Optimal 2-D (n×m, 3, 2, 1)-optical Orthogonal Codes

Optical orthogonal codes are commonly used as signature codes for optical code-division multiple access systems. So far, research on 2-D optical orthogonal codes has mainly concentrated on the same autocorrelation and cross-correlation constraints. In this paper, we are concerned about optimal 2-D optical orthogonal codes with the autocorrelation λa and the cross-correlation 1. Some combinatorial constructions for 2-D (n×m,k,λa,1) -optical orthogonal codes are presented. When k=3 and λa=2, the exact number of codewords of an optimal 2-D (n×m,3,2,1)-optical orthogonal code is determined for any positive integers n ≡ 0,1,3,6,9,10 (mod 12) and m ≡ 2(mod 4).

[1]  Guu-chang Yang,et al.  Optical orthogonal codes with unequal auto- and cross-correlation constraints , 1995, IEEE Trans. Inf. Theory.

[2]  Oscar Moreno,et al.  A Generalized Bose-Chowla Family of Optical Orthogonal Codes and Distinct Difference Sets , 2007, IEEE Transactions on Information Theory.

[3]  Jawad A. Salehi,et al.  Code division multiple-access techniques in optical fiber networks. II. Systems performance analysis , 1989, IEEE Trans. Commun..

[4]  O. Moreno,et al.  Multimedia transmission in fiber-optic LANs using optical CDMA , 1996 .

[5]  Jianxing Yin,et al.  A General Construction for Optimal Cyclic Packing Designs , 2002, J. Comb. Theory, Ser. A.

[6]  Yanxun Chang,et al.  Further results on optimal optical orthogonal codes with weight 4 , 2004, Discret. Math..

[7]  Hung-Lin Fu,et al.  Optimal Conflict-Avoiding Codes of Even Length and Weight 3 , 2010, IEEE Transactions on Information Theory.

[8]  S. V. Maric,et al.  Multirate fiber-optic CDMA: system design and performance analysis , 1998 .

[9]  Yanxun Chang,et al.  Further results on optimal (v, 4, 2, 1)-OOCs , 2012, Discret. Math..

[10]  G. Ge,et al.  Constructions for optimal (v, 4, 1) optical orthogonal codes , 2001, IEEE Trans. Inf. Theory.

[11]  Tao Feng,et al.  Constructions for rotational Steiner quadruple systems , 2009 .

[12]  Ruizhong Wei,et al.  Combinatorial Constructions for Optimal Two-Dimensional Optical Orthogonal Codes , 2009, IEEE Transactions on Information Theory.

[13]  Marco Buratti,et al.  Cyclic Designs with Block Size 4 and Related Optimal Optical Orthogonal Codes , 2002, Des. Codes Cryptogr..

[14]  Jianxing Yin,et al.  On constructions for optimal two-dimensional optical orthogonal codes , 2010, Des. Codes Cryptogr..

[15]  Catharine A. Baker,et al.  Extended skolem sequences , 1995 .

[16]  Oscar Moreno,et al.  Constructions of families with unequal autoand cross-correlation constraints , 2009, 2009 IEEE International Symposium on Information Theory.

[17]  Charles J. Colbourn,et al.  Recursive constructions for optimal (n,4,2)-OOCs , 2004 .

[18]  Vladimir D. Tonchev,et al.  On Conflict-Avoiding Codes of Length $n=4m$ for Three Active Users , 2007, IEEE Transactions on Information Theory.

[19]  László Györfi,et al.  Constructions of binary constant-weight cyclic codes and cyclically permutable codes , 1992, IEEE Trans. Inf. Theory.

[20]  Hirobumi Mizuno,et al.  Optical orthogonal codes obtained from conics on finite projective planes , 2004, Finite Fields Their Appl..

[21]  Jawad A. Salehi,et al.  Code division multiple-access techniques in optical fiber networks. I. Fundamental principles , 1989, IEEE Trans. Commun..

[22]  Alan C. H. Ling,et al.  The spectrum of cyclic group divisible designs with block size three , 1999 .

[23]  Yanxun Chang,et al.  Combinatorial constructions of optimal optical orthogonal codes with weight 4 , 2003, IEEE Trans. Inf. Theory.

[24]  Anita Pasotti,et al.  New results on optimal (v, 4, 2, 1) optical orthogonal codes , 2011, Des. Codes Cryptogr..

[25]  Tao Feng,et al.  Constructions for strictly cyclic 3-designs and applications to optimal OOCs with lambda=2 , 2008, J. Comb. Theory, Ser. A.

