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

Zhou, Xia

doi:https://doi.org/10.1155/2020/6625652

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Fuzzy Approximation and its Applications in Stabilization of Dynamic Systems

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 6625652 | https://doi.org/10.1155/2020/6625652

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

Xia Zhou¹

Academic Editor: Heng Liu

Received06 Nov 2020

Revised23 Nov 2020

Accepted04 Dec 2020

Published23 Dec 2020

Abstract

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

1. Introduction

Throughout this paper, assume that is a power of a prime number . A linear [n,k,d] code over is defined as a -dimensional subspace of with minimum distance , where is a finite field with order and denotes the -dimensional linear space over . If each if and only if , then is named as a cyclic code. Let and the ring . Hence, there is a one-to-one correspondence between a linear code over with length and a real subset of . If an ideal is not trivial, there is a monic polynomial over , satisfying , where is unique and . Then, and are called to be the generator polynomial and parity-check polynomial of , respectively. Please refer to [1] for more details on cyclic codes.

Let be a sequence with period over and . At present, the methods of constructing cyclic codes are substantial. One of the important methods is to employwhere is named as the generator polynomial of .

As is known to all, there are a good deal of results on cyclic codes in a series of papers [2–11]. Let and be two distinct prime numbers. In order to search for more residue difference sets, Whiteman [12] introduced a generalized cyclotomy regarding . 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.

The main contributions of this article are as follows: firstly, we proved that there are only three classes of generalized cyclotomies with order two over , 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 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 over , where and , and computed their minimal polynomials and linear complexities. The 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 and , see Remark 4. Particularly, if , these sequences are new. Thirdly, we employed these sequences to produce some families of cyclic codes. The 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.

2. Preliminaries

2.1. Minimal Polynomial and Linear Complexity

Suppose that the sequence , where and . The sequence is called to be linear feedback sequence, if there are constants , satisfying

It is widely known that such a positive integer for any finite sequence always exists. Here, the linear complexity of the sequence is defined as the minimal positive integer and the feedback polynomial (or characteristic polynomial) of is defined as the polynomial . Furthermore, the minimal polynomial of 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 , its minimal polynomial is given byand its linear complexity is determined by

2.2. Classical Cyclotomy with Order Two

Suppose that is an integer and is an odd prime number. Then, there exists a finite field with order . Furthermore, one can always find an element such that , where is the set of nonzero elements of . Define the cyclotomic classes with order two in :

For given integers and , , the cyclotomic number with order two regarding is defined as

Now, we will recall the properties of the classical cyclotomy with order two in [17] as follows.

Lemma 2. Suppose that is an odd prime number. Then,

Let and denote an algebraic closure of a finite field . Define and aswhere is a -th primitive root of unity; and are the cyclotomic classes with order two regarding .

When , over are identified by the following lemma.

Lemma 3 (see [18]). Let the symbols be the same as before. Then,where is the Legendre symbol.

2.3. The General Two-Prime Generalized Cyclotomic Sequence of Order Two

Suppose that is the residue class ring module and , where is a positive integer. Let denote the Euler function. For , if the multiplicative order of modulo is , then is named as a primitive root modulo .

Definition 1. A partition of is a family of sets, satisfyingIf there exist a subgroup of and of , satisfying , , then the are called to be classical cyclotomic classes with order if is a prime number, and generalized cyclotomic classes with order if is a composite number. The (generalized) cyclotomic numbers with order are defined as

Lemma 4. There are only three classes of generalized cyclotomies with order two regarding .

Proof. Let be a fixed multiplicative subgroup of with order . So, there exists such that . Obviously, and . LetWe next show that is a multiplicative subgroup of . It is straightforward to prove that is a multiplicative subgroup of . We only need to show that . For any , there exists such that . If , then . If , then there exists such that and . Hence, is a multiplicative subgroup of with order and there exists such that . Finally, . Therefore, and the corresponding . This finishes the proof of Lemma 4.
Suppose that is a common primitive root modulo and . Let be a positive integer satisfyingAssume that , where . Therefore, the multiplicative group of is as follows [12]:According to Lemma 4, we easily verify the assertion as the following.

Lemma 5. There are only three classes of generalized cyclotomies with order two over as follows. The first generalized cyclotomic classes with order two are defined as

The second generalized cyclotomic classes with order two are defined asand the third generalized cyclotomic classes with order two are defined aswhere the multiplications are operated modulo .

