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

[1]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[2]

Wenxun Xing, Feng Chen. A-shaped bin packing: Worst case analysis via simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 323-335. doi: 10.3934/jimo.2005.1.323
[3]

Shinji Imahori, Yoshiyuki Karuno, Kenju Tateishi. Pseudo-polynomial time algorithms for combinatorial food mixture packing problems. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1057-1073. doi: 10.3934/jimo.2016.12.1057
[4]

Chuanxin Zhao, Lin Jiang, Kok Lay Teo. A hybrid chaos firefly algorithm for three-dimensional irregular packing problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 409-429. doi: 10.3934/jimo.2018160
[5]

Mao Chen, Xiangyang Tang, Zhizhong Zeng, Sanya Liu. An efficient heuristic algorithm for two-dimensional rectangular packing problem with central rectangle. Journal of Industrial & Management Optimization, 2020, 16 (1) : 495-510. doi: 10.3934/jimo.2018164
[6]

Cuiling Fan, Koji Momihara. Unified combinatorial constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2014, 8 (1) : 53-66. doi: 10.3934/amc.2014.8.53
[7]

T. L. Alderson, K. E. Mellinger. Geometric constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2008, 2 (4) : 451-467. doi: 10.3934/amc.2008.2.451
[8]

Boris P. Belinskiy. Optimal design of an optical length of a rod with the given mass. Conference Publications, 2007, 2007 (Special) : 85-91. doi: 10.3934/proc.2007.2007.85
[9]

Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018
[10]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017
[11]

Gerardo Vega, Jesús E. Cuén-Ramos. The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020059
[12]

Leetika Kathuria, Madhu Raka. Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless. Advances in Mathematics of Communications, 2012, 6 (4) : 499-503. doi: 10.3934/amc.2012.6.499
[13]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[14]

Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41
[15]

Hai Huyen Dam, Kok Lay Teo. Variable fractional delay filter design with discrete coefficients. Journal of Industrial & Management Optimization, 2016, 12 (3) : 819-831. doi: 10.3934/jimo.2016.12.819
[16]

Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[17]

Lanqiang Li, Shixin Zhu, Li Liu. The weight distribution of a class of p-ary cyclic codes and their applications. Advances in Mathematics of Communications, 2019, 13 (1) : 137-156. doi: 10.3934/amc.2019008
[18]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[19]

Fatmanur Gursoy, Irfan Siap, Bahattin Yildiz. Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$. Advances in Mathematics of Communications, 2014, 8 (3) : 313-322. doi: 10.3934/amc.2014.8.313
[20]

Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences