# American Institute of Mathematical Sciences

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

## 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

Received  May 2012 Revised  June 2013 Published  January 2014

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.
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. doi: 10.1137/050632257. Google Scholar [2] T. L. Alderson and K. E. Mellinger, Optical orthogonal codes from Singer groups,, in Advances in Coding Theory and Cryptology, 3 (2007), 51. doi: 10.1142/9789812772022_0004. Google Scholar [3] T. L. Alderson and K. E. Mellinger, Classes of optical orthogonal codes from arcs in root subspaces,, Discrete Math., 308 (2008), 1093. doi: 10.1016/j.disc.2007.03.063. Google Scholar [4] T. L. Alderson and K. E. Mellinger, Families of optimal OOCs with $\lambda=2$,, IEEE Trans. Inform. Theory, 54 (2008), 3722. doi: 10.1109/TIT.2008.926394. Google Scholar [5] T. L. Alderson and K. E. Mellinger, Geometric constructions of optimal optical orthogonal codes,, Adv. Math. Commun., 2 (2008), 451. doi: 10.3934/amc.2008.2.451. Google Scholar [6] B. Berndt, R. Evans and K. S. Williams, Gauss and Jacobi Sums,, Wiley, (1997). Google Scholar [7] C. M. Bird and A. D. Keedwell, Design and applications of optical orthogonal codes-a survey,, Bull. Inst. Combin. Appl., 11 (1994), 21. Google Scholar [8] I. Bousrih, Families of rational functions over finite fields and constructions of optical orthogonal codes,, Afr. Diaspora J. Math., 3 (2005), 95. Google Scholar [9] M. Buratti and A. Pasotti, Further progress on difference families with block size $4$ or $5$,, Des. Codes Cryptogr., 56 (2010), 1. doi: 10.1007/s10623-009-9335-6. Google Scholar [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. doi: 10.1109/18.30982. Google Scholar [11] H. Chung and P. V. Kumar, Optical orthogonal codes-new bounds and an optimal construction,, IEEE Trans. Inform. Theory, 36 (1990), 866. doi: 10.1109/18.53748. Google Scholar [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. doi: 10.1109/TIT.2009.2021366. Google Scholar [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. doi: 10.1109/TIT.2010.2048504. Google Scholar [14] R. Fuji-Hara and Y. Miao, Optical orthogonal codes: their bounds and new optimal constructions,, IEEE Trans. Inform. Theory, 46 (2000), 2396. doi: 10.1109/18.887852. Google Scholar [15] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties,, IEEE Trans. Inform. Theory, 20 (1974), 90. Google Scholar [16] R. Lidl and H. Niederreiter, Finite Fields,, Cambridge Univ. Press, (1997). Google Scholar [17] F. J. MacWilliams and N. J. A. Sloan, The Theory of Error-Correcting Codes,, Twelfth editioin, (2006). Google Scholar [18] S. V. Maric, O. Moreno and C. Corrada, Multimedia transmission in fiber-optic lans using optical cdma,, J. Lightwave Technol., 14 (1996), 2149. doi: 10.1109/50.541202. Google Scholar [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. doi: 10.1109/TCOMM.2006.878835. Google Scholar [20] K. Momihara, New optimal optical orthogonal codes by restrictions to subgroups,, Finite Fields Appl., 17 (2010), 166. doi: 10.1016/j.ffa.2010.11.001. Google Scholar [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. doi: 10.1109/TIT.2007.894658. Google Scholar [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. doi: 10.1109/18.370146. Google Scholar [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. doi: 10.1109/18.135636. Google Scholar [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. Google Scholar [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. doi: 10.1109/TIT.2004.833362. Google Scholar [26] H. Stichtenoth, Algebraic Function Fields and Codes,, Second edition, (2009). Google Scholar [27] R. M. Wilson, Cyclotomy and difference families in elementary abelian groups,, J. Number Theory, 4 (1972), 17. doi: 10.1016/0022-314X(72)90009-1. Google Scholar [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. doi: 10.1109/TIT.2011.2137290. Google Scholar

