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Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes

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Abstract

The existence problem for a semicyclic group divisible design (SCGDD) of type m n with block size 4 and index unity, denoted by 4-SCGDD, has been studied for any odd integer m to construct a kind of two-dimensional optical orthogonal codes (2-D OOCs) with the AM-OPPW (at most one-pulse per wavelength) restriction. In this paper, the existence of a 4-SCGDD of type m n is determined for any even integer m, with some possible exceptions. A complete asymptotic existence result for k-SCGDDs of type m n is also obtained for all larger k and odd integer m. All these SCGDDs are used to derive new 2-D OOCs with the AM-OPPW restriction, which are optimal in the sense of their sizes.

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References

  • Abel R.J.R., Buratti M.: Difference families. In: Colbourn, C.J., Dinitz, J.H (eds) CRC Handbook of Combinatorial Designs, 2nd edn., pp. 392–410. Chapman & Hall/CRC, Boca Raton, FL (2007)

    Google Scholar 

  • Abel R.J.R., Bennett F.E., Greig M.: PBD-closure. In: Colbourn, C.J., Dinitz, J.H (eds) CRC Handbook of Combinatorial Designs, 2nd edn., pp. 247–255. Chapman & Hall/CRC, Boca Raton, FL (2007)

    Google Scholar 

  • Abel R.J.R., Colbourn C.J., Dinitz J.H.: Mutually orthogonal Latin squares (MOLS). In: Colbourn, C.J., Dinitz, J.H (eds) CRC Handbook of Combinatorial Designs, 2nd edn., pp. 160–193. Chapman & Hall/CRC, Boca Raton, FL (2007)

    Google Scholar 

  • Abel R.J.R., Chan N.H.N., Combe D., Palmer W.D.: Existence of GBRDs with block size 4 and BRDs with block size 5. Des. Codes Cryptogr. doi:10.1007/s10623-010-9477-6.

  • Bose R.C.: On the construction of balanced incomplete block designs. Ann. Engen. 9, 353–399 (1939)

    Article  Google Scholar 

  • Buratti M.: Cyclic designs with block size 4 and the related optimal optical orthogonal codes. Des. Codes Cryptogr. 26, 111–125 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Chang Y., Miao Y.: Constructions for optimal optical orthogonal codes. Discrete Math. 261, 127–139 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Chang Y., Yin J.: Further results on optimal optical orthogonal codes with weight 4. Discrete Math. 279, 135–151 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Chung F.R.K., Salehi J.A., Wei V.K.: Optical orthogonal codes: design, analysis, and applications. IEEE Trans. Inform. Theory 35, 595–604 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Colbourn C.J., Jiang Z.: The spectrum for rotational Steiner triple systems. J. Combin. Des. 4, 205–217 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • De Launey W.: (0, G)-designs and applications. Ph.D. thesis, University of Sydney (1987).

  • De Launey W.: Bhaskar Rao designs. In: Colbourn, C.J., Dinitz, J.H (eds) CRC Handbook of Combinatorial Designs, 2nd edn., pp. 299–301. Chapman & Hall/CRC, Boca Raton, FL (2007)

    Google Scholar 

  • Gallant R.P., Jiang Z., Ling A.C.H.: The spectrum of cyclic group divisible designs with block size three. J. Combin. Des. 7, 95–105 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Ge G.: On (g, 4; 1)-difference matrices. Discrete Math. 301, 164–174 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Ge G., Greig M., Seberry J.: Generalized Bhaskar Rao designs with block size 4 signed over elementary Abelian groups. J. Combin. Math. Combin. Comput. 46, 3–45 (2003)

    MathSciNet  MATH  Google Scholar 

  • Ge G., Wang J., Wei R.: MGDD with block size 4 and its application to sampling designs. Discrete Math. 272, 277–283 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • http://math.suda.edu.cn/wangjm/dcr.pdf.

  • Jiang Z.: Concerning cyclic group divisible designs with block size three. Australas. J. Combin. 13, 227–245 (1996)

    MathSciNet  MATH  Google Scholar 

  • John J.A.: Cyclic Designs. Chapman and Hall, New York (1987)

    MATH  Google Scholar 

  • Mendez A.J., Gagliardi R.M., Hernandez V.J., Bennett C.V., Lennon W.J.: Design and performance analysis of wavelength/time (W/T) matrix codes for optical CDMA. J. Lightwave Technol. 21, 2524–2533 (2003)

    Article  Google Scholar 

  • Omrani R., Kummar P.V.: Codes for Optical CDMA. In: Proc. of SETA 2006, Springer-Verlags Lecture Notes in Computer Science Series, vol. 4086, pp. 34–46 (2006).

  • Omrani R., Elia P., Kumar P.V.: New constructions and bounds for 2-D optical orthogonal codes. In: Proc. of SETA 2004, Springer-Verlags Lecture Notes in Computer Science Series, vol. 3486, pp. 389–395 (2005).

  • Salehi J.A.: Code-division multiple access techniques in optical fiber networks. Part I. Fundamental principles. IEEE Trans. Commun. 37, 824–833 (1989)

    Google Scholar 

  • Seberry J.: Regular group divisible designs and Bhaskar Rao designs with block size three. J. Stat. Plann. Infer. 10, 69–82 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Shivaleea E.S., Selvarajan A., Srinivas T.: Two-dimensional optical orthogonal codes for fiber-optic cdma networks. J. Lightwave Technol. 23, 647–654 (2005)

    Article  Google Scholar 

  • Wang J., Yin J.: Two-dimensional optical orthogonal codes and semicyclic group divisible designs. IEEE Trans. Inform. Theory 56, 2177–2187 (2010)

    Article  MathSciNet  Google Scholar 

  • Wang J., Shan X., Yin J.: On constructions for optimal two-dimensional optical orthogonal codes. Des. Codes Cryptogr. 54, 43–60 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Yang G.C., Kwong W.C.: Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks. IEEE Trans. Communun. 45, 1426–1434 (1997)

    Article  Google Scholar 

  • Yin J.: Some combinatorial constructions for optical orthogonal codes. Discrete Math. 185, 201–219 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Yin J.: A general construction for optimal cyclic packing designs. J. Combin. Theory Ser. A 97, 272–284 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Yin J.: Cyclic difference packing and covering arrays. Des. Codes Cryptogr. 37, 281–292 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Soochow University, Suzhou, 215006, China

    Kun Wang & Jianmin Wang

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  1. Kun Wang
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  2. Jianmin Wang
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Correspondence to Jianmin Wang.

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Communicated by J. D. Key.

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Wang, K., Wang, J. Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes. Des. Codes Cryptogr. 63, 305–319 (2012). https://doi.org/10.1007/s10623-011-9556-3

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  • DOI: https://doi.org/10.1007/s10623-011-9556-3

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