Constructions of optimal variable-weight OOCs via quadratic residues

Abstract

Variable-weight optical orthogonal code (OOC) was introduced by G. C. Yang [IEEE Trans. Commun., 1996, 44: 47-55] for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, seven new infinite classes of optimal (v, {3, 4, 6}, 1,Q)-OOCs are constructed.

Keywords

Cyclic packing / optical orthogonal code (OOC) / quadratic residue / relative difference family / variable-weight OOC

References

[1] Abel R J R, Buratti M. Some progress on (v, 4, 1) difference families and optical orthogonal codes. J Combin Theory, 2004, 106: 59-75

[2] Bitan S, Etzion T. Constructions for optimal constant weight cyclically permutable codes and difference families. IEEE Trans Inform Theory, 1995, 41: 77-87

[3] Brickell E F, Wei V. Optical orthogonal codes and cyclic block designs. Congr Numer, 1987, 58: 175-182

[4] Buratti M. Recursive constructions for difference matrices and relative difference families. J Combin Des, 1988, 6: 165-182

[5] Buratti M. Pairwise balanced designs from finite fields. Discrete Math, 1999, 208/209: 103-117

[6] Buratti M. Old and new designs via strong difference families. J Combin Des, 1999, 7: 406-425

[7] Buratti M. On point-regular linear spaces. J Statist Plann Inference, 2001, 94: 139-146

[8] Buratti M. Cyclic designs with block size 4 and related optimal optical orthogonal codes. Des Codes Cryptogr, 2002, 26: 111-125

[9] Buratti M, Gionfriddo L. Strong difference families over arbitrary graphs. J Combin Des, 2008, 16: 443-461

[10] Buratti M, Pasotti A. Graph decompositions with the use of difference matrices. Bull Inst Combin Appl, 2006, 47: 23-32

[11] Buratti M, Wei Y, Wu D, Fan P, Cheng M. Relative difference families with variable block sizes and their related OOCs. IEEE Trans Inform Theory, 2011, 57: 7489-7497

[12] Chang Y, Fuji-Hara R, Miao Y. Combinatorial constructions of optimal optical orthogonal codes with weight 4. IEEE Trans Inform Theory, 2003, 49: 1283-1292

[13] Chang Y, Ji L. Optimal (4up, 5, 1) optical orthogonal codes. J Combin Des, 2004, 12: 346-361

[14] Chang Y, Miao Y. Constructions for optimal optical orthogonal codes. Discrete Math, 2003, 261: 127-139

[15] Chen K, Ge G, Zhu L. Starters and related codes. J Statist Plann Inference, 2000, 86: 379-395

[16] Chen K, Zhu L. Existence of (q, 6, 1) difference families with qa prime power. Des Codes Cryptogr, 1998, 15: 167-174

[17] Chu W, Colbourn C J. Recursive constructions for optimal (n, 4, 2)-OOCs. J Combin Des, 2004, 12: 333-345

[18] Chu W, Golomb S W. A new recursive construction for optical orthogonal codes. IEEE Trans Inform Theory, 2003, 49: 3072-3076

[19] Chung F R K, Salehi J A, Wei V K. Optical orthogonal codes: Design, analysis, and applications. IEEE Trans Inform Theory, 1989, 35: 595-604

[20] Chung H, Kumar P V. Optical orthogonal codes-new bounds and an optimal construction. IEEE Trans Inform Theory, 1990, 36: 866-873

[21] Fuji-Hara F, Miao Y. Optical orthogonal codes: Their bounds and new optimal constructions. IEEE Trans Inform Theory, 2000, 46: 2396-2406

[22] Fuji-Hara F, Miao Y, Yin J. Optimal (9v, 4, 1) optical orthogonal codes. SIAM J Discrete Math, 2001, 14: 256-266

[23] Ge G, Yin J. Constructions for optimal (v, 4, 1) optical orthogonal codes. IEEE Trans Inform Theory, 2001, 47: 2998-3004

