American Institute of Mathematical Sciences

Advanced
May  2019, 13(2): 329-341. doi: 10.3934/amc.2019022

Constructions of optimal balanced $ (m, n, \{4, 5\}, 1) $-OOSPCs

Wei Li 1, , Hengming Zhao 2, , Rongcun Qin 2, and  Dianhua Wu 3,,

1. 

Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba 305-8573, Japan
2. 

Xingjian College of Science and Liberal Arts, Guangxi University, Nanning 530004, China
3. 

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

* Corresponding author: Dianhua Wu

Received  June 2018 Published  January 2019

Fund Project: The second author is supported in part by Guangxi Nature Science Foundation (No. 2018GXNSFA138038). The third author is supported by the Project of Basic Ability Improvement of Young and Middle-Aged Teachers of Universities in Guangxi (No. 2017KY1301). The last author is supported in part by NSFC (No. 11671103, 11801103), Guangxi Nature Science Foundation (No. 2017GXNSFBA198030), and Foundation of Guangxi Key Lab of Multi-Source Information Mining and Security (No. 18-A-03-01).

Kitayama proposed a novel OCDMA (called spatial CDMA) for parallel transmission of 2-D images through multicore fiber. Optical orthogonal signature pattern codes (OOSPCs) play an important role in this CDMA network. Multiple-weight (MW) optical orthogonal signature pattern code (OOSPC) was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirements. Some results had been done on optimal balanced $ (m, n, \{3, 4\}, 1) $-OOSPCs. In this paper, it is proved that there exist optimal balanced $ (2u, 16v, \{4, 5\}, 1) $-OOSPCs for odd integers $ u\geq 1 $, $ v\geq 1 $.

Keywords: Incomplete difference matrix, optical orthogonal signature pattern code, optimal, packing, quadratic residue.
Mathematics Subject Classification: Primary: 05B40; Secondary: 94C30.
Citation: Wei Li, Hengming Zhao, Rongcun Qin, Dianhua Wu. Constructions of optimal balanced $ (m, n, \{4, 5\}, 1) $-OOSPCs. Advances in Mathematics of Communications, 2019, 13 (2) : 329-341. doi: 10.3934/amc.2019022
References:
[1]

R. J. R. Abel, C. J. Colbourn and J. H. Dinitz, Mutually orthogonal latin squares (MOLS), In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook of Combinatorial Designs, New York: CRC Press, 2007,160–193.

[2]

M. Buratti, A power method for constructing difference families and optimal optical orthogonal codes, Des. Codes Cryptogr., 5 (1995), 13-25.  doi: 10.1007/BF01388501.

[3]

M. Buratti, Recursive constructions for difference matrices and relative difference families, J. Combin. Des., 6 (1998), 165-182. 

[4]

M. Buratti, Y. Wei, D. Wu, P. 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.

[5]

J. Chen, L. Ji and Y. Li, Combinatorial constructions of optimal $(m, n, 4, 2)$ optical orthogonal signature pattern codes, Des. Codes Cryptogr., 86 (2018), 1499-1525.  doi: 10.1007/s10623-017-0409-6.

[6]

J. Chen, L. Ji and Y. Li, New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, Des. Codes Cryptogr., 85 (2017), 299-318.  doi: 10.1007/s10623-016-0310-8.

[7]

C. J. Colbourn, Difference matrices, In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook of Combinatorial Designs, New York: CRC Press, 2007,411–419.

[8]

P. A. Davies and A. A. Shaar, Asynchronous multiplexing for an optical-fibre local area network, Electron. Leu., 19 (1983), 390-392.  doi: 10.1049/el:19830270.

[9]

I. B. Djordjevic, B. Vasic and J. Rorison, Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs, J. Lightw. Technol., 21 (2003), 1850-1856.  doi: 10.1109/JLT.2003.816819.

[10]

G. Ge, On $(g, 4;1)$-diffference matrices, Discrete Math., 301 (2005), 164-174.  doi: 10.1016/j.disc.2005.07.004.

[11]

F. R. Gu and J. Wu, Construction and performance analysis of variable-weight optical orthogonal codes for asynchronous optical CDMA systems, J. Lightw. Technol., 23 (2005), 740-748.  doi: 10.1109/JLT.2004.838880.

[12]

A. A. Hassan, J. E. Hershey and N. A. Riza, Spatial optical CDMA, Perspectives in Spread Spectrum, 459 (1995), 107-125.  doi: 10.1007/978-1-4615-5531-5_5.

[13]

J. Y. Hui, Pattern code modulation and optical decoding a novel code division multiplexing technique for multifiber networks, IEEE J. Select. Areas Commun., 3 (1985), 916-927.  doi: 10.1109/JSAC.1985.1146265.

[14]

L. Ji, B. Ding, X. Wang and G. Ge, Asymptotically optimal optical orthogonal signature pattern codes, IEEE Trans. Inform. Theory, 64 (2018), 5419-5431.  doi: 10.1109/TIT.2017.2787593.

