Advertisement

Constructions of Optical Orthogonal Codes Based on Cyclic t-Wise Quasi-Difference Matrices

  • Koji Momihara
Article

Abstract

In this article, some constructions of cyclic t-wise quasi-difference matrices are presented. Based on these results, many new series of optimal and asymptotically optimal optical orthogonal codes are obtained.

AMS Subject Classification

05B20 05B40 

Key-words

Cyclotomic number Jacobi sum Optical orthogonal code Relative difference family t-Wise quasi-difference matrix 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abel, R. J. R., and M. Buratti. 2006. Difference families. In The CRC handbook of combinatorial designs, 2nd ed., ed. C. J. Colbourn and J. H. Dinitz. Boca Raton, FL, Chapman & Hall/CRC. pp. 392–410.Google Scholar
  2. Alderson, T. L. 2007. Optical orthogonal codes and arcs in PG(d, q). Finite Fields Appl., 13, 762–768.MathSciNetCrossRefGoogle Scholar
  3. Alderson, T. L., and K. E. Mellinger. 2008a. Classes of optical orthogonal codes from arcs in root subspaces. Discrete Math., 308, 1093–1101.MathSciNetCrossRefGoogle Scholar
  4. Alderson, T. L., and K. E. Mellinger. 2008b. Geometric constructions of optimal optical orthogonal codes. Adv. Math. Commun., 2, 451–467.MathSciNetCrossRefGoogle Scholar
  5. Berndt, B., R. Evans, and K. S. Williams. 1997. Gauss and Jacobi sums. New York, Wiley.MATHGoogle Scholar
  6. Beth, T., D. Jungnickel, and H. Lenz. 1999. Design theory. Cambridge, Cambridge University Press.MATHGoogle Scholar
  7. Braumert, L. D., W. H. Mills, and R. L. Ward. 1982. Uniform cyclotomy. J. Number Theory, 14, 67–82.MathSciNetCrossRefGoogle Scholar
  8. Bush, K. A. 1952. Orthogonal arrays of index unity. Ann. Math. Statistics, 23, 426–434.MathSciNetCrossRefGoogle Scholar
  9. Chang, Y., and Y. Miao. 2003. Constructions for optimal optical orthogonal codes. Discrete Math., 261, 127–139.MathSciNetCrossRefGoogle Scholar
  10. Chu, W., and C. W. Golomb. 2003. A new recursive construction for optical orthogonal codes. IEEE Trans. Inform. Theory, 49, 3072–3076.MathSciNetCrossRefGoogle Scholar
  11. Chung, F. R. K., J. A. Salehi, and V. K. Wei. 1989. Optical orthogonal codes: Design analysis and applications. IEEE Trans. Inform. Theory, 35, 595–604.MathSciNetCrossRefGoogle Scholar
  12. Chung, H., and P. V. Kumar. 1990. Optical orthogonal codes—New bounds and an optimal construction. IEEE Trans. Inform. Theory, 36, 866–873.MathSciNetCrossRefGoogle Scholar
  13. Colbourn, C. J., 2006. Difference matrices. In The CRC handbook of combinatorial designs, 2nd ed., ed. C. J. Colbourn and J. H. Dinitz. Boca Raton, FL, Chapman & Hall/CRC. pp. 411–419.Google Scholar
  14. Colbourn, M. J., and C. J. Colbourn. 1984. Recursive constructions for cyclic block designs. J. Statist. Plan. Inference, 10, 97–103.MathSciNetCrossRefGoogle Scholar
  15. Feng, T., Y. Chang, and L. Ji. 2008. Constructions for strictly cyclic 3-designs and applications to optimal oocs with λ = 2. J. Combin. Theory, Ser. A, 115, 1527–1551.MathSciNetCrossRefGoogle Scholar
  16. Helleseth, T. 2006. Optical orthogonal codes. In The CRC handbook of combinatorial designs, 2nd ed., ed. C. J. Colbourn and J. H. Dinitz. Boca Raton, FL, Chapman & Hall/CRC.Google Scholar
  17. Lidl, R., and H. Niederreiter. 1997. Finite fields. Cambridge, Cambridge University Press.MATHGoogle Scholar
  18. Momihara, K. 2011. New optimal optical orthogonal codes by restrictions to sub-groups. Finite Fields Appl., 17, 166–182.MathSciNetCrossRefGoogle Scholar
  19. Moreno, O., R. Omrani, P. V. Kumar, and H. Lu. 2007. A generalized bose-chowla family of optical orthogonal codes and distinct difference sets. IEEE Trans. Inform. Theory, 53, 1907–1910.MathSciNetCrossRefGoogle Scholar
  20. Schmidt, W. M., 2004. Equations over finite fields; An elementary approach, 2nd ed. Herber City, Kendrick Press.Google Scholar
  21. Storer, T. 1967. Cyclotomy and difference sets. Lectures in Advanced Mathematics, Chicago, Markham Publishing Company.MATHGoogle Scholar
  22. Yamada, M. 2007. Supplementary difference sets constructed from (q + 1)st cyclotomic classes in GF(q2). Austr. J. Combin., 39, 73–87.MATHGoogle Scholar
  23. Yin, J. 2002. A general construction for optimal cyclic packing designs. J. Combin. Theory, Ser. A, 97, 272–284.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.Faculty of EducationKumamoto UniversityKumamotoJapan

Personalised recommendations