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doi: 10.3934/amc.2021012
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Some progress on optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal codes

1. 

School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang 315211, China

2. 

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

3. 

School of Intelligence Policing, China People's Police University, Langfang 065000, China

* Corresponding author: Xiaomiao Wang

Received  November 2020 Revised  March 2021 Early access May 2021

Fund Project: This work is supported by Zhejiang Provincial Natural Science Foundation of China under Grant LY21A010005, NSFC under Grant 11771227, the Fundamental Research Funds for the Provincial Universities of Zhejiang under Grant SJLY2020008, and Natural Science Foundation of Ningbo under Grant 202003N4141 (X. Wang), NSFC under Grant 11871095 (M. Zhang), NSFC under Grant 11771119, and NSFHB under Grant A2019507002 (L. Wang)

In this paper, we are concerned about bounds and constructions of optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal codes. The exact number of codewords of an optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal code is determined for $ n = 2 $, $ m\equiv 1 \pmod{2} $, and $ n\equiv 1 \pmod{2} $, $ m\equiv 1,3,5 \pmod{12} $, and $ n\equiv 4 \pmod{6} $, $ m\equiv 8 \pmod{16} $.

Citation: Kailu Yang, Xiaomiao Wang, Menglong Zhang, Lidong Wang. Some progress on optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021012
References:
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R. J. R. Abel and M. Buratti, Some progress on $(v, 4, 1)$ difference families and optical orthogonal codes, J. Combin. Theory (A), 106 (2004), 59-75.  doi: 10.1016/j.jcta.2004.01.003.

[2]

T. L. Alderson and K. E. Mellinger, $2$-dimensional optical orthogonal codes from Singer groups, Discrete Appl. Math., 157 (2009), 3008-3019.  doi: 10.1016/j.dam.2009.06.002.

[3]

T. L. Alderson and K. E. Mellinger, Spreads, arcs, and multiple wavelength codes, Discrete Math., 311 (2011), 1187-1196.  doi: 10.1016/j.disc.2010.06.010.

[4]

S. Bitan and T. Etzion, Constructions for optimal constant weight cyclically permutable codes and difference families, IEEE Trans. Inform. Theory, 41 (1995), 77-87.  doi: 10.1109/18.370117.

[5]

M. Buratti, Cyclic designs with block size $4$ and related optimal optical orthogonal codes, Des. Codes Cryptogr., 26 (2002), 111-125.  doi: 10.1023/A:1016505309092.

[6]

M. Buratti, On silver and golden optical orthogonal codes, Art Discret. Appl. Math., 1 (2018), #P2.02. doi: 10.26493/2590-9770.1236.ce4.

[7]

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.

[8]

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.

[9]

M. BurattiA. Pasotti 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.

[10]

H. Cao and R. Wei, Combinatorial constructions for optimal two-dimensional optical orthogonal codes, IEEE Trans. Inform. Theory, 55 (2009), 1387-1394.  doi: 10.1109/TIT.2008.2011431.

[11]

Y. ChangR. Fuji-Hara and Y. Miao, Combinatorial constructions of optimal optical orthogonal codes with weight $4$, IEEE Trans. Inform. Theory, 49 (2003), 1283-1292.  doi: 10.1109/TIT.2003.810628.

[12]

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

[13]

W. Chu and C. J. Colbourn, Recursive constructions for optimal $(n, 4, 2)$-OOCs, J. Combin. Designs, 12 (2004), 333-345.  doi: 10.1002/jcd.20003.

[14]

F. R. K. ChungJ. A. Salehi and V. K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Trans. Inform. Theory, 35 (1989), 595-604.  doi: 10.1109/18.30982.

[15]

H. Chung and P. V. Kumar, Optical orthogonal codes - new bounds and an optimal construction, IEEE Trans. Inform. Theory, 36 (1990), 866-873.  doi: 10.1109/18.53748.

[16]

T. Feng and Y. Chang, Combinatorial constructions for optimal two-dimensional optical orthogonal codes with $\lambda=2$, IEEE Trans. Inform. Theory, 57 (2011), 6796-6819.  doi: 10.1109/TIT.2011.2165805.

[17]

T. FengY. Chang and L. Ji, Constructions for strictly cyclic $3$-designs and applications to optimal OOCs with $\lambda=2$, J. Combin. Theory (A), 115 (2008), 1527-1551.  doi: 10.1016/j.jcta.2008.03.003.

