May  2019, 13(2): 297-312. doi: 10.3934/amc.2019020

Further results on optimal $ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $-OOCs

1. 

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

2. 

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

* Corresponding author: Dianhua Wu

Received  May 2018 Revised  August 2018 Published  February 2019

Fund Project: The first author is supported in part by NSFC (No. 11801103) and Guangxi Nature Science Foundation (No. 2017GXNSFBA198030). The third author is supported in part by NSFC (No. 11671103), Program on the High Level Innovation Team and Outstanding Scholars in Universities of Guangxi Province, Foundation of Guangxi Key Lab of Multi-Source Information Mining and Security (No. 18-A-03-01). The last author is supported in part by Guangxi Nature Science Foundation (No. 2018GXNSFA138038)

Let $ W = \{w_1, w_2, \cdots, w_r\} $ be a set of $ r $ integers greater than 1, $ \Lambda_a = (\lambda_a^{(1)}, \lambda_a^{(2)}, \cdots, \lambda_a^{(r)}) $ be an $ r $-tuple of positive integers, $ \lambda_c $ be a positive integer, and $ Q = (q_1, q_2, \cdots, q_r) $ be an $ r $-tuple of positive rational numbers whose sum is 1. Variable-weight optical orthogonal code ($ (n, W, \Lambda_a, \lambda_c, Q) $-OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service requirements. In this paper, tight upper bounds on the maximum code size of $ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $-OOCs are obtained, and infinite classes of optimal $ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $-OOCs are constructed.

Citation: Huangsheng Yu, Feifei Xie, Dianhua Wu, Hengming Zhao. Further results on optimal $ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $-OOCs. Advances in Mathematics of Communications, 2019, 13 (2) : 297-312. doi: 10.3934/amc.2019020
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]

T. Baicheva and S. Topalova, Optimal $(v, 4, 2, 1)$ optical orthogonal codes with small parameters, J. Combin. Des., 20 (2012), 142-160. doi: 10.1002/jcd.20296.

[3]

E. F. Brickell and V. Wei, Optical orthogonal codes and cyclic block designs, Congr. Numer., 58 (1987), 175-182.

[4]

M. BurattiK. Momihara and A. Pasotti, New results on optimal $(v, 4, 2, 1)$ optical orthogonal codes, Des. Codes Cryptogr., 58 (2011), 89-109. doi: 10.1007/s10623-010-9382-z.

[5]

M. Buratti and A. Pasotti, Further progress on difference families with block size $4$ or $5$, Des. Codes Cryptogr., 56 (2010), 1-20. doi: 10.1007/s10623-009-9335-6.

[6]

M. BurattiA. Passotti and D. Wu, On optimal $(v, 5, 2, 1)$ optical orthogonal codes, Des. Codes Cryptogr., 68 (2013), 349-371. doi: 10.1007/s10623-012-9654-x.

[7]

M. BurattiY. WeiD. WuP. 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.

[8]

Y. Chang and L. Ji, Optimal $(4up, 5, 1)$ optical orthogonal codes, J. Combin. Des., 12 (2004), 346-361. doi: 10.1002/jcd.20011.

[9]

Y. Chang and Y. Miao, Constructions for optimal optical orthogonal codes, Discr. Math., 261 (2003), 127-139. doi: 10.1016/S0012-365X(02)00464-8.

[10]

K. ChenG. Ge and L. Zhu, Starters and related codes, J. Statist. Plann. Inference, 86 (2000), 379-395. doi: 10.1016/S0378-3758(99)00119-6.

[11]

C. J. ColbournJ. H. Dinitz and D. R. Stinson, Applications of Combinatorial Designs to Communications, Cryptography, and Networking, London Math. Soc. Lecture Note Ser., 267 (1999), 37-100.

[12]

J. H. Dinitz and D. R. Stinson, Room squares and related designs, In: Contemporary Design Theory, Wiley, New York, (1992), 137–204.

[13]

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.

[14]

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.

