doi: 10.3934/amc.2021012

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 Published  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:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[44]

J. Yin, Some combinatorial constructions for optical orthogonal codes, Discrete Math., 185 (1998), 201-219.  doi: 10.1016/S0012-365X(97)00172-6.  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 (A), 106 (2004), 59-75.  doi: 10.1016/j.jcta.2004.01.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[44]

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

[1]

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

[2]

María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021018

[3]

Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2020, 14 (3) : 423-436. doi: 10.3934/amc.2020058

[4]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83

[5]

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

[6]

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

[7]

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

[8]

Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021086

[9]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399

[10]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074

[11]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889

[12]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503

[13]

Lars Lamberg, Lauri Ylinen. Two-Dimensional tomography with unknown view angles. Inverse Problems & Imaging, 2007, 1 (4) : 623-642. doi: 10.3934/ipi.2007.1.623

[14]

Elissar Nasreddine. Two-dimensional individual clustering model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 307-316. doi: 10.3934/dcdss.2014.7.307

[15]

Jerzy Gawinecki, Wojciech M. Zajączkowski. Global regular solutions to two-dimensional thermoviscoelasticity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1009-1028. doi: 10.3934/cpaa.2016.15.1009

[16]

Ibrahim Fatkullin, Valeriy Slastikov. Diffusive transport in two-dimensional nematics. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 323-340. doi: 10.3934/dcdss.2015.8.323

[17]

Min Chen. Numerical investigation of a two-dimensional Boussinesq system. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1169-1190. doi: 10.3934/dcds.2009.23.1169

[18]

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

[19]

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

[20]

R. V. Gurjar, Mariusz Koras and Peter Russell. Two dimensional quotients of $CC^n$ by a reductive group. Electronic Research Announcements, 2008, 15: 62-64. doi: 10.3934/era.2008.15.62

2019 Impact Factor: 0.734

Article outline

[Back to Top]