defined as a k-dimensional subspace of F n l with minimum distance d, where F l is a finite field with order l and F n l denotes the n-dimensional linear space over F l . If each (c 0 , c 1 , . . . , c n− 1 ) ∈ C if and only if (c n− 1 , c 0 , . . . , c n− 2 ) ∈ C, then C is named as a cyclic code. Let gcd(l, n) � 1 and the ring R � F l [x]/〈x n − 1〉. Hence, there is a one-to-one correspondence between a linear code over F l with length n and a real subset of R. If an ideal C is not trivial, there is a monic polynomial g(x) over F l , satisfying C � 〈g(x)〉, where g(x) is unique and g(x)|(x n − 1). en, g(x) and h(x) � (x n − 1)/g(x) are called to be the generator polynomial and parity-check polynomial of C, respectively. Please refer to [1] for more details on cyclic codes.
Let s n � (s i ) n− 1 i�0 be a sequence with period n over F l and S(x) � n− 1 i�0 s i x i ∈ F l [x]. At present, the methods of constructing cyclic codes are substantial. One of the important methods is to employ where h(x) is named as the generator polynomial of s n .
As is known to all, there are a good deal of results on cyclic codes in a series of papers [2][3][4][5][6][7][8][9][10][11]. Let p and q be two distinct prime numbers. In order to search for more residue difference sets, Whiteman [12] introduced a generalized cyclotomy regarding pq. In 1997, Ding [13] introduced the Whiteman's generalized cyclotomic sequence (WGCS) and studied its coding properties in [6,7,9,11]. To be more specific, Ding [6] constructed three families of cyclic codes, some of which are optimal with regard to the minimum distance. In addition, Ding [7] presented many cyclic codes and their lower bounds about the minimum weight. Furthermore, based on a number of WGCSs of order 4 and order 6, some classes of cyclic codes were produced by Sun et al. [9] and Kewat and Kumari [11], respectively, whose lower bounds on the nonzero minimum weight were also provided.
As an important measure of the quality of a sequence, its linear complexity is defined as the length of the shortest linear feedback shift register which can produce this sequence. Nowadays, pseudorandom sequences with higher linear complexity are widely used in communication systems and cryptography. e main contributions of this article are as follows: firstly, we proved that there are only three classes of generalized cyclotomies with order two over Z pq , see Lemmas 4 and 5. Specifically, the generalized cyclotomies with order two in [12] and in [14,15] are special cases of the first class and the second class, respectively. In essence, the generalized cyclotomies with order two over Z pq in [7,16] are exactly the first type and the second type, respectively. Secondly, by means of this general generalized cyclotomy, we constructed a class of the general two-prime GCSs of order two (see Definition 2) 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. Compared with the previous constructions of sequences with high linear complexity, our construction not only includes the aforementioned constructions in [14,15] as special cases but also gives more parameters with high linear complexity due to the free choice of p and q, see Remark 4. Particularly, if gcd(p − 1, q − 1) ≠ 2, these sequences are new. irdly, we employed these sequences to produce some families of cyclic codes. e idea of constructing cyclic codes employing special sequences in this paper is enlightened by [7]. Let us say it again, more optimal cyclic codes with new parameters can be generated by our construction compared to the cyclic codes obtained by [7], see Remark 3 for details.

Minimal Polynomial and Linear Complexity.
Suppose that the sequence s n � s 0 s 1 , . . . , s n− 1 , where s i ∈ F l and gcd(l, n) � 1. e sequence s n is called to be linear feedback sequence, if there are L + 1 constants c 0 � 1, c 1 , . . . , c L ∈ F l , satisfying It is widely known that such a positive integer L for any finite sequence always exists. Here, the linear complexity of the sequence s n is defined as the minimal positive integer L and the feedback polynomial (or characteristic polynomial) of s n is defined as the polynomial c(x) � c 0 + c 1 x + · · · +c L x L . Furthermore, the minimal polynomial of s n is defined as the characteristic polynomial with the minimal length. It is demonstrated that the degree of its minimal polynomial of a periodic sequence is equal to its linear complexity. So far, there are many ways to calculate the minimal polynomial and linear complexity of a periodic sequence, one of which is stated as follows.
Lemma 1 (see [1]). Define For the sequence s n , its minimal polynomial m(x) is given by and its linear complexity L s is determined by

Classical Cyclotomy with Order
Two. Suppose that f is an integer and q � 2f + 1 is an odd prime number. en, there exists a finite field G � F q with order q. Furthermore, one can always find an element g ∈ G such that G * � 〈g〉, where G * � G\ 0 { } is the set of nonzero elements of G. Define the cyclotomic classes D (q) i with order two in G: For given integers i and j, 0 ≤ i, j ≤ 1, the cyclotomic number (i, j) (q) 2 with order two regarding q is defined as Now, we will recall the properties of the classical cyclotomy with order two in [17] as follows.