[15]

B. HuangY. Wei and D. Wu, Bounds and constructions for optimal variable-weight OOCs with unequal auto- and cross-correlation constraints, Utilitas Math., 103 (2017), 3-21.

[16]

W. LiH. Yu and D. Wu, Bounds and constructions for optimal $(n, \{3, 5\}$, $\Lambda_a, 1, Q)$-OOCs, Discr. Math., 339 (2016), 21-32. doi: 10.1016/j.disc.2015.07.006.

[17]

K. Momihara and M. Buratti, Bounds and Constructions of Optimal $(n, 4, 2, 1)$ Optical Orthogonal Codes, IEEE Trans. Inform. Theory, 55 (2009), 514-523. doi: 10.1109/TIT.2008.2009852.

[18]

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

[19]

J. A. Salehi, Code division multiple access techniques in optical fiber networks-Part Ⅰ Fundamental Principles, IEEE Trans. Commun., 37 (1989), 824-833.

[20]

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

[21]

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

[22]

X. Wang and Y. Chang, Further results on optimal $(v, 4, 2, 1)$-OOCs, Discr. Math., 312 (2012), 331-340. doi: 10.1016/j.disc.2011.09.025.

[23]

D. WuH. ZhaoP. 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.

[24]

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

[25]

J. Yin, Some combinatorial constructions for optical orthogonal codes, Discr. Math., 185 (1998), 201-219. doi: 10.1016/S0012-365X(97)00172-6.

[26]

H. YuS. Dang and D. Wu, Bounds and constructions for optimal $(n, \{3, 4, 5\}, \Lambda_a, 1, Q)$-OOCs, IEEE Trans. Inform. Theory, 64 (2018), 1361-1367. doi: 10.1109/TIT.2017.2739778.

[27]

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

[28]

H. ZhaoD. Wu and R. Qin, Further results on $(v, \{3, 4\}, \Lambda_a, 1, Q)$-OOCs, Discr. Math., 337 (2014), 87-96. doi: 10.1016/j.disc.2014.08.003.

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), 59-75. doi: 10.1016/j.jcta.2004.01.003.

[2]

T. Baicheva and S. Topalova, Optimal $(v, 4, 2, 1)$ optical orthogonal codes with small parameters, J. Combin. Des., 20 (2012), 142-160. doi: 10.1002/jcd.20296.

[3]

E. F. Brickell and V. Wei, Optical orthogonal codes and cyclic block designs, Congr. Numer., 58 (1987), 175-182.

[4]

M. BurattiK. Momihara and A. Pasotti, New results on optimal $(v, 4, 2, 1)$ optical orthogonal codes, Des. Codes Cryptogr., 58 (2011), 89-109. doi: 10.1007/s10623-010-9382-z.

[5]

M. Buratti and A. Pasotti, Further progress on difference families with block size $4$ or $5$, Des. Codes Cryptogr., 56 (2010), 1-20. doi: 10.1007/s10623-009-9335-6.

[6]

M. BurattiA. Passotti and D. Wu, On optimal $(v, 5, 2, 1)$ optical orthogonal codes, Des. Codes Cryptogr., 68 (2013), 349-371. doi: 10.1007/s10623-012-9654-x.

[7]

M. BurattiY. WeiD. WuP. 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.

[8]

Y. Chang and L. Ji, Optimal $(4up, 5, 1)$ optical orthogonal codes, J. Combin. Des., 12 (2004), 346-361. doi: 10.1002/jcd.20011.

[9]

Y. Chang and Y. Miao, Constructions for optimal optical orthogonal codes, Discr. Math., 261 (2003), 127-139. doi: 10.1016/S0012-365X(02)00464-8.

[10]

K. ChenG. Ge and L. Zhu, Starters and related codes, J. Statist. Plann. Inference, 86 (2000), 379-395. doi: 10.1016/S0378-3758(99)00119-6.

[11]

C. J. ColbournJ. H. Dinitz and D. R. Stinson, Applications of Combinatorial Designs to Communications, Cryptography, and Networking, London Math. Soc. Lecture Note Ser., 267 (1999), 37-100.

