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Symmetries of weight enumerators and applications to Reed-Muller codes
Constructions of optimal balanced $ (m, n, \{4, 5\}, 1) $-OOSPCs
1. | Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba 305-8573, Japan |
2. | Xingjian College of Science and Liberal Arts, Guangxi University, Nanning 530004, China |
3. | Guangxi Key Lab of Multi-source Information Mining & Security, Department of Mathematics, Guangxi Normal University, Guilin 541004, China |
Kitayama proposed a novel OCDMA (called spatial CDMA) for parallel transmission of 2-D images through multicore fiber. Optical orthogonal signature pattern codes (OOSPCs) play an important role in this CDMA network. Multiple-weight (MW) optical orthogonal signature pattern code (OOSPC) was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirements. Some results had been done on optimal balanced $ (m, n, \{3, 4\}, 1) $-OOSPCs. In this paper, it is proved that there exist optimal balanced $ (2u, 16v, \{4, 5\}, 1) $-OOSPCs for odd integers $ u\geq 1 $, $ v\geq 1 $.
References:
[1] |
R. J. R. Abel, C. J. Colbourn and J. H. Dinitz, Mutually orthogonal latin squares (MOLS),
In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook of Combinatorial Designs, New York:
CRC Press, 2007,160–193. |
[2] |
M. Buratti,
A power method for constructing difference families and optimal optical orthogonal codes, Des. Codes Cryptogr., 5 (1995), 13-25.
doi: 10.1007/BF01388501. |
[3] |
M. Buratti,
Recursive constructions for difference matrices and relative difference families, J. Combin. Des., 6 (1998), 165-182.
|
[4] |
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. |
[5] |
J. Chen, L. Ji and Y. Li,
Combinatorial constructions of optimal $(m, n, 4, 2)$ optical orthogonal signature pattern codes, Des. Codes Cryptogr., 86 (2018), 1499-1525.
doi: 10.1007/s10623-017-0409-6. |
[6] |
J. Chen, L. Ji and Y. Li,
New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, Des. Codes Cryptogr., 85 (2017), 299-318.
doi: 10.1007/s10623-016-0310-8. |
[7] |
C. J. Colbourn, Difference matrices, In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook
of Combinatorial Designs, New York: CRC Press, 2007,411–419. |
[8] |
P. A. Davies and A. A. Shaar,
Asynchronous multiplexing for an optical-fibre local area network, Electron. Leu., 19 (1983), 390-392.
doi: 10.1049/el:19830270. |
[9] |
I. B. Djordjevic, B. Vasic and J. Rorison,
Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs, J. Lightw. Technol., 21 (2003), 1850-1856.
doi: 10.1109/JLT.2003.816819. |
[10] |
G. Ge,
On $(g, 4;1)$-diffference matrices, Discrete Math., 301 (2005), 164-174.
doi: 10.1016/j.disc.2005.07.004. |
[11] |
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.
doi: 10.1109/JLT.2004.838880. |
[12] |
A. A. Hassan, J. E. Hershey and N. A. Riza,
Spatial optical CDMA, Perspectives in Spread Spectrum, 459 (1995), 107-125.
doi: 10.1007/978-1-4615-5531-5_5. |
[13] |
J. Y. Hui,
Pattern code modulation and optical decoding a novel code division multiplexing technique for multifiber networks, IEEE J. Select. Areas Commun., 3 (1985), 916-927.
doi: 10.1109/JSAC.1985.1146265. |
[14] |
L. Ji, B. Ding, X. Wang and G. Ge,
Asymptotically optimal optical orthogonal signature pattern codes, IEEE Trans. Inform. Theory, 64 (2018), 5419-5431.
doi: 10.1109/TIT.2017.2787593. |
[15] |
J. Jiang, D. Wu and M. H. Lee,
Some infinte classes of optimal $(v, \{3, 4\}, 1, Q)$-OOCs with $Q\in\{\{1/3, 2/3\}, \{2/3, 1/3\}\}$, Graphs Combin., 29 (2013), 1795-1811.