[18]

T. FengL. WangX. Wang and Y. Zhao, Optimal two dimensional optical orthogonal codes with the best cross-correlation constrain, J. Combin. Designs, 25 (2017), 349-380.  doi: 10.1002/jcd.21554.

[19]

T. FengL. Wang and X. Wang, Optimal $2$-D $(n\times m, 3, 2, 1)$-optical orthogonal codes and related equi-difference conflict avoiding codes, Des. Codes Cryptogr., 87 (2019), 1499-1520.  doi: 10.1007/s10623-018-0549-3.

[20]

T. FengX. Wang and Y. Chang, Semi-cyclic holey group divisible designs with block size three, Des. Codes Cryptogr., 74 (2015), 301-324.  doi: 10.1007/s10623-013-9859-7.

[21]

T. FengX. Wang and R. Wei, Semi-cyclic holey group divisible designs and applications to sampling designs and optical orthogonal codes, J. Combin. Designs, 24 (2016), 201-222.  doi: 10.1002/jcd.21417.

[22]

H. L. FuY. H. Lo and K. W. Shum, Optimal conflict-avoiding codes of odd length and weight $3$, Des. Codes Cryptogr., 72 (2014), 289-309.  doi: 10.1007/s10623-012-9764-5.

[23]

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.

[24] G. Ge, Group divisible designs, in CRC Handbook of Combinatorial Designs (eds. C. J. Colbourn and J. H. Dinitz), CRC Press, 2006.  doi: 10.1201/9781420049954.
[25]

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.

[26]

Y. Huang and Y. Chang, Two classes of optimal two-dimensional OOCs, Des. Codes Cryptogr., 63 (2012), 357-363.  doi: 10.1007/s10623-011-9560-7.

[27]

Y. LinM. MishimaJ. Satoh and M. Jimbo, Optimal equi-difference conflict-avoiding codes of odd length and weight three, Finite Fields Appl., 26 (2014), 49-68.  doi: 10.1016/j.ffa.2013.11.001.

[28]

N. MiyamotoH. Mizuno and S. Shinohara, Optical orthogonal codes obtained from conics on finite projective planes, Finite Fields Appl., 10 (2004), 405-411.  doi: 10.1016/j.ffa.2003.09.004.

[29]

K. Momihara, Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three, Des. Codes Cryptogr., 45 (2007), 379-390.  doi: 10.1007/s10623-007-9139-5.

[30]

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.

[31]

R. OmraniG. GargP. V. KumarP. Elia and P. Bhambhani, Large families of asymptotically optimal two-dimensional optical orthogonal codes, IEEE Trans. Inform. Theory, 58 (2012), 1163-1185.  doi: 10.1109/TIT.2011.2169299.

[32]

R. S. Rees, Two new direct product type constructions for resolvable group divisible designs, J. Combin. Designs, 1 (1993), 15-26.  doi: 10.1002/jcd.3180010104.

[33]

J. WangX. Shan and J. Yin, On constructions for optimal two-dimentional optical orthogonal codes, Des. Codes Cryptogr., 54 (2010), 43-60.  doi: 10.1007/s10623-009-9308-9.

[34]

J. Wang and J. Yin, Two-dimensional optical orthogonal codes and semicyclic group divisible designs, IEEE Trans. Inform. Theory, 56 (2010), 2177-2187.  doi: 10.1109/TIT.2010.2043772.

[35]

L. Wang and Y. Chang, Determination of sizes of optimal three-dimensional optical orthogonal codes of weight three with the AM-OPP restriction, J. Combin. Designs, 25 (2017), 310-334.  doi: 10.1002/jcd.21550.

[36]

L. Wang and Y. Chang, Combinatorial constructions of optimal three-dimensional optical orthogonal codes, IEEE Trans. Inform. Theory, 61 (2015), 671-687.  doi: 10.1109/TIT.2014.2368133.

[37]

L. WangT. FengR. Pan and X. Wang, The spectrum of semicyclic holey group divisible designs with block size three, J. Combin. Designs, 28 (2020), 49-74.  doi: 10.1002/jcd.21680.

[38]

X. Wang and Y. Chang, Further results on $(v, 4, 1)$-perfect difference families, Discrete Math., 310 (2010), 1995-2006.  doi: 10.1016/j.disc.2010.03.017.

[39]

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

[40]

X. WangY. Chang and T. Feng, Optimal $2$-D $(n\times m, 3, 2, 1)$-optical orthogonal codes, IEEE Trans. Inform. Theory, 59 (2013), 710-725.  doi: 10.1109/TIT.2012.2214025.

