American Institute of Mathematical Sciences

Advanced
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

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.  doi: 10.1016/j.jcta.2004.01.003.  Google Scholar

[2]

M. Buratti, Cyclic designs with block size $4$ and related optimal optical orthogonal codes,, Des. Codes Cryptogr., 26 (2002), 111.  doi: 10.1023/A:1016505309092.  Google Scholar

[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.  doi: 10.1109/TIT.2011.2162225.  Google Scholar

[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.  doi: 10.1109/TIT.2003.810628.  Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1109/18.30982.  Google Scholar

[8]

H. Chung and P.V. Kumar, Optical orthogonal codes-new bounds and an optimal construction,, IEEE Trans. Inform. Theory, 36 (1990), 866.  doi: 10.1109/18.53748.  Google Scholar

[9]

J. H. Dinitz, Starters,, in The CRC Handbook of Combinatorial Designs, (2006), 622.   Google Scholar

[10]

J. H. Dinitz and D. R. Stinson, Room squares and related designs,, in Contemp. Des. Theory Wiley, (1992), 137.   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

S. W. Golomb, Digital Communication with Space Application,, Penisula, (1982).   Google Scholar

[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.   Google Scholar

[16]

J. Jiang, D. Wu and P. Fan, General constructions of optimal variable-weight optical orthogonal codes,, IEEE Trans. Inform. Theory, 57 (2011), 4488.  doi: 10.1109/TIT.2011.2146110.  Google Scholar

[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.  doi: 10.1007/s00373-012-1235-2.  Google Scholar

[18]

S. Ma and Y. Chang, Constructions of optimal optical orthogonal codes with weight five,, J. Combin. Des., 13 (2005), 54.  doi: 10.1002/jcd.20022.  Google Scholar

[19]

J. L. Massey and P. Mathys, The collision channel without feedback,, IEEE Trans. Inform. Theory, 31 (1985), 192.  doi: 10.1109/TIT.1985.1057010.  Google Scholar

[20]

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

[21]

J. A. Salehi, Emerging optical code-division multiple-access communications systems,, IEEE Netw., 3 (1989), 31.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1109/TIT.2010.2050927.  Google Scholar

[25]

G. C. Yang, Variable weight optical orthogonal codes for CDMA networks with multiple performance requirements,, in GLOBECOM '93., (1993), 488.   Google Scholar

[26]

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

[27]

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

[28]

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

[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.  doi: 10.1016/j.disc.2010.09.012.  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.  doi: 10.1016/j.jcta.2004.01.003.  Google Scholar

[2]

M. Buratti, Cyclic designs with block size $4$ and related optimal optical orthogonal codes,, Des. Codes Cryptogr., 26 (2002), 111.  doi: 10.1023/A:1016505309092.  Google Scholar

[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.  doi: 10.1109/TIT.2011.2162225.  Google Scholar

[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.  doi: 10.1109/TIT.2003.810628.  Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1109/18.30982.  Google Scholar

[8]

H. Chung and P.V. Kumar, Optical orthogonal codes-new bounds and an optimal construction,, IEEE Trans. Inform. Theory, 36 (1990), 866.  doi: 10.1109/18.53748.  Google Scholar

[9]

J. H. Dinitz, Starters,, in The CRC Handbook of Combinatorial Designs, (2006), 622.   Google Scholar

[10]

J. H. Dinitz and D. R. Stinson, Room squares and related designs,, in Contemp. Des. Theory Wiley, (1992), 137.   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

S. W. Golomb, Digital Communication with Space Application,, Penisula, (1982).   Google Scholar

[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.   Google Scholar

[16]

J. Jiang, D. Wu and P. Fan, General constructions of optimal variable-weight optical orthogonal codes,, IEEE Trans. Inform. Theory, 57 (2011), 4488.  doi: 10.1109/TIT.2011.2146110.  Google Scholar

[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.  doi: 10.1007/s00373-012-1235-2.  Google Scholar

[18]

S. Ma and Y. Chang, Constructions of optimal optical orthogonal codes with weight five,, J. Combin. Des., 13 (2005), 54.  doi: 10.1002/jcd.20022.  Google Scholar

[19]

J. L. Massey and P. Mathys, The collision channel without feedback,, IEEE Trans. Inform. Theory, 31 (1985), 192.  doi: 10.1109/TIT.1985.1057010.  Google Scholar

[20]

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

[21]

J. A. Salehi, Emerging optical code-division multiple-access communications systems,, IEEE Netw., 3 (1989), 31.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1109/TIT.2010.2050927.  Google Scholar

[25]

G. C. Yang, Variable weight optical orthogonal codes for CDMA networks with multiple performance requirements,, in GLOBECOM '93., (1993), 488.   Google Scholar

[26]

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

[27]

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

[28]

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

[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.  doi: 10.1016/j.disc.2010.09.012.  Google Scholar

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