Constructions of Optimal Optical Orthogonal Codes with Weights Set <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M1" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 35.4241 15.2545" width="35.4241pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-124" d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z"/></g><g transform="matrix(.017,0,0,-0.017,5.955,0)"><path id="g113-54" d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"/></g><g transform="matrix(.017,0,0,-0.017,14.191,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,20.98,0)"><path id="g113-56" d="M447 623H65C61 580 56 530 47 475H76C100 541 106 550 172 550H388C308 376 196 170 91 -1L98 -12L172 -2C268 204 360 408 455 611L447 623Z"/></g><g transform="matrix(.017,0,0,-0.017,29.221,0)"><path id="g113-126" d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"/></g></svg> via Cyclic Packing

Huang, Bichang; Luo, Dan; Zhu, Wenxing

doi:https://doi.org/10.1155/2022/3368661

Mathematical Problems in Engineering

On this page

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

Research Article | Open Access

Volume 2022 | Article ID 3368661 | https://doi.org/10.1155/2022/3368661

Constructions of Optimal Optical Orthogonal Codes with Weights Set via Cyclic Packing

Bichang Huang,¹Dan Luo,¹and Wenxing Zhu²

Academic Editor: Fazal M. Mahomed

Received04 Jul 2021

Revised25 Nov 2021

Accepted01 Dec 2021

Published03 Jan 2022

Abstract

Double-weight optical orthogonal codes are variable-weight optical orthogonal codes (OOCs), which have been widely applied in optical networks and systems. Some works have been devoted to optimal -OOCs with . So far, there is no explicit construction of optimal -OOCs with . It is known that heavier-weight codewords have better code performance than lighter-weight codewords. So, in this paper, we use cyclic packing to construct two infinite classes of optimal OOCs with weights set explicitly, for any prime and . In addition, for , by breaking blocks of size 7 into 3 of -OOCs and -OOCs, we obtain new infinite classes of optimal -OOCs and -OOCs, respectively.

1. Introduction

In 1989, Salehi [1] defined the constant-weight optical orthogonal codes (OOCs), and applied them in the optical code division multiple access (OCDMA) network. Later on, Yang et al. [2] introduced the variable-weight optical orthogonal codes (variable-weight OOCs) in order to meet the requirements of differential quality-of-service (QoS). That is, the codewords of low code weight are assigned to the low-QoS (quality-of-services) requirement applications and high code weight codewords are assigned to high-QoS requirement applications. Because of its desirable features, OOCs have found wide range of applications in many communications fields, such as mobile radio, frequency-hopping spread-spectrum communications, radar, sonar signal design, and so on [3–7].

In [8], suppose is an ordering of a set of integers greater than 1 and is an -tuple (autocorrelation sequence) of positive integers. Moreover, suppose that is a positive integer (cross-correlation parameter) and is an -tuple (weight distribution sequence) of positive rational numbers whose sum is 1. Then, an optical orthogonal code (OOC) (briefly, -OOC) is a set of subsets (called codeword-sets) of with sizes (weights) from satisfying the following three properties:(1)Weight distribution property: the number of codewords with weight is exactly , , and , where (2)The autocorrelation property: for any with weight and (3)The cross-correlation property: for any , with and

If , then the notation -OOC is used to denote -OOC. is normalized if it is written in the form with . And, an -OOC is said to be balanced if .

If and , an -OOC is a conflict-avoiding code [9].

It is well known that constructing the optimal OOCs with and minimal is preferred due to its correlation property. Namely, the smaller the value of the correlation is, the smaller the interference is.

The following lemma provides an upper bound on the size of an -OOC.

Lemma 1. ([8]) If is an -OOC with and normalized weight distribution sequence , then we have

An -OOC is optimal if reaches the upper bound.

2-CP is a generalization of 2- packing. It is a useful tool for constructing -OOC. The details of the concept of 2-CP were introduced in [10].

Let be an Abelian group, and assume that , , and . Define , , and , where , , and are multisets.

Let be a family of subsets of , , . A 2-CP (W,1,n) is (base blocks set), such that the differences in , i.e., , cover each element of at most once. A 2-CP is a 2-CP with the property that the number of blocks of size is , where , .

A 2-CP is g-regular if covers each element of exactly once, and each element of is not covered.

The following results are the relationships between a 2-CP and an -OOC.

Lemma 2. An optimal 2- is equivalent to an optimal -OOC.

Lemma 3. Let , where . If , then a -regular 2-CP is optimal.