Remark 1. By definition, when , the first generalized cyclotomy with order two over is exactly Whiteman’s generalized cyclotomy [12]. In essence, it is in accord with the one introduced by Ding [7]. When , the second generalized cyclotomy with order two over is identical to Ding-Helleseth’s generalized cyclotomy [2] in the case of . Furthermore, this cyclotomy is the same as the extended generalized cyclotomy with order two presented by Wang and Lin [16]. For fixed , , and defined by (14), the third generalized cyclotomy is new. In [7], the linear complexity and minimal polynomial of generalized cyclotomic sequence with period over 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.
The generalized cyclotomic numbers with order two are defined byDefineThen,

Definition 2. The two-prime general generalized cyclotomic sequences (GGCS) of order two are defined bywhere 0 and 1 are the zero element and identity element, respectively, in .

Remark 2. In [16], 0 and 1 in equation (22) are both in . Hence, our sequence and the sequence defined by Wang and Lin are totally different.

3. A Family of New Cyclic Codes Defined by Our Sequences

3.1. The Properties on the Generalized Cyclotomy with Order Two Over

In this subsection, the following lemmas follow from [16].

Lemma 6. When , . When , .

Lemma 7. If , thenIf , then

Lemma 8.

Lemma 9. When ,When ,

3.2. The Parameters of the Code , Minimal Polynomial, and Linear Complexity of the Sequence

Let and be two difference odd prime numbers with and be a power of a prime number . We always assume that and is the order of modulo . Define

Our main objective in this section is to compute the generator polynomialof the cyclic code defined by the sequence , where is defined as equation (28). With this purpose, we need to seek out the , satisfying , where is a fixed -th primitive root of unity of the finite field . The following auxiliary results are important for our calculation. Obviously, we have

Lemma 10.

Proof. Obviously, , since is a common primitive root modulo and , and . Furthermore, one hasHence, takes on every element of exactly times, when ranges among and ranges among . If , it follows from equations (30) and (32) thatSimilarly, we haveIf , it follows thatHence, we get the conclusions that, for and ,and for and ,

Lemma 11.

Proof. If , then the conclusion is straightforward.
If , then and . By equation (30), we getIf , we have and , since . So, we haveIf , then . So by Lemma 10,If , then . By equation (30) and Lemma 10, one has

Lemma 12. (1) If , then and thus ; (2) if , then .

Proof. (1)If , then by Lemma 11, we have Hence, .(2)If , then by Lemma 11, we have similarlyUnder this assumption, if is even, and may be in if is odd.

Lemma 13. When , we haveWhen , we have

Proof. According to equation (30) and the definition of , one hasThen, we getSuppose that . According to Lemma 6, and . It follows from Lemmas 2, 7, 8, and 9 thatCombining equations (48) and (49), one can get the first conclusion of this lemma.
Suppose that now. According to Lemma 6, and . We have similarlyCombining equations (48) and (50), we can arrive at the second conclusion of this lemma.

Lemma 14. (1) and , if with ; (2) and , if with .

Proof. Obviously, if . Now, we only give the proof of the first conclusion since the proof of the second part is similar to it. Let with .
Firstly, we consider the case that . Therefore, and . Suppose that on the contrary. Then, 2 can be written as for and . According to the definition of and , it follows that . Therefore, 2 is not a quadratic residue modulo . The contradiction proves the conclusion for the case .
Secondly, when is an odd prime number, we prove this lemma. Since , . Hence, . Now, one hasSuppose that on the contrary. By definition of , for and . Then, we have . Therefore, is not a quadratic residue modulo . This is contrary to equation (69). So, we arrive at the conclusion.
In order to calculate the minimal polynomial and linear complexity of , we need to study the factorization of over . It can be checked that the are -th roots of unity and are -th roots of unity. Therefore,For any , definewhere denotes the general generalized cyclotomic classes with order two. If , then it is obvious that for any .
Let . After the above preparations, one has

Theorem 1. Assume that . We have the assertions as follows:(1)When , the linear complexity of is , and the cyclic code defined by has the parameters , whose generator polynomial is .(2)When , the linear complexity of is , and the cyclic code defined by has the parameters , whose generator polynomial is .(3)When , the linear complexity of is , and the cyclic code defined by has the parameters , whose generator polynomial is(4)When , the linear complexity of is , and the cyclic code defined by has the parameters , whose generator polynomial is

Proof. From , we get . According to Lemma 11, one hasNow, we only prove Case 1, since the others are similar.
For Case 1, by Lemma 11,Therefore, and . Then, the linear complexity of the sequence is . Furthermore, the definition of the code can lead to the parameters of the code .