doi: 10.1007/s00373-012-1235-2. |
[16] |
K. Kitayama,
Novel spatial spread spectrum based fiber optic CDMA networks for image transmission, IEEE J. Select. Areas Commun., 12 (1994), 762-772.
doi: 10.1109/49.286683. |
[17] |
W. C. Kwong and G. C. Yang,
Double-weight signature pattern codes for multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 5 (2001), 203-205.
doi: 10.1109/4234.922760. |
[18] |
W. Kwong and G. C. Yang,
Image transmission in multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 2 (1998), 285-287.
doi: 10.1109/4234.725225. |
[19] |
R. Pan and Y. Chang,
A note on difference matrices over non-cyclic finite abelian groups, Discrete Math., 339 (2016), 822-830.
doi: 10.1016/j.disc.2015.10.028. |
[20] |
R. Pan and Y. Chang,
Combinatorial constructions for maximum optical orthogonal signature pattern codes, Discrete Math., 33 (2013), 2918-2931.
doi: 10.1016/j.disc.2013.09.005. |
[21] |
R. Pan and Y. Chang,
Determination of the sizes of optimal $(m, n, k, \lambda, k-1)$-OOSPCs for $\lambda = k-1, k$, Discrete Math., 313 (2013), 1327-1337.
doi: 10.1016/j.disc.2013.02.019. |
[22] |
R. Pan and Y. Chang,
Further results on optimal $(m, n, 4, 1)$ optical orthogonal signature pattern codes (in Chinese), Sci. Sin. Math., 44 (2014), 1141-1152.
|
[23] |
R. Pan and Y. Chang, $(m, n, 3, 1)$ optical orthogonal signature pattern codes with maximum possible size, IEEE Trans. Inform. Theorey, 61 (2015), 443-461. |
[24] |
E. Park, A. J. Mendez and E. M. Garmire,
Temporal/spatial optical CDMA networks-design, demonstration, and comparison with temporal networks, IEEE Photon. Technol. Lett., 4 (1992), 1160-1162.
doi: 10.1109/68.163765. |
[25] |
P. R. Prucnal, M. A. Santoro and T. R. Fan,
Spread spectrum fiberoptic local network using optical processing, IEEE J. Lightwave Technol., LT-4 (1986), 547-554.
|
[26] |
J. A. Salehi,
Emerging optical code-division multiple access communications systems, IEEE Network, 3 (1989), 31-39.
doi: 10.1109/65.21908. |
[27] |
M. Sawa, Optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, IEEE Trans. Inform. Theorey, 56 (2010), 3613- = 3620.
doi: 10.1109/TIT.2010.2048487. |
[28] |
M. Sawa and S. Kageyama,
Optimal optical orthogonal signature pattern codes of weight 3, Biom. Lett., 46 (2009), 89-102.
|
[29] |
S. Tamura, S. Nakano and K. Okazaki,
Optical codemultiplex transmission by gold sequences, IEEE J. Lightwave Technol., LT-3 (1985), 121-127.
|
[30] |
D. Wu, H. Zhao, P. Fan and S. Shinohara,
Optimal variable-weight optical orthogonal codes via difference packings, IEEE Trans. Inform. Theorey, 53 (2010), 4053-4060.
doi: 10.1109/TIT.2010.2050927. |
[31] |
G. C. Yang,
Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements, IEEE Trans. Commun., 44 (1996), 47-55.
|
[32] |
G. C. Yang and W. C. Kwong,
Two-dimensional spatial signature patterns, IEEE Trans. Commun., 44 (1996), 184-191.
|
[33] |
H. Zhao,
New optimal $(v, \{4, 5\}, 1, \{1/2, 1/2\})$-OOCs, J. Guangxi Teachers Edu. University, 28 (2011), 17-22.