[41]

S. L. Wu and H. L. Fu, Optimal tight equi-difference conflict-avoiding codes of length $n=2^k\pm1$ and weight $3$, J. Combin. Designs, 21 (2013), 223-231.  doi: 10.1002/jcd.21332.

[42]

G. C. Yang and T. E. Fuja, Optical orthogonal codes with unequal auto- and cross-correlation constraints, IEEE Trans. Inform. Theory, 41 (1995), 96-106. 

[43]

G. C. Yang and W. C. Kwong, Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks, IEEE Trans. Communications, 45 (1997), 1426-1434. 

[44]

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

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 (A), 106 (2004), 59-75.  doi: 10.1016/j.jcta.2004.01.003.

[2]

T. L. Alderson and K. E. Mellinger, $2$-dimensional optical orthogonal codes from Singer groups, Discrete Appl. Math., 157 (2009), 3008-3019.  doi: 10.1016/j.dam.2009.06.002.

[3]

T. L. Alderson and K. E. Mellinger, Spreads, arcs, and multiple wavelength codes, Discrete Math., 311 (2011), 1187-1196.  doi: 10.1016/j.disc.2010.06.010.

[4]

S. Bitan and T. Etzion, Constructions for optimal constant weight cyclically permutable codes and difference families, IEEE Trans. Inform. Theory, 41 (1995), 77-87.  doi: 10.1109/18.370117.

[5]

M. Buratti, Cyclic designs with block size $4$ and related optimal optical orthogonal codes, Des. Codes Cryptogr., 26 (2002), 111-125.  doi: 10.1023/A:1016505309092.

[6]

M. Buratti, On silver and golden optical orthogonal codes, Art Discret. Appl. Math., 1 (2018), #P2.02. doi: 10.26493/2590-9770.1236.ce4.

[7]

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.

[8]

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.

[9]

M. BurattiA. Pasotti 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.

[10]

H. Cao and R. Wei, Combinatorial constructions for optimal two-dimensional optical orthogonal codes, IEEE Trans. Inform. Theory, 55 (2009), 1387-1394.  doi: 10.1109/TIT.2008.2011431.

[11]

Y. ChangR. Fuji-Hara and Y. Miao, Combinatorial constructions of optimal optical orthogonal codes with weight $4$, IEEE Trans. Inform. Theory, 49 (2003), 1283-1292.  doi: 10.1109/TIT.2003.810628.

[12]

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

[13]

W. Chu and C. J. Colbourn, Recursive constructions for optimal $(n, 4, 2)$-OOCs, J. Combin. Designs, 12 (2004), 333-345.  doi: 10.1002/jcd.20003.

[14]

F. R. K. ChungJ. A. Salehi and V. K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Trans. Inform. Theory, 35 (1989), 595-604.  doi: 10.1109/18.30982.

[15]

H. Chung and P. V. Kumar, Optical orthogonal codes - new bounds and an optimal construction, IEEE Trans. Inform. Theory, 36 (1990), 866-873.  doi: 10.1109/18.53748.

[16]

T. Feng and Y. Chang, Combinatorial constructions for optimal two-dimensional optical orthogonal codes with $\lambda=2$, IEEE Trans. Inform. Theory, 57 (2011), 6796-6819.  doi: 10.1109/TIT.2011.2165805.

[17]

T. FengY. Chang and L. Ji, Constructions for strictly cyclic $3$-designs and applications to optimal OOCs with $\lambda=2$, J. Combin. Theory (A), 115 (2008), 1527-1551.  doi: 10.1016/j.jcta.2008.03.003.

[18]

T. FengL. WangX. Wang and Y. Zhao, Optimal two dimensional optical orthogonal codes with the best cross-correlation constrain, J. Combin. Designs, 25 (2017), 349-380.  doi: 10.1002/jcd.21554.

[19]

T. FengL. Wang and X. Wang, Optimal $2$-D $(n\times m, 3, 2, 1)$-optical orthogonal codes and related equi-difference conflict avoiding codes, Des. Codes Cryptogr., 87 (2019), 1499-1520.  doi: 10.1007/s10623-018-0549-3.

[20]

T. FengX. Wang and Y. Chang, Semi-cyclic holey group divisible designs with block size three, Des. Codes Cryptogr., 74 (2015), 301-324.  doi: 10.1007/s10623-013-9859-7.