Example 1. Assume that . Then, , , the minimal polynomial is , and the linear complexity of is 12. The cycle code defined by is a cyclic code over . According to the database [19], the best binary linear code known with the parameters has minimum distance 8.

Example 2. Assume that . In this case, , , the minimal polynomial is , and the linear complexity of is 14. The cycle code defined by is a cyclic code over , which is an optimal linear code according to the database [19].

Remark 3. In [7], example 3.20 presents a cyclic code over , whose generator polynomial is . This is bad because of its poor minimum distance. Obviously, our cyclic code is more optimal.

Example 3. Assume that . Then, , , the minimal polynomial is , and the linear complexity of is 60. The cycle code defined by is a cyclic code over .

Theorem 2. Assume that and .(1)When and , or and , the linear complexity of is , and has the minimal polynomial .(2)When and , the linear complexity of is , and has the minimal polynomial . The cyclic code defined by has the parameters .(3)When and , and for by Lemma 14. Hence, . The linear complexity of is , and has the minimal polynomial The cyclic code defined by has the parameters .(4)When and , and for by Lemma 14. Hence, . The linear complexity of is , and has the minimal polynomialThe cyclic code defined by has the parameters .

Proof. According to Lemma 11 and , one has . Now, we only prove Case 3, since the others are similar.
For Case 3, if , by Lemma 11,Therefore, . If , by Lemma 11,Therefore, . Then, the linear complexity of is . Moreover, the definition of the cyclic code can lead to the parameters of .

Example 4. Assume that . Then, , and , the minimal polynomial is , and the linear complexity of is 47. The cycle code defined by is a cyclic code over .

Lemma 15 (see [15]. Let . If and , then .
DefineIf , then it is straightforward to prove that for . Therefore, when and , one has by Lemma 15.

Theorem 3. Assume that .(1)When , , and or , , and , the linear complexity of is , and has the minimal polynomial . The parameters of defined by are .(2)When , , and or , , and , the linear complexity of is and has the minimal polynomial The cyclic code defined by has the parameters .(3)When and , the linear complexity of is and has the minimal polynomial The cyclic code defined by has the parameters .(4)When and , the linear complexity of is , and has the minimal polynomial The cyclic code defined by has the parameters , where the minimum weight follows from Theorem 5.(5)When , , and , the linear complexity of is , and has the minimal polynomial The cyclic code defined by has the parameters .(6)When , , and , the linear complexity of is , and has the minimal polynomial The cyclic code defined by has the parameters .

Proof. From , we get that is an odd prime. DefineWe will use the following property, which is easily seen to hold.

Property 1. if and only if ; or if and only if . If , then .

Property 2. and cannot exist simultaneously, since and .

Property 3. or if and only if .

Property 4. , if and .

Property 5. , if and , or and .
Now, we only prove Case 3, since the others are similar. One has and , since and . Hence, and . According to Properties 1–5, if , by Lemma 11,Therefore, . If , by Lemma 11,Hence, . If , by Lemma 11,Therefore, . Then, and the definition of the cyclic code can lead to the parameters of .

Example 5. Assume that . Then, , , , , , the minimal polynomial is , and the linear complexity of is 30. The cycle code defined by is a cyclic code over .

Remark 4. Theorems 1, 2, and 3 show that the linear complexity of the two-prime GGCSs defined by equation (22) is very high. Without the limit of , this paper can obtain some binary sequences with new parameters and higher linear complexity. Furthermore, when , Theorem 1 in our paper is the same with Theorem 1 in [14]. When , Theorem 2 in our paper is in accord with Theorems 13 and 16 in [15], and Theorem 3 in our paper is exactly identical with Theorems 14 and 18 in [15]. However, if is not equal to 2, the conclusions in Theorems 1, 2, and 3 of our paper are new, which can be seen from Example 3. Therefore, the sequences defined by equation (22) will be more attractive than those in [14, 15] in some cryptographic applications.

4. The Minimum Distance of the Cycle Codes Obtained in This Paper

Theorem 4 (see [7]). If the generator polynomial of over is , then has parameters . Also, if the generator polynomial of over is , then has parameters .

Theorem 5. Let . Let be the cyclic code over generated by the polynomial

Assume that is the minimum distance of this code. The parameters of the cycle code arewhere

If , we have

Proof. Assume that , whose Hamming weight is . Take any . Then, is a code word in , whose Hamming weight is also . Therefore, we have .
Suppose that , whose Hamming weight is minimum. Then, is a code word in , whose Hamming weight is also minimum. Furthermore, for any , is a code word in the cyclic codewhose generator polynomial is and minimum distance is (see Theorem 4). Hence, we have thatif .