Lemma 2.
Suppose that q � 2f + 1 is an odd prime number. en, Let gcd(q, l) � 1 and Ω l denote an algebraic closure of a finite field F l . Define η 0 and η 1 as where ζ q ∈ Ω l is a q-th primitive root of unity; D (q) are the cyclotomic classes with order two regarding q. When 2 ∤ l, η i (i � 0, 1) over Ω l are identified by the following lemma.
Lemma 3 (see [18]). Let the symbols be the same as before. en, where (÷) is the Legendre symbol.

e General Two-Prime Generalized Cyclotomic Sequence of Order Two.
Suppose that Z n is the residue class ring module n and Z * n � x: is a positive integer. Let ϕ(n) denote the Euler function. For gcd(a, n) � 1, if the multiplicative order of a modulo n is ϕ(n), then a is named as a primitive root modulo n.
If there exist a subgroup W 0 of Z * n and g 1 , then the W i are called to be classical cyclotomic classes with order h if n is a prime number, and generalized cyclotomic classes with order h if n is a composite number. e (generalized) cyclotomic numbers with order h are defined as

Lemma 4.
ere are only three classes of generalized cyclotomies with order two regarding pq.
We next show that H is a multiplicative subgroup of D 0 . It is straightforward to prove that H is a multiplicative subgroup of Z * n . We only need to show that H⊆D 0 . For any is finishes the proof of Lemma 4. Suppose that g is a common primitive root modulo p and q. Let x be a positive integer satisfying Assume . erefore, the multiplicative group of Z n is as follows [12]: According to Lemma 4, we easily verify the assertion as the following.

Lemma 5.
ere are only three classes of generalized cyclotomies with order two over Z pq as follows. e first generalized cyclotomic classes H (n) i (0 ≤ i ≤ 1) with order two are defined as e second generalized cyclotomic classes D (n) i (0 ≤ i ≤ 1) with order two are defined as and the third generalized cyclotomic classes W (n) i (0 ≤ i ≤ 1) with order two are defined as where the multiplications are operated modulo n.
Remark 1. By definition, when gcd(p − 1, q − 1) � 2, the first generalized cyclotomy with order two over Z pq is exactly Whiteman's generalized cyclotomy [12]. In essence, it is in accord with the one introduced by Ding [7]. When gcd(p − 1, q − 1) � 2, the second generalized cyclotomy with order two over Z pq is identical to Ding-Helleseth's generalized cyclotomy [2] in the case of v � pq. Furthermore, this cyclotomy is the same as the extended generalized cyclotomy with order two presented by Wang and Lin [16]. For fixed p, q, and x defined by (14), the third generalized cyclotomy is new. In [7], the linear complexity and minimal polynomial of generalized cyclotomic sequence with period n over F l based on the first generalized cyclotomy of order two have been determined. In addition, cyclic codes defined by this sequence were analyzed. Here, we only study the second generalized cyclotomy with order two in this paper. e generalized cyclotomic numbers with order two are defined by Define en, Definition 2. e two-prime general generalized cyclotomic sequences (GGCS) of order two are defined by Journal of Mathematics where 0 and 1 are the zero element and identity element, respectively, in F l .
Remark 2. In [16], 0 and 1 in equation (22) are both in F 2 . Hence, our sequence and the sequence defined by Wang and Lin are totally different.

e Properties on the Generalized Cyclotomy with Order
Two Over Z n . In this subsection, the following lemmas follow from [16].

e Parameters of the Code C s , Minimal Polynomial, and
Linear Complexity of the Sequence s n . Let p and q be two difference odd prime numbers with n � pq and l be a power of a prime number r. We always assume that gcd(n, l) � 1 and m is the order of l modulo n. Define Our main objective in this section is to compute the generator polynomial of the cyclic code C s defined by the sequence s n , where S(x) is defined as equation (28). With this purpose, we need to seek out the a(0 ≤ a ≤ n − 1), satisfying S(ζ a n ) � 0, where ζ n is a fixed n-th primitive root of unity of the finite field F l m . e following auxiliary results are important for our calculation. Obviously, we have

(34)
If a ∈ P, it follows that Hence, we get the conclusions that, for a ∈ P and ((a/p)/q) � 1, and for a ∈ P and ((a/p)/q) � − 1,

Lemma 11
S ζ a n � if a ∈ P and a/p q � 1.

Journal of Mathematics
Proof. If a � 0, then the conclusion is straightforward. If a ∈ D (n) 1 , then aD (n) 1 � D (n) 0 and aP(mod n) � P. By equation (30), we get S ζ a n � i∈P + i∈D (n) If a ∈ D (n) 0 , we have aD (n) 1 � D (n) 1 and aP(mod n) � P, since gcd(a, q) � 1. So, we have If a ∈ Q, then aP(mod n) � 0. So by Lemma 10, If a ∈ P, then aP(mod n) � P. By equation (30) and Lemma 10, one has Lemma 12.