Recently, many researchers have tried to construct OOCs via various methods, such as the search methods using the greedy strategy, the outer-product matrix algorithm [11], projective geometry [12], the finite field theory [13, 14], the combinational design theory [15–21], and so on. In particular, researchers have obtained many optimal OOCs with and by constructing 2-CP s. For the case of constant-weight OOCs, see [22–24] for some of the examples. For the case of double-weight OOCs, some works have been done on , see [10, 25–27] for some of the examples.

For the case of double-weight optimal OOCs with and , little work has been done. The representative articles are [28–31]. So far, as the authors are aware, no explicit result of OOCs with has been obtained. Based on the results of performance analysis in and [32], it is easy to see that heavier-weight codewords have better code performance than lighter-weight codewords due to higher autocorrelation peaks. So, in this paper, our aim is to present explicit results of optimal OOCs with .

In [8], Buratti et al. presented many explicit results of optimal OOCs with and by using elementary facts of the quadratic residues and cyclic packing. In this paper, continue using the construction method in [8] and further elementary conclusions of the quadratic residues, two new infinite classes of OOCs with are presented. The main results are as follows.

Theorem 4. For any prime and , there exist an optimal 2-CP and an optimal -OOC.

Theorem 5. For any prime and , there exist an optimal 2-CP and an optimal -OOC.

An outline of this paper is as follows. In Section 2, the basic knowledge of the quadratic residues is introduced. Theorem 4 and Theorem 5 are proved in Section 3 and Section 4, respectively, Conclusions are presented in Section 5.

2. Preliminaries

For a fixed prime , let be a primitive element of , and denote the quadratic residues and the quadratic nonresidues of , respectively.

For any , , , define(1)(2)

Suppose that and each element of is greater than 1. Assume that . The difference lists are defined as follows:

, .

In order to prove the main results, we need to present the elementary facts of numbers of in . The interested readers may refer to [33] for the details.

Lemma 6. Suppose that is a prime, we have(1), if and only if (2), if and only if (3), if and only if (4), if and only if (5), if and only if (6), if and only if (7), if and only if (8), if and only if (9), if and only if (10), if and only if (11), if and only if (12), if and only if (13), if and only if (14), if and only if (15), if and only if (16), if and only if

3. Proof of Theorem 4

In this section, Theorem 4 will be proved by considering the prime in fourteen cases, where explicit constructions are presented, respectively. According to the Chinese Remainder Theorem, for any and , since , there exists a one-to-one and onto mapping from to .

3.1. The Cases of ≡ 71, 191, 239, 359, 431, 599, 311, 479, 551, 671, 719, 839

Lemma 7. Suppose that ≡ 71, 191, 239, 359, 431, 599, 311, 479, 551, 671, 719, 839 is a prime, and . Then, forms a 2-CP , where

Proof. We compute the difference lists , , from and . It is not difficult to see that , . So, we only need to compute the difference lists for as follows:
, , , , , , , , , , and .
According to (1) and (2) of Lemma 6, we know that , , , and . It is easy to check that , , . Hence, covers every element in exactly once, and each element of is not covered. So, forms a 2-CP .

3.2. The Cases of ≡ 43, 67, 163, 403, 547, 667, 187, 283, 307, 523, 643, 787

Lemma 8. Suppose that 43, 67, 163, 403, 547, 667, 187, 283, 307, 523, 643, 787 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , and .
According to (3) and (4) of Lemma 6, we know that , , and . It is easy to check that forms a 2-CP .

3.3. The Cases of ≡ 23, 263, 407, 527, 743, 767, 47, 143, 167, 383, 503, 647

Lemma 9. Suppose that 23, 263, 407, 527, 743, 767, 47, 143, 167, 383, 503, 647 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , and .
According to (5) and (6) of Lemma 6, we know that , , and . It is easy to check that forms a 2-CP .

3.4. The Cases of ≡ 211, 331, 379, 499, 571, 739

Lemma 10. Suppose that ≡ 211, 331, 379, 499, 571, 739 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , and .
According to (7) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.5. The Cases of ≡ 19, 139, 451, 619, 691, 811

Lemma 11. Suppose that ≡ 19, 139, 451, 619, 691, 811 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , and .
According to (8) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.6. The Cases of ≡ 11, 179, 491, 611, 659, 779

Lemma 12. Suppose that ≡ 11, 179, 491, 611, 659, 779 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , and .
According to (13) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.7. The Cases of ≡ 59, 131, 251, 299, 419, 731

Lemma 13. Suppose that ≡ 59, 131, 251, 299, 419, 731 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for from as follows:
, , , , , , , , , , , and .
According to (14) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.8. The Cases of ≡ 127, 247, 463, 487, 583, 823, 103, 223, 367, 607, 703, 727

Lemma 14. Suppose that ≡ 127, 247, 463, 487, 583, 823, 103, 223, 367, 607, 703, 727 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , and .
According to (15) and (16) of Lemma 6, we know that , , and . It is easy to check that forms a 2-CP .