Example 6. Assume that . In this case, , the cyclic code generated by the polynomial has parameters . The actual minimum distance is 10, but the lower bound of (77) is 2.

5. Conclusion

In this paper, we studied the uniform representation of general generalized cyclotomy with order two over . Based on the general generalized cyclotomy, the general two-prime generalized cyclotomic sequences were proposed. The 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

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This work was supported by the Foundation of Nanjing Institute of Technology (no. QKJ201804).

References

L. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, UK, 1997.
C. Ding and T. Helleseth, “New generalized cyclotomy and its applications,” Finite Fields and their Applications, vol. 4, no. 2, pp. 140–166, 1998.
View at: Publisher Site | Google Scholar
C. Ding and V. Pless, “Cyclotomy and duadic codes of prime lengths,” IEEE Transactions on Information Theory, vol. 45, no. 2, pp. 453–466, 1999.
View at: Publisher Site | Google Scholar
C. Ding and T. Helleseth, “Generalized cyclotomic codes of length ,” IEEE Transactions on Information Theory, vol. 45, no. 2, pp. 467–474, 1999.
View at: Google Scholar
E. Betti and M. Sala, “A new bound for the minimum distance of a cyclic code from its defining set,” IEEE Transactions on Information Theory, vol. 52, no. 8, pp. 3700–3706, 2006.
View at: Publisher Site | Google Scholar
C. Ding, “Cyclotomic constructions of cyclic codes with length being the product of two primes,” IEEE Transactions on Information Theory, vol. 58, no. 4, pp. 2231–2236, 2012.
View at: Publisher Site | Google Scholar
C. Ding, “Cyclic codes from the two-prime sequences,” IEEE Transactions on Information Theory, vol. 58, no. 6, pp. 3881–3891, 2012.
View at: Publisher Site | Google Scholar
C. Ding, “Cyclic codes from cyclotomic sequences of order four,” Finite Fields and Their Applications, vol. 23, pp. 8–34, 2013.
View at: Publisher Site | Google Scholar
Y. Sun, T. Yan, and H. Li, “Cyclic code from the first class Whiteman’s generalized cyclotomic sequence with order 4,” 2013, http://arxiv.org/abs/1303.6378.
View at: Google Scholar
C. Ding, X. Du, and Z. Zhou, “The bose and minimum distance of a class of BCH codes,” IEEE Transactions on Information Theory, vol. 61, no. 5, pp. 2351–2356, 2015.
View at: Google Scholar
P. Kewat and P. Kumari, “Cyclic codes from the first class two-prime Whiteman’s generalized cyclotomic sequence with order 6,” 2015, http://arxiv.org/abs/1509.07714.
View at: Google Scholar
A. L. Whiteman, “A family of difference sets,” Illinois Journal of Mathematics, vol. 6, no. 6, pp. 107–121, 1962.
View at: Publisher Site | Google Scholar
C. Ding, “Linear complexity of generalized cyclotomic binary sequences of order 2,” Finite Fields and Their Applications, vol. 3, no. 2, pp. 159–174, 1997.
View at: Publisher Site | Google Scholar
S. Li, Z. Chen, R. Sun, and G. Xiao, “On the randomness of generalized cyclotomic sequences of order two and length pq,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E90-A, no. 9, pp. 2037–2041, 2007.
View at: Publisher Site | Google Scholar
Q. Wang, Y. Jiang, and D. Lin, “Linear complexity of Ding-Helleseth sequences of order 2 over GF(l),” Cryptography and Communications, vol. 8, no. 1, pp. 33–49, 2016.
View at: Publisher Site | Google Scholar
Q. Wang and D. Lin, “Generalized cyclotomic numbers of order two and their applications,” Cryptography and Communications, vol. 8, no. 4, pp. 605–616, 2016.
View at: Publisher Site | Google Scholar
T. Storer, Cyclotomy and Difference Sets, Markham Publishing Company, Markham, Canada, 1967.
Q. Wang, Y. Jiang, and D. Lin, “Linear complexity of binary generalized cyclotomic sequences over GF(q),” Journal of Complexity, vol. 31, no. 5, pp. 731–740, 2015.
View at: Publisher Site | Google Scholar
M. Grassl, “Bounds on the minimum distance of linear codes,” http://www.codetables.de.
View at: Google Scholar

Copyright

Copyright © 2020 Xia Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

506

Downloads

652

Citations