May  2019, 13(2): 297-312. doi: 10.3934/amc.2019020

Further results on optimal $ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $-OOCs

1. 

Guangxi Key Lab of Multi-source Information Mining & Security, Department of Mathematics, Guangxi Normal University, Guilin 541004, China

2. 

Xingjian College of Science and Liberal Arts, Guangxi University, Nanning 530004, China

* Corresponding author: Dianhua Wu

Received  May 2018 Revised  August 2018 Published  February 2019

Fund Project: The first author is supported in part by NSFC (No. 11801103) and Guangxi Nature Science Foundation (No. 2017GXNSFBA198030). The third author is supported in part by NSFC (No. 11671103), Program on the High Level Innovation Team and Outstanding Scholars in Universities of Guangxi Province, Foundation of Guangxi Key Lab of Multi-Source Information Mining and Security (No. 18-A-03-01). The last author is supported in part by Guangxi Nature Science Foundation (No. 2018GXNSFA138038)

Let $ W = \{w_1, w_2, \cdots, w_r\} $ be a set of $ r $ integers greater than 1, $ \Lambda_a = (\lambda_a^{(1)}, \lambda_a^{(2)}, \cdots, \lambda_a^{(r)}) $ be an $ r $-tuple of positive integers, $ \lambda_c $ be a positive integer, and $ Q = (q_1, q_2, \cdots, q_r) $ be an $ r $-tuple of positive rational numbers whose sum is 1. Variable-weight optical orthogonal code ($ (n, W, \Lambda_a, \lambda_c, Q) $-OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service requirements. In this paper, tight upper bounds on the maximum code size of $ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $-OOCs are obtained, and infinite classes of optimal $ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $-OOCs are constructed.

Citation: Huangsheng Yu, Feifei Xie, Dianhua Wu, Hengming Zhao. Further results on optimal $ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $-OOCs. Advances in Mathematics of Communications, 2019, 13 (2) : 297-312. doi: 10.3934/amc.2019020
References:
[1]

R. J. R. Abel and M. Buratti, Some progress on $(v, 4, 1)$ difference families and optical orthogonal codes, J. Combin. Theory, 106 (2004), 59-75. doi: 10.1016/j.jcta.2004.01.003. Google Scholar

[2]

T. Baicheva and S. Topalova, Optimal $(v, 4, 2, 1)$ optical orthogonal codes with small parameters, J. Combin. Des., 20 (2012), 142-160. doi: 10.1002/jcd.20296. Google Scholar

[3]

E. F. Brickell and V. Wei, Optical orthogonal codes and cyclic block designs, Congr. Numer., 58 (1987), 175-182. Google Scholar

[4]

M. BurattiK. Momihara and A. Pasotti, New results on optimal $(v, 4, 2, 1)$ optical orthogonal codes, Des. Codes Cryptogr., 58 (2011), 89-109. doi: 10.1007/s10623-010-9382-z. Google Scholar

[5]

M. Buratti and A. Pasotti, Further progress on difference families with block size $4$ or $5$, Des. Codes Cryptogr., 56 (2010), 1-20. doi: 10.1007/s10623-009-9335-6. Google Scholar

[6]

M. BurattiA. Passotti and D. Wu, On optimal $(v, 5, 2, 1)$ optical orthogonal codes, Des. Codes Cryptogr., 68 (2013), 349-371. doi: 10.1007/s10623-012-9654-x. Google Scholar

[7]

M. BurattiY. WeiD. WuP. Fan and M. Cheng, Relative difference families with variable block sizes and their related OOCs, IEEE Trans. Inform. Theory, 57 (2011), 7489-7497. doi: 10.1109/TIT.2011.2162225. Google Scholar

[8]

Y. Chang and L. Ji, Optimal $(4up, 5, 1)$ optical orthogonal codes, J. Combin. Des., 12 (2004), 346-361. doi: 10.1002/jcd.20011. Google Scholar

[9]

Y. Chang and Y. Miao, Constructions for optimal optical orthogonal codes, Discr. Math., 261 (2003), 127-139. doi: 10.1016/S0012-365X(02)00464-8. Google Scholar

[10]

K. ChenG. Ge and L. Zhu, Starters and related codes, J. Statist. Plann. Inference, 86 (2000), 379-395. doi: 10.1016/S0378-3758(99)00119-6. Google Scholar

[11]

C. J. ColbournJ. H. Dinitz and D. R. Stinson, Applications of Combinatorial Designs to Communications, Cryptography, and Networking, London Math. Soc. Lecture Note Ser., 267 (1999), 37-100. Google Scholar

[12]

J. H. Dinitz and D. R. Stinson, Room squares and related designs, In: Contemporary Design Theory, Wiley, New York, (1992), 137–204. Google Scholar

[13]

R. Fuji-Hara and Y. Miao, Optical orthogonal codes: Their bounds and new optimal constructions, IEEE Trans. Inform. Theory, 46 (2000), 2396-2406. doi: 10.1109/18.887852. Google Scholar

[14]