[21]

T. FengX. Wang and R. Wei, Semi-cyclic holey group divisible designs and applications to sampling designs and optical orthogonal codes, J. Combin. Designs, 24 (2016), 201-222.  doi: 10.1002/jcd.21417.

[22]

H. L. FuY. H. Lo and K. W. Shum, Optimal conflict-avoiding codes of odd length and weight $3$, Des. Codes Cryptogr., 72 (2014), 289-309.  doi: 10.1007/s10623-012-9764-5.

[23]

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.

[24] G. Ge, Group divisible designs, in CRC Handbook of Combinatorial Designs (eds. C. J. Colbourn and J. H. Dinitz), CRC Press, 2006.  doi: 10.1201/9781420049954.
[25]

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.

[26]

Y. Huang and Y. Chang, Two classes of optimal two-dimensional OOCs, Des. Codes Cryptogr., 63 (2012), 357-363.  doi: 10.1007/s10623-011-9560-7.

[27]

Y. LinM. MishimaJ. Satoh and M. Jimbo, Optimal equi-difference conflict-avoiding codes of odd length and weight three, Finite Fields Appl., 26 (2014), 49-68.  doi: 10.1016/j.ffa.2013.11.001.

[28]

N. MiyamotoH. Mizuno and S. Shinohara, Optical orthogonal codes obtained from conics on finite projective planes, Finite Fields Appl., 10 (2004), 405-411.  doi: 10.1016/j.ffa.2003.09.004.

[29]

K. Momihara, Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three, Des. Codes Cryptogr., 45 (2007), 379-390.  doi: 10.1007/s10623-007-9139-5.

[30]

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.

[31]

R. OmraniG. GargP. V. KumarP. Elia and P. Bhambhani, Large families of asymptotically optimal two-dimensional optical orthogonal codes, IEEE Trans. Inform. Theory, 58 (2012), 1163-1185.  doi: 10.1109/TIT.2011.2169299.

[32]

R. S. Rees, Two new direct product type constructions for resolvable group divisible designs, J. Combin. Designs, 1 (1993), 15-26.  doi: 10.1002/jcd.3180010104.

[33]

J. WangX. Shan and J. Yin, On constructions for optimal two-dimentional optical orthogonal codes, Des. Codes Cryptogr., 54 (2010), 43-60.  doi: 10.1007/s10623-009-9308-9.

[34]

J. Wang and J. Yin, Two-dimensional optical orthogonal codes and semicyclic group divisible designs, IEEE Trans. Inform. Theory, 56 (2010), 2177-2187.  doi: 10.1109/TIT.2010.2043772.

[35]

L. Wang and Y. Chang, Determination of sizes of optimal three-dimensional optical orthogonal codes of weight three with the AM-OPP restriction, J. Combin. Designs, 25 (2017), 310-334.  doi: 10.1002/jcd.21550.

[36]

L. Wang and Y. Chang, Combinatorial constructions of optimal three-dimensional optical orthogonal codes, IEEE Trans. Inform. Theory, 61 (2015), 671-687.  doi: 10.1109/TIT.2014.2368133.

[37]

L. WangT. FengR. Pan and X. Wang, The spectrum of semicyclic holey group divisible designs with block size three, J. Combin. Designs, 28 (2020), 49-74.  doi: 10.1002/jcd.21680.

[38]

X. Wang and Y. Chang, Further results on $(v, 4, 1)$-perfect difference families, Discrete Math., 310 (2010), 1995-2006.  doi: 10.1016/j.disc.2010.03.017.

[39]

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

[40]

X. WangY. Chang and T. Feng, Optimal $2$-D $(n\times m, 3, 2, 1)$-optical orthogonal codes, IEEE Trans. Inform. Theory, 59 (2013), 710-725.  doi: 10.1109/TIT.2012.2214025.

[41]

S. L. Wu and H. L. Fu, Optimal tight equi-difference conflict-avoiding codes of length $n=2^k\pm1$ and weight $3$, J. Combin. Designs, 21 (2013), 223-231.  doi: 10.1002/jcd.21332.

[42]

G. C. Yang and T. E. Fuja, Optical orthogonal codes with unequal auto- and cross-correlation constraints, IEEE Trans. Inform. Theory, 41 (1995), 96-106. 

[43]

G. C. Yang and W. C. Kwong, Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks, IEEE Trans. Communications, 45 (1997), 1426-1434. 

[44]

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

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