Proof
(1) If l(mod n) ∈ D (n) 0 , then by Lemma 11, we have S ζ n l � S ζ l n � S ζ n .
Hence, S(ζ n ) ∈ F l . (2) If l(mod n) ∈ D (n) 1 , then by Lemma 11, we have similarly Under this assumption, S(ζ n ) ∉ F l if l is even, and S(ζ n ) may be in F l if l is odd. Lemma 13. When q ≡ 1(mod 4), we have When q ≡ 3(mod 4), we have Proof. According to equation (30) and the definition of S(x), one has en, we get Suppose that q ≡ 1(mod 4). According to Lemma 6, − 1 ∈ D (n) 0 and − D (n) 1 � D (n) 1 . It follows from Lemmas 2, 7, 8, and 9 that 6 Journal of Mathematics Combining equations (48) and (49), one can get the first conclusion of this lemma.
Proof. Obviously, l(mod n) ∈ D (n) 0 if r ∈ D (n) 0 . Now, we only give the proof of the first conclusion since the proof of the second part is similar to it. Let q ≡ 1(mod 4r) with q ≡ 1(mod 4).
Secondly, when r is an odd prime number, we prove this lemma. Since q ≡ 1(mod 4r), q ≡ 1(mod r). Hence, (q/r) � 1. Now, one has Suppose that r ∈ D (n) 1 on the contrary. By definition of D (n) 1 , r � g 2s+1 x t for 0 ≤ s ≤ e/2 − 1 and 0 ≤ t ≤ d − 1. en, we have r ≡ g 2s+1 (mod q). erefore, r is not a quadratic residue modulo q. is is contrary to equation (69). So, we arrive at the conclusion.
In order to calculate the minimal polynomial and linear complexity of s n , we need to study the factorization of x n − 1 over F l . It can be checked that the ζ i n (i ∈ P ∪ 0 { }) are q-th roots of unity and ζ i n (i ∈ Q ∪ 0 { }) are p-th roots of unity. erefore, For any i ∈ 0, 1 { }, define Journal of Mathematics 7 where D (n) i denotes the general generalized cyclotomic classes with order two. If l(mod n) ∈ D (n) 0 , then it is obvious . After the above preparations, one has Theorem 1. Assume that r � 2. We have the assertions as follows: (1) When q ≡ 5(mod 8), the linear complexity of s n is L s � n − p, and the cyclic code C s defined by s n has the parameters [n, p, q], whose generator polynomial is m(x) � (x n − 1/x p − 1). (2) When q ≡ 3(mod 8), the linear complexity of s n is L s � n − 1, and the cyclic code C s defined by s n has the parameters [n, 1, n], whose generator polynomial is m(x) � (x n − 1/x − 1). (3) When q ≡ 1(mod 8), the linear complexity of s n is L s � ((p + 1)(q − 1)/2), and the cyclic code C s defined by s n has the parameters [n, ((p − 1)(q + 1)/2) + 1, d], whose generator polynomial is (4) When q ≡ 7(mod 8), the linear complexity of s n is L s � ((p + 1)(q + 1)/2) − 2, and the cyclic code C s defined by s n has the parameters [n, ((p − 1)(q − 1)/2) + 1, d], whose generator polynomial is , if S ζ n � 0, , if S ζ n � 1.
Now, we only prove Case 1, since the others are similar. For Case 1, by Lemma 11, erefore, gcd(x n − 1, S(x)) � x p − 1 and m(x) � (x n − 1/x p − 1). en, the linear complexity of the sequence s n is L s � deg(m(x)) � n − p. Furthermore, the definition of the code can lead to the parameters of the code C s . Example 1. Assume that (l, p, q) � (2, 3, 5). en, n � 15, q ≡ 5(mod 8), the minimal polynomial is x 12 + x 9 + x 6 + x 3 + 1, and the linear complexity of s n is 12.
e cycle code C s defined by s n is a [15,3,5] cyclic code over F 2 . According to the database [19], the best binary linear code known with the parameters [15, 3, d] has minimum distance 8.
is is bad because of its poor minimum distance. Obviously, our cyclic code is more optimal.
(1) When q ≡ ± 1(mod r) and q ≡ 3(mod 4), or q ≡ 1(mod r) and q ≡ 1(mod 4), the linear complexity of s n is L s � n, and s n has the minimal polynomial m(x) � x n − 1.
(2) When q ≡ 1(mod r) and q ≡ 3(mod 4), the linear complexity of s n is L s � n − p, and s n has the minimal polynomial m(x) � (x n − 1/x p − 1). e cyclic code C s defined by s n has the parameters [n, p, q].
(3) When q ≡ 1(mod r) and q ≡ 1(mod 4), l(mod n) ∈ D (n) 0 and d i (x) ∈ F l [x] for i � 0, 1 by Lemma 14. Hence, S(ζ n ) ∈ 0, − 1 { }. e linear complexity of s n is L s � ((p + 1)(q − 1)/2), and s n has the minimal polynomial sequences in our paper have high linear complexity. Notably, our construction can generate more sequences with new parameters and high linear complexity. Furthermore, inspired by the idea of [7], we constructed some cyclic codes by virtue of special classes of sequences. Our results show that several cyclic codes are optimal, such as Example 2.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e author declares no conflicts of interest.