G. Ge and J. Yin, Constructions for optimal $(v, 4, 1)$ optical orthogonal codes, IEEE Trans. Inform. Theory, 47 (2001), 2998-3004. doi: 10.1109/18.959278. Google Scholar

[15]

B. HuangY. Wei and D. Wu, Bounds and constructions for optimal variable-weight OOCs with unequal auto- and cross-correlation constraints, Utilitas Math., 103 (2017), 3-21. Google Scholar

[16]

W. LiH. Yu and D. Wu, Bounds and constructions for optimal $(n, \{3, 5\}$, $\Lambda_a, 1, Q)$-OOCs, Discr. Math., 339 (2016), 21-32. doi: 10.1016/j.disc.2015.07.006. Google Scholar

[17]

K. Momihara and M. Buratti, Bounds and Constructions of Optimal $(n, 4, 2, 1)$ Optical Orthogonal Codes, IEEE Trans. Inform. Theory, 55 (2009), 514-523. doi: 10.1109/TIT.2008.2009852. Google Scholar

[18]

P. R. PrucnalM. A. Santoro and T. R. Fan, Spread spectrum fiberoptic local network using optical processing, IEEE J. Lightwave Technol., LT-4 (1986), 547-554. Google Scholar

[19]

J. A. Salehi, Code division multiple access techniques in optical fiber networks-Part Ⅰ Fundamental Principles, IEEE Trans. Commun., 37 (1989), 824-833. Google Scholar

[20]

J. A. Salehi, Emerging optical code-division multiple-access communications systems, IEEE Network, 3 (1989), 31-39. doi: 10.1109/65.21908. Google Scholar

[21]

S. TamuraS. Nakano and K. Okazaki, Optical codemultiplex transmission by gold sequences, IEEE J. Lightwave Technol., LT-3 (1985), 121-127. Google Scholar

[22]

X. Wang and Y. Chang, Further results on optimal $(v, 4, 2, 1)$-OOCs, Discr. Math., 312 (2012), 331-340. doi: 10.1016/j.disc.2011.09.025. Google Scholar

[23]

D. WuH. ZhaoP. Fan and S. Shinohara, Optimal variable-weight optical orthogonal codes via difference packings, IEEE Trans. Inform. Theory, 56 (2010), 4053-4060. doi: 10.1109/TIT.2010.2050927. Google Scholar

[24]

G. C. Yang, Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements, IEEE Trans. Commun., 44 (1996), 47-55. Google Scholar

[25]

J. Yin, Some combinatorial constructions for optical orthogonal codes, Discr. Math., 185 (1998), 201-219. doi: 10.1016/S0012-365X(97)00172-6. Google Scholar

[26]

H. YuS. Dang and D. Wu, Bounds and constructions for optimal $(n, \{3, 4, 5\}, \Lambda_a, 1, Q)$-OOCs, IEEE Trans. Inform. Theory, 64 (2018), 1361-1367. doi: 10.1109/TIT.2017.2739778. Google Scholar

[27]

H. ZhaoD. Wu and P. Fan, Constructions of optimal variable-weight optical orthogonal codes, J. Combin. Des., 18 (2010), 274-291. doi: 10.1002/jcd.20246. Google Scholar

[28]

H. ZhaoD. Wu and R. Qin, Further results on $(v, \{3, 4\}, \Lambda_a, 1, Q)$-OOCs, Discr. Math., 337 (2014), 87-96. doi: 10.1016/j.disc.2014.08.003. Google Scholar

show all references

References:
[1]

R. J. R. Abel and M. Buratti, Some progress on $(v, 4, 1)$ difference families and optical orthogonal codes, J. Combin. Theory, 106 (2004), 59-75. doi: 10.1016/j.jcta.2004.01.003. Google Scholar

[2]

T. Baicheva and S. Topalova, Optimal $(v, 4, 2, 1)$ optical orthogonal codes with small parameters, J. Combin. Des., 20 (2012), 142-160. doi: 10.1002/jcd.20296. Google Scholar

[3]

E. F. Brickell and V. Wei, Optical orthogonal codes and cyclic block designs, Congr. Numer., 58 (1987), 175-182. Google Scholar

[4]

M. BurattiK. Momihara and A. Pasotti, New results on optimal $(v, 4, 2, 1)$ optical orthogonal codes, Des. Codes Cryptogr., 58 (2011), 89-109. doi: 10.1007/s10623-010-9382-z. Google Scholar

[5]

M. Buratti and A. Pasotti, Further progress on difference families with block size $4$ or $5$, Des. Codes Cryptogr., 56 (2010), 1-20. doi: 10.1007/s10623-009-9335-6. Google Scholar

[6]

M. BurattiA. Passotti and D. Wu, On optimal $(v, 5, 2, 1)$ optical orthogonal codes, Des. Codes Cryptogr., 68 (2013), 349-371. doi: 10.1007/s10623-012-9654-x. Google Scholar

[7]