|
[34] |
H. Zhao,
On balanced optimal $(18u, \{3, 4\}, 1)$ optical orthogonal codes, J. Combin. Des., 20 (2012), 290-303.
doi: 10.1002/jcd.21303. |
[35] |
H. Zhao and R. Qin,
Combinatorial constructions for optimal multiple-weight optical orthogonal signature pattern codes, Discrete Math., 339 (2016), 179-193.
doi: 10.1016/j.disc.2015.08.005. |
[36] |
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. |
show all references
References:
[1] |
R. J. R. Abel, C. J. Colbourn and J. H. Dinitz, Mutually orthogonal latin squares (MOLS),
In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook of Combinatorial Designs, New York:
CRC Press, 2007,160–193. |
[2] |
M. Buratti,
A power method for constructing difference families and optimal optical orthogonal codes, Des. Codes Cryptogr., 5 (1995), 13-25.
doi: 10.1007/BF01388501. |
[3] |
M. Buratti,
Recursive constructions for difference matrices and relative difference families, J. Combin. Des., 6 (1998), 165-182.
|
[4] |
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. |
[5] |
J. Chen, L. Ji and Y. Li,
Combinatorial constructions of optimal $(m, n, 4, 2)$ optical orthogonal signature pattern codes, Des. Codes Cryptogr., 86 (2018), 1499-1525.
doi: 10.1007/s10623-017-0409-6. |
[6] |
J. Chen, L. Ji and Y. Li,
New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, Des. Codes Cryptogr., 85 (2017), 299-318.
doi: 10.1007/s10623-016-0310-8. |
[7] |
C. J. Colbourn, Difference matrices, In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook
of Combinatorial Designs, New York: CRC Press, 2007,411–419. |
[8] |
P. A. Davies and A. A. Shaar,
Asynchronous multiplexing for an optical-fibre local area network, Electron. Leu., 19 (1983), 390-392.
doi: 10.1049/el:19830270. |
[9] |
I. B. Djordjevic, B. Vasic and J. Rorison,
Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs, J. Lightw. Technol., 21 (2003), 1850-1856.
doi: 10.1109/JLT.2003.816819. |
[10] |
G. Ge,
On $(g, 4;1)$-diffference matrices, Discrete Math., 301 (2005), 164-174.
doi: 10.1016/j.disc.2005.07.004. |
[11] |
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.
doi: 10.1109/JLT.2004.838880. |
[12] |
A. A. Hassan, J. E. Hershey and N. A. Riza,
Spatial optical CDMA, Perspectives in Spread Spectrum, 459 (1995), 107-125.
doi: 10.1007/978-1-4615-5531-5_5. |
[13] |
J. Y. Hui,
Pattern code modulation and optical decoding a novel code division multiplexing technique for multifiber networks, IEEE J. Select. Areas Commun., 3 (1985), 916-927.
doi: 10.1109/JSAC.1985.1146265. |
[14] |
L. Ji, B. Ding, X. Wang and G. Ge,
Asymptotically optimal optical orthogonal signature pattern codes, IEEE Trans. Inform. Theory, 64 (2018), 5419-5431.
doi: 10.1109/TIT.2017.2787593. |
[15] |
J. Jiang, D. Wu and M. H. Lee,
Some infinte classes of optimal $(v, \{3, 4\}, 1, Q)$-OOCs with $Q\in\{\{1/3, 2/3\}, \{2/3, 1/3\}\}$, Graphs Combin., 29 (2013), 1795-1811.
doi: 10.1007/s00373-012-1235-2. |
[16] |
K. Kitayama,
Novel spatial spread spectrum based fiber optic CDMA networks for image transmission, IEEE J. Select. Areas Commun., 12 (1994), 762-772.
doi: 10.1109/49.286683. |
[17] |
W. C. Kwong and G. C. Yang,
Double-weight signature pattern codes for multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 5 (2001), 203-205.