[15]

J. Jiang, D. Wu and M. H. Lee, Some infinte classes of optimal $(v, \{3, 4\}, 1, Q)$-OOCs with $Q\in\{\{1/3, 2/3\}, \{2/3, 1/3\}\}$, Graphs Combin., 29 (2013), 1795-1811.  doi: 10.1007/s00373-012-1235-2.

[16]

K. Kitayama, Novel spatial spread spectrum based fiber optic CDMA networks for image transmission, IEEE J. Select. Areas Commun., 12 (1994), 762-772.  doi: 10.1109/49.286683.

[17]

W. C. Kwong and G. C. Yang, Double-weight signature pattern codes for multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 5 (2001), 203-205.  doi: 10.1109/4234.922760.

[18]

W. Kwong and G. C. Yang, Image transmission in multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 2 (1998), 285-287.  doi: 10.1109/4234.725225.

[19]

R. Pan and Y. Chang, A note on difference matrices over non-cyclic finite abelian groups, Discrete Math., 339 (2016), 822-830.  doi: 10.1016/j.disc.2015.10.028.

[20]

R. Pan and Y. Chang, Combinatorial constructions for maximum optical orthogonal signature pattern codes, Discrete Math., 33 (2013), 2918-2931.  doi: 10.1016/j.disc.2013.09.005.

[21]

R. Pan and Y. Chang, Determination of the sizes of optimal $(m, n, k, \lambda, k-1)$-OOSPCs for $\lambda = k-1, k$, Discrete Math., 313 (2013), 1327-1337.  doi: 10.1016/j.disc.2013.02.019.

[22]

R. Pan and Y. Chang, Further results on optimal $(m, n, 4, 1)$ optical orthogonal signature pattern codes (in Chinese), Sci. Sin. Math., 44 (2014), 1141-1152. 

[23]

R. Pan and Y. Chang, $(m, n, 3, 1)$ optical orthogonal signature pattern codes with maximum possible size, IEEE Trans. Inform. Theorey, 61 (2015), 443-461.

[24]

E. Park, A. J. Mendez and E. M. Garmire, Temporal/spatial optical CDMA networks-design, demonstration, and comparison with temporal networks, IEEE Photon. Technol. Lett., 4 (1992), 1160-1162.  doi: 10.1109/68.163765.

[25]

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

[26]

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

[27]

M. Sawa, Optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, IEEE Trans. Inform. Theorey, 56 (2010), 3613- = 3620. doi: 10.1109/TIT.2010.2048487.

[28]

M. Sawa and S. Kageyama, Optimal optical orthogonal signature pattern codes of weight 3, Biom. Lett., 46 (2009), 89-102. 

[29]

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

[30]

D. Wu, H. Zhao, P. Fan and S. Shinohara, Optimal variable-weight optical orthogonal codes via difference packings, IEEE Trans. Inform. Theorey, 53 (2010), 4053-4060.  doi: 10.1109/TIT.2010.2050927.

[31]

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

[32]

G. C. Yang and W. C. Kwong, Two-dimensional spatial signature patterns, IEEE Trans. Commun., 44 (1996), 184-191. 

[33]

H. Zhao, New optimal $(v, \{4, 5\}, 1, \{1/2, 1/2\})$-OOCs, J. Guangxi Teachers Edu. University, 28 (2011), 17-22. 

[34]

H. Zhao, On balanced optimal $(18u, \{3, 4\}, 1)$ optical orthogonal codes, J. Combin. Des., 20 (2012), 290-303.  doi: 10.1002/jcd.21303.

[35]

H. Zhao and R. Qin, Combinatorial constructions for optimal multiple-weight optical orthogonal signature pattern codes, Discrete Math., 339 (2016), 179-193.  doi: 10.1016/j.disc.2015.08.005.

[36]

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

show all references

References:
[1]

R. J. R. Abel, C. J. Colbourn and J. H. Dinitz, Mutually orthogonal latin squares (MOLS), In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook of Combinatorial Designs, New York: CRC Press, 2007,160–193.

[2]

M. Buratti, A power method for constructing difference families and optimal optical orthogonal codes, Des. Codes Cryptogr., 5 (1995), 13-25.  doi: 10.1007/BF01388501.

[3]

M. Buratti, Recursive constructions for difference matrices and relative difference families, J. Combin. Des., 6 (1998), 165-182. 

[4]

M. Buratti, Y. Wei, D. Wu, P. 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.

[5]

J. Chen, L. Ji and Y. Li, Combinatorial constructions of optimal $(m, n, 4, 2)$ optical orthogonal signature pattern codes, Des. Codes Cryptogr., 86 (2018), 1499-1525.  doi: 10.1007/s10623-017-0409-6.

[6]

J. Chen, L. Ji and Y. Li, New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, Des. Codes Cryptogr., 85 (2017), 299-318.  doi: 10.1007/s10623-016-0310-8.

[7]

C. J. Colbourn, Difference matrices, In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook of Combinatorial Designs, New York: CRC Press, 2007,411–419.

[8]

P. A. Davies and A. A. Shaar, Asynchronous multiplexing for an optical-fibre local area network, Electron. Leu., 19 (1983), 390-392.  doi: 10.1049/el:19830270.