[26]  Vladimir D. Tonchev,et al.  Optimal conflict-avoiding codes for three active users , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[27]  Yanxun Chang,et al.  Constructions for optimal optical orthogonal codes , 2003, Discret. Math..

[28]  Jianxing Yin,et al.  Some combinatorial constructions for optical orthogonal codes , 1998, Discret. Math..

[29]  Charles J. Colbourn,et al.  Optimal (n, 4, 2)-OOC of small orders , 2004, Discret. Math..

[30]  Gennian Ge,et al.  Perfect difference families, perfect difference matrices, and related combinatorial structures , 2010 .

[31]  Yanxun Chang,et al.  Constructions of optimal optical orthogonal codes with weight five , 2005 .

[32]  Ryoh Fuji-Hara,et al.  Optical orthogonal codes: Their bounds and new optimal constructions , 2000, IEEE Trans. Inf. Theory.

[33]  Jianxing Yin,et al.  Two-Dimensional Optical Orthogonal Codes and Semicyclic Group Divisible Designs , 2010, IEEE Transactions on Information Theory.

[34]  P. Vijay Kumar,et al.  Large Families of Asymptotically Optimal Two-Dimensional Optical Orthogonal Codes , 2012, IEEE Transactions on Information Theory.

[35]  Zhen Zhang,et al.  New constructions of optimal cyclically permutable constant weight codes , 1995, IEEE Trans. Inf. Theory.

[36]  Koji Momihara,et al.  Strong difference families, difference covers, and their applications for relative difference families , 2009, Des. Codes Cryptogr..

[37]  Hongxi Yin,et al.  A new family of 2-D optical orthogonal codes and analysis of its performance in optical CDMA access networks , 2006 .

[38]  Kun Wang,et al.  Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes , 2012, Des. Codes Cryptogr..

[39]  Hung-Lin Fu,et al.  Optimal conflict-avoiding codes of length n ≡ 0 (mod 16) and weight 3 , 2009, Des. Codes Cryptogr..

[40]  Keith E. Mellinger,et al.  2-dimensional Optical Orthogonal Codes from Singer Groups , 2009, Discret. Appl. Math..

[41]  R. Julian R. Abel,et al.  Some progress on (v, 4, 1) difference families and optical orthogonal codes , 2004, J. Comb. Theory, Ser. A.

[42]  Keith E. Mellinger,et al.  Families of Optimal OOCs With $\lambda = 2$ , 2008, IEEE Transactions on Information Theory.

[43]  Tuvi Etzion,et al.  Constructions for optimal constant weight cyclically permutable codes and difference families , 1995, IEEE Trans. Inf. Theory.

[44]  Tao Feng,et al.  Combinatorial Constructions for Optimal Two-Dimensional Optical Orthogonal Codes With $\lambda=2$ , 2011, IEEE Transactions on Information Theory.

[45]  Yanxun Chang,et al.  Two classes of optimal two-dimensional OOCs , 2012, Des. Codes Cryptogr..

[46]  Wing C. Kwong,et al.  Performance comparison of asynchronous and synchronous code-division multiple-access techniques for fiber-optic local area networks , 1991, IEEE Trans. Commun..

[47]  Yanxun Chang,et al.  A New Class of Optimal Optical Orthogonal Codes With Weight Five , 2004, IEEE Trans. Inf. Theory.

[48]  Keith E. Mellinger,et al.  Spreads, arcs, and multiple wavelength codes , 2011, Discret. Math..

[49]  Yanxun Chang,et al.  Optimal (4up, 5, 1) optical orthogonal codes , 2004 .

[50]  Seung-Woo Seo,et al.  New construction of multiwavelength optical orthogonal codes , 2002, IEEE Trans. Commun..

[51]  Koji Momihara,et al.  Bounds and Constructions of Optimal ($n, 4, 2, 1$) Optical Orthogonal Codes , 2009, IEEE Transactions on Information Theory.

[52]  Fan Chung Graham,et al.  Optical orthogonal codes: Design, analysis, and applications , 1989, IEEE Trans. Inf. Theory.

[53]  P. Vijay Kumar,et al.  Optical orthogonal codes-New bounds and an optimal construction , 1990, IEEE Trans. Inf. Theory.

[54]  Yanxun Chang,et al.  Further results on (v, 4, 1)-perfect difference families , 2010, Discret. Math..

[55]  Guu-chang Yang,et al.  Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks , 1997, IEEE Trans. Commun..

[56]  Anita Pasotti,et al.  Further progress on difference families with block size 4 or 5 , 2010, Des. Codes Cryptogr..

[57]  Keith E. Mellinger,et al.  Geometric constructions of optimal optical orthogonal codes , 2008, Adv. Math. Commun..