Unified combinatorial constructions of optimal optical orthogonal codes

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February 2014, 8(1): 53-66. doi: 10.3934/amc.2014.8.53

Unified combinatorial constructions of optimal optical orthogonal codes

Cuiling Fan ^1, and Koji Momihara ^2,

1.	Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China
2.	Faculty of Education, Kumamoto University, 2-40-1 Kurokami, Kumamoto 860-8555, Japan

Received May 2012 Revised June 2013 Published January 2014

Abstract
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We present unified constructions of optical orthogonal codes (OOCs) using other combinatorial objects such as cyclic linear codes and frequency hopping sequences. Some of the obtained OOCs are optimal or asymptotically optimal with respect to the Johnson bound. Also, we are able to show the existence of new optimal frequency hopping sequences (FHSs) with respect to the Singleton bound from our observation on a relation between OOCs and FHSs. The last construction is based on residue rings of polynomials over finite fields, and it yields a new large class of asymptotically optimal $(q-1,k,k-2)$-OOCs for any prime power $q$ with $\gcd{(q-1,k)}=1$. Some infinite families of optimal ones are included as a subclass.

Keywords: Optical orthogonal code, frequency hopping sequence., cyclic linear code.

Mathematics Subject Classification: Primary: 94B25; Secondary: 05B4.

Citation: 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

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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