# American Institute of Mathematical Sciences

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

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

 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 $v$.
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

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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. 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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