[12]

J. H. Dinitz and D. R. Stinson, Room squares and related designs, In: Contemporary Design Theory, Wiley, New York, (1992), 137–204.

[13]

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.

[14]

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.

[15]

B. HuangY. Wei and D. Wu, Bounds and constructions for optimal variable-weight OOCs with unequal auto- and cross-correlation constraints, Utilitas Math., 103 (2017), 3-21.

[16]

W. LiH. Yu and D. Wu, Bounds and constructions for optimal $(n, \{3, 5\}$, $\Lambda_a, 1, Q)$-OOCs, Discr. Math., 339 (2016), 21-32. doi: 10.1016/j.disc.2015.07.006.

[17]

K. Momihara and M. Buratti, Bounds and Constructions of Optimal $(n, 4, 2, 1)$ Optical Orthogonal Codes, IEEE Trans. Inform. Theory, 55 (2009), 514-523. doi: 10.1109/TIT.2008.2009852.

[18]

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

[19]

J. A. Salehi, Code division multiple access techniques in optical fiber networks-Part Ⅰ Fundamental Principles, IEEE Trans. Commun., 37 (1989), 824-833.

[20]

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

[21]

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

[22]

X. Wang and Y. Chang, Further results on optimal $(v, 4, 2, 1)$-OOCs, Discr. Math., 312 (2012), 331-340. doi: 10.1016/j.disc.2011.09.025.

[23]

D. WuH. ZhaoP. 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.

[24]

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

[25]

J. Yin, Some combinatorial constructions for optical orthogonal codes, Discr. Math., 185 (1998), 201-219. doi: 10.1016/S0012-365X(97)00172-6.

[26]

H. YuS. Dang and D. Wu, Bounds and constructions for optimal $(n, \{3, 4, 5\}, \Lambda_a, 1, Q)$-OOCs, IEEE Trans. Inform. Theory, 64 (2018), 1361-1367. doi: 10.1109/TIT.2017.2739778.

[27]

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

[28]

H. ZhaoD. Wu and R. Qin, Further results on $(v, \{3, 4\}, \Lambda_a, 1, Q)$-OOCs, Discr. Math., 337 (2014), 87-96. doi: 10.1016/j.disc.2014.08.003.

[1]

Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023

[2]

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

[3]

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

[4]

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

[5]

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

[6]

Yong Xia. Convex hull of the orthogonal similarity set with applications in quadratic assignment problems. Journal of Industrial & Management Optimization, 2013, 9 (3) : 689-701. doi: 10.3934/jimo.2013.9.689

[7]

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

[8]

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

[9]

Bavo Langerock. Optimal control problems with variable endpoints. Conference Publications, 2003, 2003 (Special) : 507-516. doi: 10.3934/proc.2003.2003.507

[10]

Rui Gao, Weixiao Shen. Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2013-2036. doi: 10.3934/dcds.2014.34.2013

[11]

Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027

[12]

Karim Samei, Arezoo Soufi. Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$. Advances in Mathematics of Communications, 2017, 11 (4) : 791-804. doi: 10.3934/amc.2017058

[13]

Pieter Moree. On the distribution of the order over residue classes. Electronic Research Announcements, 2006, 12: 121-128.

[14]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1

[15]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003

[16]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45

[17]

Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764

[18]

Wei Feng, Shuhua Hu, Xin Lu. Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective. Conference Publications, 2003, 2003 (Special) : 544-553. doi: 10.3934/proc.2003.2003.544

[19]

K.H. Wong, C. Myburgh, L. Omari. A gradient flow approach for computing jump linear quadratic optimal feedback gains. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 803-808. doi: 10.3934/dcds.2000.6.803

[20]

X. X. Huang, D. Li, Xiaoqi Yang. Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints. Journal of Industrial & Management Optimization, 2006, 2 (3) : 287-296. doi: 10.3934/jimo.2006.2.287

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (40)
  • HTML views (223)
  • Cited by (0)

[Back to Top]