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An improved lower bound for $(1,\leq 2)$-identifying codes in the king grid
Unified combinatorial constructions of optimal optical orthogonal codes
1. | Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China |
2. | Faculty of Education, Kumamoto University, 2-40-1 Kurokami, Kumamoto 860-8555, Japan |
References:
[1] |
T. L. Alderson and K. E. Mellinger, Constructions of optical orthogonal codes from finite geometry, SIAM J. Discrete Math., 21 (2007), 785-793.
doi: 10.1137/050632257. |
[2] |
T. L. Alderson and K. E. Mellinger, Optical orthogonal codes from Singer groups, in Advances in Coding Theory and Cryptology, 3 (2007), 51-70.
doi: 10.1142/9789812772022_0004. |
[3] |
T. L. Alderson and K. E. Mellinger, Classes of optical orthogonal codes from arcs in root subspaces, Discrete Math., 308 (2008), 1093-1101.
doi: 10.1016/j.disc.2007.03.063. |
[4] |
T. L. Alderson and K. E. Mellinger, Families of optimal OOCs with $\lambda=2$, IEEE Trans. Inform. Theory, 54 (2008), 3722-3724.
doi: 10.1109/TIT.2008.926394. |
[5] |
T. L. Alderson and K. E. Mellinger, Geometric constructions of optimal optical orthogonal codes, Adv. Math. Commun., 2 (2008), 451-467.
doi: 10.3934/amc.2008.2.451. |
[6] |
B. Berndt, R. Evans and K. S. Williams, Gauss and Jacobi Sums, Wiley, 1997. |
[7] |
C. M. Bird and A. D. Keedwell, Design and applications of optical orthogonal codes-a survey, Bull. Inst. Combin. Appl., 11 (1994), 21-44. |
[8] |
I. Bousrih, Families of rational functions over finite fields and constructions of optical orthogonal codes, Afr. Diaspora J. Math., 3 (2005), 95-105. |
[9] |
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. |
[10] |
F. R. K. Chung, J. A. Salehi and V. K. Wei, Optical orthogonal codes: design, analysis, and applications, IEEE Trans. Inform. Theory, 35 (1989), 595-604.
doi: 10.1109/18.30982. |
[11] |
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. |
[12] |
C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima, Sets of optimal frequency hopping sequences: bounds and optimal constructions, IEEE Trans. Inform. Theory, 55 (2009), 3297-3304.
doi: 10.1109/TIT.2009.2021366. |
[13] |
C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inform. Theory, 56 (2010), 3605-3612.
doi: 10.1109/TIT.2010.2048504. |
[14] |
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. |
[15] |
A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inform. Theory, 20 (1974), 90-94. |
[16] |
R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1997. |
[17] |
F. J. MacWilliams and N. J. A. Sloan, The Theory of Error-Correcting Codes, Twelfth editioin, North-Holland Mathematical Library, 2006. |
[18] |
S. V. Maric, O. Moreno and C. Corrada, Multimedia transmission in fiber-optic lans using optical cdma, J. Lightwave Technol., 14 (1996), 2149-2153.
doi: 10.1109/50.541202. |
[19] |
S. Mashhadi and J. A. Salehi, Code-division multiple-access techniques in optical fiber networks-part iii: optical and logic gate receiver structure with generalized optical orthogonal codes, IEEE Trans. Commun., 54 (2006), 1457-1468.
doi: 10.1109/TCOMM.2006.878835. |
[20] |
K. Momihara, New optimal optical orthogonal codes by restrictions to subgroups, Finite Fields Appl., 17 (2010), 166-182.
doi: 10.1016/j.ffa.2010.11.001. |
[21] |
O. Moreno, R. Omrani, P. V. Kumar and H.-F. Lu, A generalized Bose-Chowla family of optical orthogonal codes and distinct difference sets, IEEE Trans. Inform. Theory, 53 (2007), 1907-1910.
doi: 10.1109/TIT.2007.894658. |
[22] |
O. Moreno, Z. Zhang, P. V. Kumar and A. Zinoviev, New constructions of optimal cyclically permutable constant weight codes, IEEE Trans. Inform. Theory, 41 (1995), 448-454.
doi: 10.1109/18.370146. |
[23] |
Q. A. Nguyen, L. Györfi and J. L. Massey, Constructions of binary constant-weight cyclic codes and cyclically permutable codes, IEEE Trans. Inform. Theory, 38 (1992), 940-949.
doi: 10.1109/18.135636. |
[24] |
R. Omrani, O. Moreno and P. V. Kumar, Improved Johnson bounds for optical orthogonal codes with $\lambda>1$ and some optimal constructions, in Proc. Int. Symp. Inform. Theory, 2005, 259-263. |
[25] |
D. Peng and P. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences, IEEE Trans. Inform. Theory, 50 (2004), 2149-2154.
doi: 10.1109/TIT.2004.833362. |
[26] |
H. Stichtenoth, Algebraic Function Fields and Codes, Second edition, Springer, 2009. |
[27] |
R. M. Wilson, Cyclotomy and difference families in elementary abelian groups, J. Number Theory, 4 (1972), 17-47.
doi: 10.1016/0022-314X(72)90009-1. |
[28] |
Z. Zhou, X. Tang, D. Peng and U. Parampall, New constructions for optimal sets of frequency-hopping sequences, IEEE Trans. Inform. Theory, 57 (2011), 3831-3840.