3.9. The Cases of ≡ 79, 151, 319, 631, 751, 799

Lemma 15. Suppose that ≡ 79, 151, 319, 631, 751, 799 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , and .
According to (9) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.10. The Cases of ≡ 31, 199, 271, 391, 439, 559

Lemma 16. Suppose that ≡ 31, 199, 271, 391, 439, 559 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , and .
According to (10) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.11. The Cases of ≡ 107, 323, 347, 443, 683, 827

Lemma 17. Suppose that ≡ 107, 323, 347, 443, 683, 827 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , and .
According to (11) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.12. The Cases of ≡ 83, 227, 467, 563, 587, 803

Lemma 18. Suppose that ≡ 83, 227, 467, 563, 587, 803 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , and .
According to (12) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .
We are now in a position to prove Theorem 4.

Proof of Theorem 4. For each prime , , and , we obtain a 2-CP from Lemmas 7–18. Let ,where and . Then, forms a 2-CP for .
By Lemmas 2 and 3, we obtain an optimal 2-CP and an optimal -OOC for any prime and . This completes the proof.

4. Proof of Theorem 5

In this section, we prove Theorem 5 by considering fourteen cases, and explicit constructions are presented, respectively.

For any , since , according to the Chinese Remainder Theorem, there exists a one-to-one and onto mapping from to .

4.1. The Cases of ≡ 71, 191, 239, 359, 431, 599, 311, 479, 551, 671, 719, 839

Lemma 19. Suppose that ≡ 71, 191, 239, 359, 431, 599, 311, 479, 551, 671, 719, 839 is a prime, . Then, forms a 2-CP where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , and .
According to (1) and (2) of Lemma 6, , , , and . It is easy to check that , , . Hence, covers every element in exactly once, and each element of is not covered. So, forms a 2-CP .

4.2. The Cases of ≡ 43, 67, 163, 403, 547, 667, 187, 283, 307, 523, 643, 787

Lemma 20. Suppose that ≡ 43, 67, 163, 403, 547, 667, 187, 283, 307, 523, 643, 787 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , and .
According to (3) and (4) of Lemma 6, we know that , , and . It is easy to check that forms a 2-CP .

4.3. The Cases of ≡ 23, 263, 407, 527, 743, 767, 47, 143, 167, 383, 503, 647

Lemma 21. Suppose that ≡ 23, 263, 407, 527, 743, 767, 47, 143, 167, 383, 503, 647 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , and .
According to (5) and (6) of Lemma 6, we know that , , and . It is easy to check that forms a 2-CP .

4.4. The Cases of ≡ 211, 331, 379, 499, 571, 739

Lemma 22. Suppose that ≡ 211, 331, 379, 499, 571, 739 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , , and .
According to (7) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.5. The Cases of ≡ 19, 139, 451, 619, 691, 811

Lemma 23. Suppose that ≡ 19, 139, 451, 619, 691, 811 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , , and .
According to (8) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.6. The Cases of ≡ 11, 179, 491, 611, 659, 779

Lemma 24. Suppose that ≡ 11, 179, 491, 611, 659, 779 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , and .
According to (13) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.7. The Cases of ≡ 59, 131, 251, 299, 419, 731

Lemma 25. Suppose that ≡ 59, 131, 251, 299, 419, 731 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , and .
According to (14) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.8. The Cases of ≡ 127, 247, 463, 487, 583, 823

Lemma 26. Suppose that ≡ 127, 247, 463, 487, 583, 823 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , , , , and .
According to (15) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.9. The Cases of ≡ 103, 223, 367, 607, 703, 727

Lemma 27. Suppose that ≡ 103, 223, 367, 607, 703, 727 is a prime. Then, forms a 2-CP where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , , and .
According to (16) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.10. The Cases of ≡ 79, 151, 319, 631, 751, 799

Lemma 28. Suppose that ≡ 79, 151, 319, 631, 751, 799 is a prime. Then, forms a 2-CP where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , and .
According to (9) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.11. The Cases of ≡ 31, 199, 271, 391, 439, 559

Lemma 29. Suppose that ≡ 31, 199, 271, 391, 439, 559 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , , and .
According to (10) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.12. The Cases of ≡ 107, 323, 347, 443, 683, 827

Lemma 30. Suppose that ≡ 107, 323, 347, 443, 683, 827 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , , , , , and .
According to (11) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

4.13. The Cases of ≡ 83, 227, 467, 563, 587, 803

Lemma 31. Suppose that ≡ 83, 227, 467, 563, 587, 803 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from , , and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:

, , , , , , , , , , , , , , , , , and .