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##### References:
 [1] T. L. Alderson and K. E. Mellinger, Constructions of optical orthogonal codes from finite geometry,, SIAM J. Discrete Math., 21 (2007), 785. doi: 10.1137/050632257. Google Scholar [2] T. L. Alderson and K. E. Mellinger, Optical orthogonal codes from Singer groups,, in Advances in Coding Theory and Cryptology, 3 (2007), 51. doi: 10.1142/9789812772022_0004. Google Scholar [3] T. L. Alderson and K. E. Mellinger, Classes of optical orthogonal codes from arcs in root subspaces,, Discrete Math., 308 (2008), 1093. doi: 10.1016/j.disc.2007.03.063. Google Scholar [4] T. L. Alderson and K. E. Mellinger, Families of optimal OOCs with $\lambda=2$,, IEEE Trans. Inform. Theory, 54 (2008), 3722. doi: 10.1109/TIT.2008.926394. Google Scholar [5] T. L. Alderson and K. E. Mellinger, Geometric constructions of optimal optical orthogonal codes,, Adv. Math. Commun., 2 (2008), 451. doi: 10.3934/amc.2008.2.451. Google Scholar [6] B. Berndt, R. Evans and K. S. Williams, Gauss and Jacobi Sums,, Wiley, (1997). Google Scholar [7] C. M. Bird and A. D. Keedwell, Design and applications of optical orthogonal codes-a survey,, Bull. Inst. Combin. Appl., 11 (1994), 21. Google Scholar [8] I. Bousrih, Families of rational functions over finite fields and constructions of optical orthogonal codes,, Afr. Diaspora J. Math., 3 (2005), 95. Google Scholar [9] M. Buratti and A. Pasotti, Further progress on difference families with block size $4$ or $5$,, Des. Codes Cryptogr., 56 (2010), 1. doi: 10.1007/s10623-009-9335-6. Google Scholar [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. doi: 10.1109/18.30982. Google Scholar [11] H. Chung and P. V. Kumar, Optical orthogonal codes-new bounds and an optimal construction,, IEEE Trans. Inform. Theory, 36 (1990), 866. doi: 10.1109/18.53748. Google Scholar [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. doi: 10.1109/TIT.2009.2021366. Google Scholar [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. doi: 10.1109/TIT.2010.2048504. Google Scholar [14] R. Fuji-Hara and Y. Miao, Optical orthogonal codes: their bounds and new optimal constructions,, IEEE Trans. Inform. Theory, 46 (2000), 2396. doi: 10.1109/18.887852. Google Scholar [15] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties,, IEEE Trans. Inform. Theory, 20 (1974), 90. Google Scholar [16] R. Lidl and H. Niederreiter, Finite Fields,, Cambridge Univ. Press, (1997). Google Scholar [17] F. J. MacWilliams and N. J. A. Sloan, The Theory of Error-Correcting Codes,, Twelfth editioin, (2006). Google Scholar [18] S. V. Maric, O. Moreno and C. Corrada, Multimedia transmission in fiber-optic lans using optical cdma,, J. Lightwave Technol., 14 (1996), 2149. doi: 10.1109/50.541202. Google Scholar [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. doi: 10.1109/TCOMM.2006.878835. Google Scholar [20] K. Momihara, New optimal optical orthogonal codes by restrictions to subgroups,, Finite Fields Appl., 17 (2010), 166. doi: 10.1016/j.ffa.2010.11.001. Google Scholar [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. doi: 10.1109/TIT.2007.894658. Google Scholar [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. doi: 10.1109/18.370146. Google Scholar [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. doi: 10.1109/18.135636. Google Scholar [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. Google Scholar [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. doi: 10.1109/TIT.2004.833362. Google Scholar [26] H. Stichtenoth, Algebraic Function Fields and Codes,, Second edition, (2009). Google Scholar [27] R. M. Wilson, Cyclotomy and difference families in elementary abelian groups,, J. Number Theory, 4 (1972), 17. doi: 10.1016/0022-314X(72)90009-1. Google Scholar [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. doi: 10.1109/TIT.2011.2137290. Google Scholar
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