doi: 10.1109/TIT.2011.2137290. |
show all references
References:
[1] |
T. L. Alderson and K. E. Mellinger, Constructions of optical orthogonal codes from finite geometry, SIAM J. Discrete Math., 21 (2007), 785-793.
doi: 10.1137/050632257. |
[2] |
T. L. Alderson and K. E. Mellinger, Optical orthogonal codes from Singer groups, in Advances in Coding Theory and Cryptology, 3 (2007), 51-70.
doi: 10.1142/9789812772022_0004. |
[3] |
T. L. Alderson and K. E. Mellinger, Classes of optical orthogonal codes from arcs in root subspaces, Discrete Math., 308 (2008), 1093-1101.
doi: 10.1016/j.disc.2007.03.063. |
[4] |
T. L. Alderson and K. E. Mellinger, Families of optimal OOCs with $\lambda=2$, IEEE Trans. Inform. Theory, 54 (2008), 3722-3724.
doi: 10.1109/TIT.2008.926394. |
[5] |
T. L. Alderson and K. E. Mellinger, Geometric constructions of optimal optical orthogonal codes, Adv. Math. Commun., 2 (2008), 451-467.
doi: 10.3934/amc.2008.2.451. |
[6] |
B. Berndt, R. Evans and K. S. Williams, Gauss and Jacobi Sums, Wiley, 1997. |
[7] |
C. M. Bird and A. D. Keedwell, Design and applications of optical orthogonal codes-a survey, Bull. Inst. Combin. Appl., 11 (1994), 21-44. |
[8] |
I. Bousrih, Families of rational functions over finite fields and constructions of optical orthogonal codes, Afr. Diaspora J. Math., 3 (2005), 95-105. |
[9] |
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. |
[10] |
F. R. K. Chung, J. A. Salehi and V. K. Wei, Optical orthogonal codes: design, analysis, and applications, IEEE Trans. Inform. Theory, 35 (1989), 595-604.
doi: 10.1109/18.30982. |
[11] |
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. |
[12] |
C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima, Sets of optimal frequency hopping sequences: bounds and optimal constructions, IEEE Trans. Inform. Theory, 55 (2009), 3297-3304.
doi: 10.1109/TIT.2009.2021366. |
[13] |
C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inform. Theory, 56 (2010), 3605-3612.
doi: 10.1109/TIT.2010.2048504. |
[14] |
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. |
[15] |
A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inform. Theory, 20 (1974), 90-94. |
[16] |
R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, 1997. |
[17] |
F. J. MacWilliams and N. J. A. Sloan, The Theory of Error-Correcting Codes, Twelfth editioin, North-Holland Mathematical Library, 2006. |
[18] |
S. V. Maric, O. Moreno and C. Corrada, Multimedia transmission in fiber-optic lans using optical cdma, J. Lightwave Technol., 14 (1996), 2149-2153.
doi: 10.1109/50.541202. |
[19] |
S. Mashhadi and J. A. Salehi, Code-division multiple-access techniques in optical fiber networks-part iii: optical and logic gate receiver structure with generalized optical orthogonal codes, IEEE Trans. Commun., 54 (2006), 1457-1468.
doi: 10.1109/TCOMM.2006.878835. |
[20] |
K. Momihara, New optimal optical orthogonal codes by restrictions to subgroups, Finite Fields Appl., 17 (2010), 166-182.
doi: 10.1016/j.ffa.2010.11.001. |
[21] |
O. Moreno, R. Omrani, P. V. Kumar and H.-F. Lu, A generalized Bose-Chowla family of optical orthogonal codes and distinct difference sets, IEEE Trans. Inform. Theory, 53 (2007), 1907-1910.
doi: 10.1109/TIT.2007.894658. |
[22] |
O. Moreno, Z. Zhang, P. V. Kumar and A. Zinoviev, New constructions of optimal cyclically permutable constant weight codes, IEEE Trans. Inform. Theory, 41 (1995), 448-454.
doi: 10.1109/18.370146. |
[23] |
Q. A. Nguyen, L. Györfi and J. L. Massey, Constructions of binary constant-weight cyclic codes and cyclically permutable codes, IEEE Trans. Inform. Theory, 38 (1992), 940-949.
doi: 10.1109/18.135636. |
[24] |
R. Omrani, O. Moreno and P. V. Kumar, Improved Johnson bounds for optical orthogonal codes with $\lambda>1$ and some optimal constructions, in Proc. Int. Symp. Inform. Theory, 2005, 259-263. |
[25] |
D. Peng and P. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences, IEEE Trans. Inform. Theory, 50 (2004), 2149-2154.
doi: 10.1109/TIT.2004.833362. |
[26] |
H. Stichtenoth, Algebraic Function Fields and Codes, Second edition, Springer, 2009. |
[27] |
R. M. Wilson, Cyclotomy and difference families in elementary abelian groups, J. Number Theory, 4 (1972), 17-47.
doi: 10.1016/0022-314X(72)90009-1. |
[28] |
Z. Zhou, X. Tang, D. Peng and U. Parampall, New constructions for optimal sets of frequency-hopping sequences, IEEE Trans. Inform. Theory, 57 (2011), 3831-3840.
doi: 10.1109/TIT.2011.2137290. |
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