[9]

I. B. Djordjevic, B. Vasic and J. Rorison, Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs, J. Lightw. Technol., 21 (2003), 1850-1856.  doi: 10.1109/JLT.2003.816819.

[10]

G. Ge, On $(g, 4;1)$-diffference matrices, Discrete Math., 301 (2005), 164-174.  doi: 10.1016/j.disc.2005.07.004.

[11]

F. R. Gu and J. Wu, Construction and performance analysis of variable-weight optical orthogonal codes for asynchronous optical CDMA systems, J. Lightw. Technol., 23 (2005), 740-748.  doi: 10.1109/JLT.2004.838880.

[12]

A. A. Hassan, J. E. Hershey and N. A. Riza, Spatial optical CDMA, Perspectives in Spread Spectrum, 459 (1995), 107-125.  doi: 10.1007/978-1-4615-5531-5_5.

[13]

J. Y. Hui, Pattern code modulation and optical decoding a novel code division multiplexing technique for multifiber networks, IEEE J. Select. Areas Commun., 3 (1985), 916-927.  doi: 10.1109/JSAC.1985.1146265.

[14]

L. Ji, B. Ding, X. Wang and G. Ge, Asymptotically optimal optical orthogonal signature pattern codes, IEEE Trans. Inform. Theory, 64 (2018), 5419-5431.  doi: 10.1109/TIT.2017.2787593.

[15]

J. Jiang, D. Wu and M. H. Lee, Some infinte classes of optimal $(v, \{3, 4\}, 1, Q)$-OOCs with $Q\in\{\{1/3, 2/3\}, \{2/3, 1/3\}\}$, Graphs Combin., 29 (2013), 1795-1811.  doi: 10.1007/s00373-012-1235-2.

[16]

K. Kitayama, Novel spatial spread spectrum based fiber optic CDMA networks for image transmission, IEEE J. Select. Areas Commun., 12 (1994), 762-772.  doi: 10.1109/49.286683.

[17]

W. C. Kwong and G. C. Yang, Double-weight signature pattern codes for multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 5 (2001), 203-205.  doi: 10.1109/4234.922760.

[18]

W. Kwong and G. C. Yang, Image transmission in multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 2 (1998), 285-287.  doi: 10.1109/4234.725225.

[19]

R. Pan and Y. Chang, A note on difference matrices over non-cyclic finite abelian groups, Discrete Math., 339 (2016), 822-830.  doi: 10.1016/j.disc.2015.10.028.

[20]

R. Pan and Y. Chang, Combinatorial constructions for maximum optical orthogonal signature pattern codes, Discrete Math., 33 (2013), 2918-2931.  doi: 10.1016/j.disc.2013.09.005.

[21]

R. Pan and Y. Chang, Determination of the sizes of optimal $(m, n, k, \lambda, k-1)$-OOSPCs for $\lambda = k-1, k$, Discrete Math., 313 (2013), 1327-1337.  doi: 10.1016/j.disc.2013.02.019.

[22]

R. Pan and Y. Chang, Further results on optimal $(m, n, 4, 1)$ optical orthogonal signature pattern codes (in Chinese), Sci. Sin. Math., 44 (2014), 1141-1152. 

[23]

R. Pan and Y. Chang, $(m, n, 3, 1)$ optical orthogonal signature pattern codes with maximum possible size, IEEE Trans. Inform. Theorey, 61 (2015), 443-461.

[24]

E. Park, A. J. Mendez and E. M. Garmire, Temporal/spatial optical CDMA networks-design, demonstration, and comparison with temporal networks, IEEE Photon. Technol. Lett., 4 (1992), 1160-1162.  doi: 10.1109/68.163765.

[25]

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

[26]

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

[27]

M. Sawa, Optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, IEEE Trans. Inform. Theorey, 56 (2010), 3613- = 3620. doi: 10.1109/TIT.2010.2048487.

[28]

M. Sawa and S. Kageyama, Optimal optical orthogonal signature pattern codes of weight 3, Biom. Lett., 46 (2009), 89-102. 

[29]

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

[30]

D. Wu, H. Zhao, P. Fan and S. Shinohara, Optimal variable-weight optical orthogonal codes via difference packings, IEEE Trans. Inform. Theorey, 53 (2010), 4053-4060.  doi: 10.1109/TIT.2010.2050927.

[31]

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

[32]

G. C. Yang and W. C. Kwong, Two-dimensional spatial signature patterns, IEEE Trans. Commun., 44 (1996), 184-191. 

[33]

H. Zhao, New optimal $(v, \{4, 5\}, 1, \{1/2, 1/2\})$-OOCs, J. Guangxi Teachers Edu. University, 28 (2011), 17-22. 

[34]

H. Zhao, On balanced optimal $(18u, \{3, 4\}, 1)$ optical orthogonal codes, J. Combin. Des., 20 (2012), 290-303.  doi: 10.1002/jcd.21303.

[35]

H. Zhao and R. Qin, Combinatorial constructions for optimal multiple-weight optical orthogonal signature pattern codes, Discrete Math., 339 (2016), 179-193.  doi: 10.1016/j.disc.2015.08.005.

[36]

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

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