M. BurattiY. WeiD. WuP. Fan and M. Cheng, Relative difference families with variable block sizes and their related OOCs, IEEE Trans. Inform. Theory, 57 (2011), 7489-7497. doi: 10.1109/TIT.2011.2162225. Google Scholar

[8]

Y. Chang and L. Ji, Optimal $(4up, 5, 1)$ optical orthogonal codes, J. Combin. Des., 12 (2004), 346-361. doi: 10.1002/jcd.20011. Google Scholar

[9]

Y. Chang and Y. Miao, Constructions for optimal optical orthogonal codes, Discr. Math., 261 (2003), 127-139. doi: 10.1016/S0012-365X(02)00464-8. Google Scholar

[10]

K. ChenG. Ge and L. Zhu, Starters and related codes, J. Statist. Plann. Inference, 86 (2000), 379-395. doi: 10.1016/S0378-3758(99)00119-6. Google Scholar

[11]

C. J. ColbournJ. H. Dinitz and D. R. Stinson, Applications of Combinatorial Designs to Communications, Cryptography, and Networking, London Math. Soc. Lecture Note Ser., 267 (1999), 37-100. Google Scholar

[12]

J. H. Dinitz and D. R. Stinson, Room squares and related designs, In: Contemporary Design Theory, Wiley, New York, (1992), 137–204. Google Scholar

[13]

R. Fuji-Hara and Y. Miao, Optical orthogonal codes: Their bounds and new optimal constructions, IEEE Trans. Inform. Theory, 46 (2000), 2396-2406. doi: 10.1109/18.887852. Google Scholar

[14]

G. Ge and J. Yin, Constructions for optimal $(v, 4, 1)$ optical orthogonal codes, IEEE Trans. Inform. Theory, 47 (2001), 2998-3004. doi: 10.1109/18.959278. Google Scholar

[15]

B. HuangY. Wei and D. Wu, Bounds and constructions for optimal variable-weight OOCs with unequal auto- and cross-correlation constraints, Utilitas Math., 103 (2017), 3-21. Google Scholar

[16]

W. LiH. Yu and D. Wu, Bounds and constructions for optimal $(n, \{3, 5\}$, $\Lambda_a, 1, Q)$-OOCs, Discr. Math., 339 (2016), 21-32. doi: 10.1016/j.disc.2015.07.006. Google Scholar

[17]

K. Momihara and M. Buratti, Bounds and Constructions of Optimal $(n, 4, 2, 1)$ Optical Orthogonal Codes, IEEE Trans. Inform. Theory, 55 (2009), 514-523. doi: 10.1109/TIT.2008.2009852. Google Scholar

[18]

P. R. PrucnalM. A. Santoro and T. R. Fan, Spread spectrum fiberoptic local network using optical processing, IEEE J. Lightwave Technol., LT-4 (1986), 547-554. Google Scholar

[19]

J. A. Salehi, Code division multiple access techniques in optical fiber networks-Part Ⅰ Fundamental Principles, IEEE Trans. Commun., 37 (1989), 824-833. Google Scholar

[20]

J. A. Salehi, Emerging optical code-division multiple-access communications systems, IEEE Network, 3 (1989), 31-39. doi: 10.1109/65.21908. Google Scholar

[21]

S. TamuraS. Nakano and K. Okazaki, Optical codemultiplex transmission by gold sequences, IEEE J. Lightwave Technol., LT-3 (1985), 121-127. Google Scholar

[22]

X. Wang and Y. Chang, Further results on optimal $(v, 4, 2, 1)$-OOCs, Discr. Math., 312 (2012), 331-340. doi: 10.1016/j.disc.2011.09.025. Google Scholar

[23]

D. WuH. ZhaoP. Fan and S. Shinohara, Optimal variable-weight optical orthogonal codes via difference packings, IEEE Trans. Inform. Theory, 56 (2010), 4053-4060. doi: 10.1109/TIT.2010.2050927. Google Scholar

[24]

G. C. Yang, Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements, IEEE Trans. Commun., 44 (1996), 47-55. Google Scholar

[25]

J. Yin, Some combinatorial constructions for optical orthogonal codes, Discr. Math., 185 (1998), 201-219. doi: 10.1016/S0012-365X(97)00172-6. Google Scholar

[26]

H. YuS. Dang and D. Wu, Bounds and constructions for optimal $(n, \{3, 4, 5\}, \Lambda_a, 1, Q)$-OOCs, IEEE Trans. Inform. Theory, 64 (2018), 1361-1367. doi: 10.1109/TIT.2017.2739778. Google Scholar

[27]

H. ZhaoD. Wu and P. Fan, Constructions of optimal variable-weight optical orthogonal codes, J. Combin. Des., 18 (2010), 274-291. doi: 10.1002/jcd.20246. Google Scholar

[28]

H. ZhaoD. Wu and R. Qin, Further results on $(v, \{3, 4\}, \Lambda_a, 1, Q)$-OOCs, Discr. Math., 337 (2014), 87-96. doi: 10.1016/j.disc.2014.08.003. Google Scholar

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