doi: 10.1109/4234.922760. |
[18] |
W. Kwong and G. C. Yang,
Image transmission in multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 2 (1998), 285-287.
doi: 10.1109/4234.725225. |
[19] |
R. Pan and Y. Chang,
A note on difference matrices over non-cyclic finite abelian groups, Discrete Math., 339 (2016), 822-830.
doi: 10.1016/j.disc.2015.10.028. |
[20] |
R. Pan and Y. Chang,
Combinatorial constructions for maximum optical orthogonal signature pattern codes, Discrete Math., 33 (2013), 2918-2931.
doi: 10.1016/j.disc.2013.09.005. |
[21] |
R. Pan and Y. Chang,
Determination of the sizes of optimal $(m, n, k, \lambda, k-1)$-OOSPCs for $\lambda = k-1, k$, Discrete Math., 313 (2013), 1327-1337.
doi: 10.1016/j.disc.2013.02.019. |
[22] |
R. Pan and Y. Chang,
Further results on optimal $(m, n, 4, 1)$ optical orthogonal signature pattern codes (in Chinese), Sci. Sin. Math., 44 (2014), 1141-1152.
|
[23] |
R. Pan and Y. Chang, $(m, n, 3, 1)$ optical orthogonal signature pattern codes with maximum possible size, IEEE Trans. Inform. Theorey, 61 (2015), 443-461. |
[24] |
E. Park, A. J. Mendez and E. M. Garmire,
Temporal/spatial optical CDMA networks-design, demonstration, and comparison with temporal networks, IEEE Photon. Technol. Lett., 4 (1992), 1160-1162.
doi: 10.1109/68.163765. |
[25] |
P. R. Prucnal, M. A. Santoro and T. R. Fan,
Spread spectrum fiberoptic local network using optical processing, IEEE J. Lightwave Technol., LT-4 (1986), 547-554.
|
[26] |
J. A. Salehi,
Emerging optical code-division multiple access communications systems, IEEE Network, 3 (1989), 31-39.
doi: 10.1109/65.21908. |
[27] |
M. Sawa, Optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, IEEE Trans. Inform. Theorey, 56 (2010), 3613- = 3620.
doi: 10.1109/TIT.2010.2048487. |
[28] |
M. Sawa and S. Kageyama,
Optimal optical orthogonal signature pattern codes of weight 3, Biom. Lett., 46 (2009), 89-102.
|
[29] |
S. Tamura, S. Nakano and K. Okazaki,
Optical codemultiplex transmission by gold sequences, IEEE J. Lightwave Technol., LT-3 (1985), 121-127.
|
[30] |
D. Wu, H. Zhao, P. Fan and S. Shinohara,
Optimal variable-weight optical orthogonal codes via difference packings, IEEE Trans. Inform. Theorey, 53 (2010), 4053-4060.
doi: 10.1109/TIT.2010.2050927. |
[31] |
G. C. Yang,
Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements, IEEE Trans. Commun., 44 (1996), 47-55.
|
[32] |
G. C. Yang and W. C. Kwong,
Two-dimensional spatial signature patterns, IEEE Trans. Commun., 44 (1996), 184-191.
|
[33] |
H. Zhao,
New optimal $(v, \{4, 5\}, 1, \{1/2, 1/2\})$-OOCs, J. Guangxi Teachers Edu. University, 28 (2011), 17-22.
|
[34] |
H. Zhao,
On balanced optimal $(18u, \{3, 4\}, 1)$ optical orthogonal codes, J. Combin. Des., 20 (2012), 290-303.
doi: 10.1002/jcd.21303. |
[35] |
H. Zhao and R. Qin,
Combinatorial constructions for optimal multiple-weight optical orthogonal signature pattern codes, Discrete Math., 339 (2016), 179-193.
doi: 10.1016/j.disc.2015.08.005. |
[36] |
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. |
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