New optimal (v, \{3,5\}, 1, Q) optical orthogonal codes

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November 2016, 10(4): 811-823. doi: 10.3934/amc.2016042

New optimal $(v, \{3,5\}, 1, Q)$ optical orthogonal codes

Huangsheng Yu ^1, , Dianhua Wu ^2, and Jinhua Wang ^3,

1.	Guangxi Key Lab of Multi-source Information Mining & Security, Guilin 541004, China
2.	Department of Mathematics, Guangxi Normal University, Guilin, Guangxi 541004
3.	School of Sciences, Nantong University, Nantong, Jiangsu 226007

Received January 2015 Revised January 2016 Published November 2016

Abstract
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Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. It is proved that optimal $(v, \{3, 5\}, 1, (1/2, 1/2))$-OOCs exist for some complete congruence classes of $v$. In this paper, for $Q\in \{(2/3,1/3), (3/4,1/4)\}$, by using skew starters, it is also proved that optimal $(v, \{3,5\}, 1, Q)$-OOCs exist for some complete congruence classes of $v$.

Keywords: variable-weight OOC., packing design, Optical orthogonal code, skew starter, Cyclic packing.

Mathematics Subject Classification: Primary: 05B40; Secondary: 94C3.

Citation: Huangsheng Yu, Dianhua Wu, Jinhua Wang. New optimal $(v, \{3,5\}, 1, Q)$ optical orthogonal codes. Advances in Mathematics of Communications, 2016, 10 (4) : 811-823. doi: 10.3934/amc.2016042

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.
[2]	M. Buratti, Cyclic designs with block size $4$ and related optimal optical orthogonal codes, Des. Codes Cryptogr., 26 (2002), 111-125. doi: 10.1023/A:1016505309092.
[3]	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.
[4]	Y. Chang, R. Fuji-Hara and Y. Miao, Combinatorial constructions of optimal optical orthogonal codes with weight $4$, IEEE Trans. Inform. Theory, 49 (2003), 1283-1292. doi: 10.1109/TIT.2003.810628.
[5]	Y. Chang and L. Ji, Optimal $(4up, 5, 1)$ optical orthogonal codes, J. Combin. Des., 12 (2004), 346-361. doi: 10.1002/jcd.20011.
[6]	K. Chen, G. Ge and L. Zhu, Starters and related codes, J. Statist. Plann. Inference, 86 (2000), 379-395. doi: 10.1016/S0378-3758(99)00119-6.
[7]	F. R. K. Chung, J. A. Salehi and V. K. Wei, Optical orthogonal codes: Design, analysis, and applications, IEEE Trans. Inform. Theory, 35 (1989), 595-604. doi: 10.1109/18.30982.
[8]	H. Chung and P.V. Kumar, Optical orthogonal codes-new bounds and an optimal construction, IEEE Trans. Inform. Theory, 36 (1990), 866-873. doi: 10.1109/18.53748.
[9]	J. H. Dinitz, Starters, in The CRC Handbook of Combinatorial Designs, Chapman and Hall/CRC, Boca Raton, 2006, 622-628.
[10]	J. H. Dinitz and D. R. Stinson, Room squares and related designs, in Contemp. Des. Theory Wiley, New York, 1992, 137-204.
[11]	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.
[12]	G. Ge, On $(g,4;1)$-diffference matrices, Discrete Math., 301 (2005), 164-174. doi: 10.1016/j.disc.2005.07.004.
[13]	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.
[14]	S. W. Golomb, Digital Communication with Space Application, Penisula, Los Altos, 1982.
[15]	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.
[16]	J. Jiang, D. Wu and P. Fan, General constructions of optimal variable-weight optical orthogonal codes, IEEE Trans. Inform. Theory, 57 (2011), 4488-4496. doi: 10.1109/TIT.2011.2146110.
[17]	J. Jiang, D. Wu and M. H. Lee, Some infinite classes of optimal $(v, \{3,4\}, 1, Q)$-OOCs with $Q\in \{(1/3, 2/3), (2/3, 1/3)\}$, Graphs Combin., 29 (2013), 1795-1812. doi: 10.1007/s00373-012-1235-2.
[18]	S. Ma and Y. Chang, Constructions of optimal optical orthogonal codes with weight five, J. Combin. Des., 13 (2005), 54-69. doi: 10.1002/jcd.20022.
[19]	J. L. Massey and P. Mathys, The collision channel without feedback, IEEE Trans. Inform. Theory, 31 (1985), 192-204. doi: 10.1109/TIT.1985.1057010.
[20]	J. A. Salehi, Code division multiple access techniques in optical fiber networks-Part I. Fundamental Principles, IEEE Trans. Commun., {37} (1989), 824-833.
[21]	J. A. Salehi, Emerging optical code-division multiple-access communications systems, IEEE Netw., 3 (1989), 31-39.
[22]	J. A. Salehi and C. A. Brackett, Code division multiple access techniques in optical fiber networks-Part II. Systems performance analysis, IEEE Trans. Commun., 37 (1989), 834-842.
[23]	M. P. Vecchi and J. A. Salehi, Neuromorphic networks based on sparse optical orthogonal codes, in Neural Inform. Proc. Systems-Natural Synth., 1988, 814-823.
[24]	D. Wu, H. Zhao, P. 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.
[25]	G. C. Yang, Variable weight optical orthogonal codes for CDMA networks with multiple performance requirements, in GLOBECOM '93., IEEE, 1993, 488-492.
[26]	G. C. Yang, Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements, IEEE Trans. Commun., 44 (1996), 47-55.
[27]	J. Yin, Some combinatorial constructions for optical orthogonal codes, Discrete Math., 185 (1998), 201-219. doi: 10.1016/S0012-365X(97)00172-6.
[28]	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.
[29]	H. Zhao, D. Wu and Z. Mo, Further results on optimal $(v,\{3,k\},1,\{1/2,$ $1/2\})$-OOCs for $k=4,5$, Discrete Math., 311 (2011), 16-23. doi: 10.1016/j.disc.2010.09.012.