[24] Golomb S W. Digital Communication with Space Application. Los Altos: Penisula, 1982

[25] Gu F R, Wu J. Construction and performance analysis of variable-weight optical orthogonal codes for asynchronous optical CDMA systems. J Lightw Technol, 2005, 23: 740-748

[26] Jiang J, Wu D, Fan P. General constructions of optimal variable-weight optical orthogonal codes. IEEE Trans Inform Theory, 2011, 7: 4488-4496

[27] Ma S, Chang Y. A new class of optimal optical orthogonal codes with weight five. IEEE Trans Inform Theory, 2004, 50: 1848-1850

[28] Ma S, Chang Y. Constructions of optimal optical orthogonal codes with weight five. J Combin Des, 2005, 13: 54-69

[29] Massey J L, Mathys P. The collision channel without feedback. IEEE Trans Inform Theory, 1985, 31: 192-204

[30] Mishima M, Fu H L, Uruno S. Optimal conflict-avoiding codes of length n ≡ 0 (mod 16) and weight 3. Des Codes Cryptogr, 2009, 52: 275-291

[31] Momihara K. Necessary and sufficient conditions for tight equi-difference conflict avoiding codes of weight three. Des Codes Cryptogr, 2007, 45: 379-390

[32] Momihara K. On cyclic 2(k-1)-support (n, k)_k-1 difference families. Finite Fields Appl, 2009, 15: 415-427

[33] Momihara K. Strong difference families, difference covers, and their applications for relative difference families. Des Codes Cryptogr, 2009, 51: 253-273

[34] Momihara K. New optimal optical orthogonal codes by restrictions to subgroups. Finite Fields Appl, 2011, 17: 166-182

[35] Momihara K, Buratti M. Bounds and constructions of optimal (n, 4, 2, 1) optical orthogonal codes. IEEE Trans Inform Theory, 2009, 55: 514-523

[36] Momihara K, Müller M, Satoh J, Jimbo M. Constant weight conflict-avoiding codes. SIAM J Discrete Math, 2007, 21: 959-979

[37] Salehi J A. Code division multiple access techniques in optical fiber networks—Part I Fundamental principles. IEEE Trans Commun, 1989, 37: 824-833

[38] Salehi J A. Emerging optical code-division multiple-access communications systems. IEEE Network, 1989, 3: 31-39

[39] Salehi J A, Brackett C A. Code division multiple access techniques in optical fiber networks—Part II Systems performance analysis. IEEE Trans Commun, 1989, 37: 834-842

[40] Vecchi M P, Salehi J A. Neuromorphic networks based on sparse optical orthogonal codes. In: Neural Information Processing Systems—Natural and Synthetic. New York: Amer Inst Phys, 1988, 814-823

[41] Wu D, Fan P, Li H, Parampalli U. Optimal variable-weight optical orthogonal codes via cyclic difference families. In: Proc 2009 IEEE International Symposium on Information Theory ISIT’9, June 28-July 3. 2009, 448-452

[42] Wu D, Zhao H, Fan P, Shinohara S. Optimal variable-weight optical orthogonal codes via difference packings. IEEE Trans Inform Theory, 2010, 56: 4053-4060

[43] Yang G C. Variable weight optical orthogonal codes for CDMA networks with multiple performance requirements. GLOBECOM ’93, IEEE, 1993, 1: 488-492

[44] Yang G C. Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements. IEEE Trans Commun, 1996, 44: 47-55

[45] Yang G C, Fuja T E. Optical orthogonal codes with unequal auto- and cross-correlation constraints. IEEE Trans Inform Theory, 1995, 41: 96-106

[46] Yin J. Some combinatorial constructions for optical orthogonal codes. Discrete Math, 1998, 185: 201-219

[47] Zhao H, Wu D, Fan P. Construction of optimal variable-weight optical orthogonal codes. J Combin Des, 2010, 18: 274-291

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