According to (12) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

We are now in a position to prove Theorem 5.

Proof of Theorem 5. For each prime and , we obtain a 2-CP from Lemmas 19–31. Let .
Then, forms a 2-CP for .
By Lemmas 2 and 3, we obtain an optimal 2-CP and an optimal -OOC for any prime and . This completes the proof.

5. Conclusions

In this paper, by constructing explicit 2-CP , we have obtained the explicit constructions of optimal -OOCs and -OOCs, for any prime and . In addition, referring to [34, 35], for , if we break blocks of size 7 into 3 of -OOCs and -OOCs, then we will get new infinite classes of optimal -OOCs and -OOCs, respectively.

Data Availability

The underlying data supporting the results of our study can be found in the proof of every theorem.

Disclosure

Dan Luo is the co-first author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Guangxi Science Foundation (2018GXNSFAA281259).

References

J. A. Salehi, “Code division multiple-access techniques in optical fiber networks. I. Fundamental principles,” IEEE Transactions on Communications, vol. 37, no. 8, pp. 824–833, 1989.
View at: Publisher Site | Google Scholar
G. C. Yang, “Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements,” IEEE Transactions on Communications, vol. 44, no. 1, pp. 47–55, 1996.
View at: Publisher Site | Google Scholar
F. R. K. Chung, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes: design, analysis and applications,” IEEE Transactions on Information Theory, vol. 35, no. 3, pp. 595–604, 1989.
View at: Publisher Site | Google Scholar
K. Fouli and M. Maier, “OCDMA and optical coding: principles, applications, and Challenges [Topics in optical communications],” IEEE Communications Magazine, vol. 45, no. 8, pp. 27–34, 2007.
View at: Publisher Site | Google Scholar
P. R. Prucnal, Optical Code Division Multiple Access: Fundamental and Applications, CRC Press, Taylor & Francis, 2006.
J. A. Salehi, “Emerging optical code-division multiple access communication systems,” IEEE Network, vol. 3, no. 2, pp. 31–39, 1989.
View at: Publisher Site | Google Scholar
G. C. Yang and W. C. Kwong, Prime Codes with Applications to CDMA Optical and Wireless Networks, Artech House, Norwood, MA. United States, 2002.
M. Buratti, Y. Wei, D. Wu, P. Fan, and M. Cheng, “Relative difference families with variable block sizes and their related OOCs,” IEEE Transactions on Information Theory, vol. 57, no. 11, pp. 7489–7497, 2011.
View at: Publisher Site | Google Scholar
T. Baicheva and S. Topalova, “Optimal conflict-avoiding codes for 3, 4 and 5 active users,” Problems of Information Transmission, vol. 53, no. 1, pp. 42–50, 2017.
View at: Publisher Site | Google Scholar
D. Wu, H. Zhao, P. Fan, and S. Shinohara, “Optimal variable-weight optical orthogonal codes via difference packings,” IEEE Transactions on Information Theory, vol. 56, no. 8, pp. 4053–4060, 2010.
View at: Publisher Site | Google Scholar
H. Charmchi and J. A. Salehi, “Outer-product matrix representation of optical orthogonal codes,” IEEE Transactions on Communications, vol. 54, no. 6, pp. 983–989, 2006.
View at: Publisher Site | Google Scholar
N. Miyamoto, H. Mizuno, and S. Shinohara, “Optical orthogonal codes obtained from conics on finite projective planes,” Finite Fields and Their Applications, vol. 10, no. 3, pp. 405–411, 2004.
View at: Publisher Site | Google Scholar
H. Chung and P. V. Kumar, “Optical orthogonal codes-New bounds and an optimal construction,” IEEE Transactions on Information Theory, vol. 36, no. 4, pp. 866–873, 1990.
View at: Publisher Site | Google Scholar
C. S. Chi-Shun Weng and J. Jingshown Wu, “Optical orthogonal codes with nonideal cross correlation,” Journal of Lightwave Technology, vol. 19, no. 12, pp. 1856–1863, 2001.
View at: Publisher Site | Google Scholar
R. J. R. Abel and M. Buratti, “Some progress on (v,4,1) difference families and optical orthogonal codes,” Journal of Combinatorial Theory - Series A, vol. 106, no. 1, pp. 59–75, 2004.
View at: Publisher Site | Google Scholar