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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.
[2]	M. Buratti, Cyclic designs with block size $4$ and related optimal optical orthogonal codes, Des. Codes Cryptogr., 26 (2002), 111-125. doi: 10.1023/A:1016505309092.
[3]	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.
[4]	Y. Chang, R. Fuji-Hara and Y. Miao, Combinatorial constructions of optimal optical orthogonal codes with weight $4$, IEEE Trans. Inform. Theory, 49 (2003), 1283-1292. doi: 10.1109/TIT.2003.810628.
[5]	Y. Chang and L. Ji, Optimal $(4up, 5, 1)$ optical orthogonal codes, J. Combin. Des., 12 (2004), 346-361. doi: 10.1002/jcd.20011.
[6]	K. Chen, G. Ge and L. Zhu, Starters and related codes, J. Statist. Plann. Inference, 86 (2000), 379-395. doi: 10.1016/S0378-3758(99)00119-6.
[7]	F. R. K. Chung, J. A. Salehi and V. K. Wei, Optical orthogonal codes: Design, analysis, and applications, IEEE Trans. Inform. Theory, 35 (1989), 595-604. doi: 10.1109/18.30982.
[8]	H. Chung and P.V. Kumar, Optical orthogonal codes-new bounds and an optimal construction, IEEE Trans. Inform. Theory, 36 (1990), 866-873. doi: 10.1109/18.53748.
[9]	J. H. Dinitz, Starters, in The CRC Handbook of Combinatorial Designs, Chapman and Hall/CRC, Boca Raton, 2006, 622-628.
[10]	J. H. Dinitz and D. R. Stinson, Room squares and related designs, in Contemp. Des. Theory Wiley, New York, 1992, 137-204.
[11]	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.
[12]	G. Ge, On $(g,4;1)$-diffference matrices, Discrete Math., 301 (2005), 164-174. doi: 10.1016/j.disc.2005.07.004.
[13]	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.
[14]	S. W. Golomb, Digital Communication with Space Application, Penisula, Los Altos, 1982.
[15]	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.
[16]	J. Jiang, D. Wu and P. Fan, General constructions of optimal variable-weight optical orthogonal codes, IEEE Trans. Inform. Theory, 57 (2011), 4488-4496. doi: 10.1109/TIT.2011.2146110.
[17]	J. Jiang, D. Wu and M. H. Lee, Some infinite classes of optimal $(v, \{3,4\}, 1, Q)$-OOCs with $Q\in \{(1/3, 2/3), (2/3, 1/3)\}$, Graphs Combin., 29 (2013), 1795-1812. doi: 10.1007/s00373-012-1235-2.
[18]	S. Ma and Y. Chang, Constructions of optimal optical orthogonal codes with weight five, J. Combin. Des., 13 (2005), 54-69. doi: 10.1002/jcd.20022.
[19]	J. L. Massey and P. Mathys, The collision channel without feedback, IEEE Trans. Inform. Theory, 31 (1985), 192-204. doi: 10.1109/TIT.1985.1057010.
[20]	J. A. Salehi, Code division multiple access techniques in optical fiber networks-Part I. Fundamental Principles, IEEE Trans. Commun., {37} (1989), 824-833.
[21]	J. A. Salehi, Emerging optical code-division multiple-access communications systems, IEEE Netw., 3 (1989), 31-39.
[22]	J. A. Salehi and C. A. Brackett, Code division multiple access techniques in optical fiber networks-Part II. Systems performance analysis, IEEE Trans. Commun., 37 (1989), 834-842.
[23]	M. P. Vecchi and J. A. Salehi, Neuromorphic networks based on sparse optical orthogonal codes, in Neural Inform. Proc. Systems-Natural Synth., 1988, 814-823.
[24]	D. Wu, H. Zhao, P. 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.
[25]	G. C. Yang, Variable weight optical orthogonal codes for CDMA networks with multiple performance requirements, in GLOBECOM '93., IEEE, 1993, 488-492.
[26]	G. C. Yang, Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements, IEEE Trans. Commun., 44 (1996), 47-55.
[27]	J. Yin, Some combinatorial constructions for optical orthogonal codes, Discrete Math., 185 (1998), 201-219. doi: 10.1016/S0012-365X(97)00172-6.
[28]	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.
[29]	H. Zhao, D. Wu and Z. Mo, Further results on optimal $(v,\{3,k\},1,\{1/2,$ $1/2\})$-OOCs for $k=4,5$, Discrete Math., 311 (2011), 16-23. doi: 10.1016/j.disc.2010.09.012.

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