Designs, Codes and Cryptography

, Volume 58, Issue 1, pp 89–109

New results on optimal (v, 4, 2, 1) optical orthogonal codes

Article

Abstract

We investigate further the existence question regarding optimal (v, 4, 2, 1) optical orthogonal codes begun in Momihara and Buratti (IEEE Trans Inform Theory 55:514–523, 2009). We give some non-existence results for infinitely many values of v ≡ ± 3 (mod 9) and several explicit constructions for infinite classes of perfect optical orthogonal codes.

Keywords

Optical orthogonal code Difference family Theorem of Weil on multiplicative character sums 

Mathematics Subject Classification (2000)

05B30 94B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abel R.J.R., Buratti M.: Some progress on (v, 4, 1) difference families and optical orthogonal codes. J. Combin. Theory Ser. A 106, 59–75 (2004).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abel, R.J.R., Buratti, M.: Difference families. In: Colbourn, C.J., Dinitz, J.H. (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn., pp. 392–409. Chapman & Hall/CRC Press, Boca Raton, FL (2006).Google Scholar
  3. 3.
    Alderson T.L., Mellinger K.E.: Families of optimal OOCs with λ = 2. IEEE Trans. Inform. Theory 54, 3722–3724 (2008).CrossRefMathSciNetGoogle Scholar
  4. 4.
    Alderson T.L., Mellinger K.E.: Geometric constructions of optimal optical orthogonal codes. Adv. Math. Commun. 2, 451–467 (2008).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Beth T., Jungnickel D., Lenz H.: Design Theory. Cambridge University Press, Cambridge (1999).Google Scholar
  6. 6.
    Bird I.C.M., Keedwell A.D.: Design and applications of optical orthogonal codes—a survey. Bull. Inst. Combin. Appl. 11, 21–44 (1994).MATHMathSciNetGoogle Scholar
  7. 7.
    Bonisoli A., Buratti M., Rinaldi G.: Sharply transitive decompositions of complete graphs into generalized Petersen graphs. Innov. Incidence Geom. 6(7), 95–109 (2009).MathSciNetGoogle Scholar
  8. 8.
    Bose R.C.: On the construction of balanced incomplete block designs. Ann. Eugenics 9, 353–399 (1939).MathSciNetGoogle Scholar
  9. 9.
    Brickell E.F., Wei V.: Optical orthogonal codes and cyclic block designs. Congr. Numer. 58, 175–182 (1987).MathSciNetGoogle Scholar
  10. 10.
    Buratti M.: Recursive constructions for difference matrices and relative difference families. J. Combin. Des. 6, 165–182 (1998).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Buratti M.: Cyclic designs with block size 4 and related optimal optical orthogonal codes. Des. Codes Cryptogr. 26, 111–125 (2002).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Buratti M., Pasotti A.: Combinatorial designs and the theorem of Weil on multiplicative character sums. Finite Fields Appl. 15, 332–344 (2009).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Buratti M., Pasotti A.: Further progress on difference families with block size 4 or 5. Des. Codes Cryptogr. Published online (2009). doi:10.1007/s10623-009-9335-6.
  14. 14.
    Chang Y.X., Ji L.: Optimal (4up,5,1) optical orthogonal codes. J. Combin. Des. 12, 135–151 (2004).MathSciNetGoogle Scholar
  15. 15.
    Chen K., Zhu L.: Existence of (q,6,1) difference families with q a prime power. Des. Codes Cryptogr. 15, 167–173 (1998).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Chen K., Zhu L.: Existence of (q,k,1) difference families with q a prime power and k = 4, 5. J. Combin. Des. 7, 21–30 (1999).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Chu W., Colbourn C.J.: Recursive constructions for optimal (n, 4, 2)-OOCs. J. Combin. Des. 12, 333–345 (2004).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    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).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Colbourn C.J., Dinitz J.H., Stinson D.R.: Applications of combinatorial designs to communications, cryptography, and networking. In: Lamb J.D., Preece D.A. (eds.) Surveys in Combinatorics, pp. 37–100. Cambridge University Press, London (1999).Google Scholar
  20. 20.
    Feng T., Chang Y., Ji L.: Constructions for strictly cyclic 3-designs and applications to optimal OOCs with λ = 2. J. Combin. Theory Ser. A 115, 1527–1551 (2008).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ma S., Chang Y.: Constructions of optimal optical orthogonal codes with weight five. J. Combin. Des. 13, 54–69 (2005).MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Mishima M., Fu H.L., Uruno S.: Optimal conflict-avoiding codes of length n ≡ 0 (mod 16) and weight 3. Des. Codes Cryptogr. 52, 275–291 (2009).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Momihara K.: On cyclic 2(k −1)-support (n, k)k−1 difference families. Finite Fields Appl. 15, 415–427 (2009).MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Momihara K.: Strong difference families, difference covers, and their applications for relative difference families. Des. Codes Cryptogr. 51, 253–273 (2009).CrossRefMathSciNetGoogle Scholar
  25. 25.
    Momihara K., Buratti M.: Bounds and constructions of optimal (n, 4, 2, 1) Optical orthogonal codes. IEEE Trans. Inform. Theory 55, 514–523 (2009).CrossRefMathSciNetGoogle Scholar
  26. 26.
    Wilson R.M.: Cyclotomic and difference families in elementary abelian groups. J. Number Theory 4, 17–47 (1972).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Yang G.C., Fuja T.E.: Optical orthogonal codes with unequal auto- and cross-correlation constraints. IEEE Trans. Inform. Theory 41, 96–106 (1995).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di PerugiaPerugiaItaly
  2. 2.Graduate School of Information ScienceNagoya UniversityNagoyaJapan
  3. 3.Dipartimento di Matematica, Facoltà di IngegneriaUniversità degli Studi di BresciaBresciaItaly

Personalised recommendations