Advertisement

Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 365–374 | Cite as

Classification of optimal (vk, 1) binary cyclically permutable constant weight codes with \(k=5,\) 6 and 7 and small lengths

  • Tsonka BaichevaEmail author
  • Svetlana Topalova
Article
  • 84 Downloads
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

We classify optimal (vk, 1) binary cyclically permutable constant weight (CPCW) codes with small v and \(k=5, 6\) and 7. Binary CPCW codes have multiple applications in contemporary communications and are closely related to cyclic binary constant weight codes, difference packings, optical orthogonal codes, cyclic difference families and cyclic block designs. The presented small length codes can be used in relevant applications, as well as with recursive constructions for bigger lengths. We also establish that a (127, 7, 1) cyclic difference family does not exist.

Keywords

Binary cyclically permutable constant weight (CPCW) codes Difference packings Cyclic difference families Optical orthogonal codes (OOC) 

Mathematics Subject Classification

94B15 05B10 05B40 

Notes

Acknowledgements

The authors are grateful to the anonymous referee whose suggestions improved the paper.

References

  1. 1.
    Baicheva T., Topalova S.: Classification of optimal (v,4,1) binary cyclically permutable constant weight codes and cyclic S(2,4, v) designs with \(v \le 76\). Probl. Inf. Transm. 47(3), 224–231 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baicheva T., Topalova S.: Classification results for (v, k, 1) cyclic difference families with small parameters. In: Deza M., Petitjean M., Markov K. (eds.) Mathematics of Distances and Applications, pp. 24–30. International Book Series: Information Science and Computing, Book 25ITHEA Publication, Sofia (2012).Google Scholar
  3. 3.
    Baicheva T., Topalova S.: Classification of maximal optical orthogonal codes of weight 3 and small lengths. Serdica J. Comput. 9(1), 83–92 (2015).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bitan S., Etzion J.: Constructions for optimal constant weight cyclically permutable codes and difference families. IEEE Trans. Inf. Theory 41(1), 77–87 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brickell E.F., Wei V.K.: Optical orthogonal codes and cyclic block designs. Congr. Numer. 58, 175–192 (1987).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Buratti M., Pasotti A.: Further progress on difference families with block size 4 or 5. Des. Codes Cryptogr. 56, 1–20 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buratti M., Momihara K., Pasotti A.: New results on optimal (v, 4, 2, 1) optical orthogonal codes. Des. Codes Cryptogr. 58, 89–109 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chang Y., Ji L.: Optimal (4up, 5, 1) optical orthogonal codes. J. Comb. Des. 12, 346–361 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen K., Zhu L.: Existence of (q, 6, 1) difference families with q a prime power. Des. Codes Cryptogr. 15, 167–173 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen K., Zhu L.: Existence of (q, k, 1) difference families with q a prime power and k=4,5. J. Comb. Des. 7(1), 21–30 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen K., Wei R., Zhu L.: Existence of (q, 7, 1) difference families with q a prime power. J. Comb. Des. 10, 126–138 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chu W., Colbourn C.J.: Optimal (n, 4, 2)- OOC of small order. Discret. Math. 279, 163–172 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chu W., Golomb S.W.: A new recursive construction for optical orthogonal codes. IEEE Trans. Inf. Theory 49, 3072–3076 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Colbourn C.J.: On cyclic Steiner systems S(2,6,91). Abstr. Am. Math. Soc. 2, 463 (1981).Google Scholar
  15. 15.
    Colbourn Ch., Dinitz J. (eds.): Handbook of Combinatorial Designs, 2nd edn (Discrete Mathematics and Its Applications, ser. ed. K. Rosen). CRC Press, Boca Raton (2007).Google Scholar
  16. 16.
    Colbourn M.J., Mathon R.A.: On cyclic Steiner 2-designs. Ann. Discret. Math. 7, 215–253 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fuji-Hara R., Miao Y.: Optical orthogonal codes: their bounds and new optimal constructions. IEEE Trans. Inf. Theory 46, 2396–2406 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hanani H.: Balanced incomplete block designs and related designs. Discret. Math. 11, 255–369 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Janko Z., Tonchev V.D.: Cyclic 2-(91, 6, 1) designs with multiplier automorphisms. Discret. Math. 97, 265–268 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Julian R., Abel R., Costa S., Finizio N.: Directed-ordered whist tournaments and (v,5,1) difference families: existence results and some new classes of Z-cyclic solutions. Discret. Appl. Math. 143, 43–53 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ma S., Chang Y.: A new class of optimal optical orthogonal codes with weight five. IEEE Trans. Inf. Theory 50, 1848–1850 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ma S., Chang Y.: Constructions of optimal optical orthogonal codes with weight five. J. Comb. Des. 13, 54–69 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Moreno O., Zhang Z., Kumar P.V., Zinoviev V.A.: New constructions of optimal cyclically permutable constant weight codes. IEEE Trans. Inf. Theory 41, 448–455 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nguyen Q.A., Gyöfri L., Massey J.L.: Constructions of binary constant weight cyclic codes and cyclically permutable codes. IEEE Trans. Inf. Theory 38(3), 940–949 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Stinson D.R., Wei R., Yin J.: Packings. In: Colbourn C.J., Dinitz J.H. (eds.) The CRC Handbook of Combinatorial Designs, 2nd edn, pp. 550–556. Chapman and Hall/CRC Press, Boca Raton (2006).Google Scholar
  26. 26.
    Tang Y., Yin J.: Combinatorial constructions for a class of optimal OOCs. Sci. China (A) 45, 1268–1275 (2002).zbMATHGoogle Scholar
  27. 27.
    Wang S., Wang L., Wang J.: A new class of optimal optical orthogonal codes with weight six. In: Proceedings of Seventh International Workshop of Signal Design and its Applications in Communications (2016).Google Scholar
  28. 28.
    Yin J.: Some combinatorial constructions for optical orthogonal codes. Discret. Math. 185, 201–219 (1998).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.D. A. Tsenov Academy of EconomicsSvishtovBulgaria

Personalised recommendations