S. Bitan and T. Etzion, “Constructions for optimal constant weight cyclically permutable codes and difference families,” IEEE Transactions on Information Theory, vol. 41, no. 1, pp. 77–87, 1995.
View at: Publisher Site | Google Scholar
M. Buratti, “Cyclic designs with block size 4 and related optimal optical orthogonal codes, Des,” Designs, Codes and Cryptography, vol. 26, no. 1/3, pp. 111–125, 2002.
View at: Publisher Site | Google Scholar
Y. Chang, R. Fuji-Hara, and Y. Miao, “Combinatorial constructions of optimal optical orthogonal codes with weight 4,” IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1283–1292, 2003.
View at: Google Scholar
Y. Chang and L. Ji, “Optimal (4up, 5, 1) optical orthogonal codes,” Journal of Combinatorial Designs, vol. 12, no. 5, pp. 346–361, 2004.
View at: Publisher Site | Google Scholar
Y. Chang and Y. Miao, “Constructions for optimal optical orthogonal codes,” Discrete Mathematics, vol. 261, no. 1-3, pp. 127–139, 2003.
View at: Publisher Site | Google Scholar
K. Chen, G. Ge, and L. Zhu, “Starters and related codes,” Journal of Statistical Planning and Inference, vol. 86, no. 2, pp. 379–395, 2000.
View at: Publisher Site | Google Scholar
R. Fuji-Hara, Y. Miao, and J. Yin, “Optimal (9v, 4, 1) optical orthogonal codes,” SIAM Journal on Discrete Mathematics, vol. 14, no. 2, pp. 256–266, 2001.
View at: Publisher Site | Google Scholar
G. Ge and J. Yin, “Constructions for optimal (υ, 4, 1) optical orthogonal codes,” IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 2998–3004, 2001.
View at: Publisher Site | Google Scholar
S. Ma and Y. Chang, “A new class of optimal optical orthogonal codes with weight five,” IEEE Transactions on Information Theory, vol. 50, no. 8, pp. 1848–1850, 2004.
View at: Publisher Site | Google Scholar
J. Yin, “Some combinatorial constructions for optical orthogonal codes,” Discrete Mathematics, vol. 185, no. 1-3, pp. 201–219, 1998.
View at: Publisher Site | Google Scholar
H. Zhao, D. Wu, and P. Fan, “Constructions of optimal variable-weight optical orthogonal codes,” Journal of Combinatorial Designs, vol. 18, no. 4, pp. 274–291, 2010.
View at: Publisher Site | Google Scholar
X. Zhong, D. Wu, and P. Fan, “New infinite classes of optimal (v,{k,6},1,Q) optical orthogonal codes via quadratic residues,” IEICE - Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E95-A, no. 11, pp. 1827–1834, 2012.
View at: Publisher Site | Google Scholar
K. Chen, R. Wei, and L. Zhu, “Existence of (q, 7, 1) difference families with q a prime power,” Journal of Combinatorial Designs, vol. 10, no. 2, pp. 126–138, 2002.
View at: Publisher Site | Google Scholar
J. H. Chung and K. Yang, “A new class of optimal variable-weight optical orthogonal codes,” in Proceedings of the International Workshop on Coding and Cryptography, pp. 285–293, WCC, Bergen (Norway), April 2013.
View at: Google Scholar
J. Jiang, D. Wu, and P. Fan, “General constructions of optimal variable-weight optical orthogonal codes,” IEEE Transactions on Information Theory, vol. 57, no. 7, pp. 4488–4496, 2011.
View at: Publisher Site | Google Scholar
G.-C. Yang, “Some new families of optical orthogonal codes for code-division multiple-access fibre-optic networks,” IEE Proceedings - Communications, vol. 142, no. 6, pp. 363–368, 1995.
View at: Publisher Site | Google Scholar
V. Baby, W. C. Kwong, C.-Y. Chang, G.-C. Yang, and P. R. Prucnal, “Performance analysis of variable-weight multilength optical codes for wavelength-time O-CDMA multimedia systems,” IEEE Transactions on Communications, vol. 55, no. 7, pp. 1325–1333, 2007.
View at: Publisher Site | Google Scholar
M. B. Nathanson, Elementary Methods in Number Theory, Springer-Verlag, NewYork, 2000.
H. Zhao, D. Wu, and Z. Mo, “Further results on optimal (v,{3,k},1,{1/2,1/2}),” Discrete Mathematics, vol. 311, pp. 16–23, 2011.
View at: Publisher Site | Google Scholar
D. Wu, J. Cao, and P. Fan, “New optimal variable-weight optical orthogonal codes,” Sequences and Their Applications - SETA 2010, vol. 6338, pp. 102–112, 2010.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Bichang Huang et al. 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