Designs, Codes and Cryptography

, Volume 57, Issue 3, pp 383–397 | Cite as

Existence of directed BIBDs with block size 7 and related perfect 5-deletion-correcting codes of length 7



Wang and Yin have established the existence of a directed BIBD with block size 7 and index 1 (a DBIBD(v, 7, 1)) for all v ≡ 1, 7 (mod 21) except for v = 22 and possibly for 68 other cases. In this paper, we reduce the number of possible exceptions to 4, namely v = 274, 358, 400, 526. Correspondingly, for all such v, our results establish the existence of a T(2, 7, v)-code or equivalently a perfect 5-deletion-correcting code with words of length 7 over an alphabet of size v, where all the coordinates must be different. In the process, we also reduce the possible exceptions for (v, 7, 2)-BIBDs to 2 cases, v = 274 and 358 (in addition to the non-existent (22, 7, 2)-BIBD).


BIBD Directed BIBD (DBIBD) Directed GDD (DGDD) 

Mathematics Subject Classification (2000)

Primary 05B05 


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  1. 1.
    Abel R.J.R.: Some new BIBDs with block size 7. J. Combin. Des. 8, 146–150 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abel R.J.R., Colbourn C.J., Dinitz J.H.: Mutually orthogonal latin squares (MOLS). In: Colbourn, C.J., Dinitz, J.H. (eds) The CRC Handbook of Combinatorial Designs, 2nd edn, pp. 160–193. CRC Press, Boca Raton, FL (2006)Google Scholar
  3. 3.
    Abel R.J.R., Greig M.: Balanced incomplete block designs with block size 7. Des. Codes Cryptogr. 13, 5–30 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baker R.D.: An elliptic semiplane. J. Combin. Theory Ser. A 25, 193–195 (1978)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bennett F.E., Shalaby N., Yin J.: Existence of directed GDDs with block size five. J. Combin. Des. 6, 389–402 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bennett F.E., Wei R., Yin J., Mahmoodi A.: Existence of DBIBDs with block size six. Util. Math. 43, 205–217 (1993)MATHMathSciNetGoogle Scholar
  7. 7.
    Bours P.A.H.: On the construction of perfect deletion-correcting codes using design theory. Des. Codes Cryptogr. 6, 5–20 (1995)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Colbourn C.J., Rosa A.: Directed and Mendelsohn triple systems. In: Dinitz, J.H., Stinson, D.R. (eds) Combinatorial Design Theory: A Collection of Surveys, pp. 97–136. Wiley, New York (1992)Google Scholar
  9. 9.
    Hanani H.: Balanced incomplete block designs and related designs. Discrete Math. 11, 255–369 (1975)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Mahmoodi A.: Combinatorial and algorithmic aspects of directed designs. PhD Thesis, University of Toronto (1996).Google Scholar
  11. 11.
    Sarvate D.G.: All directed GDDs with block size three, λ1 = 0, exist. Util. Math. 26, 311–317 (1984)MATHMathSciNetGoogle Scholar
  12. 12.
    Sarvate D.G.: Some results on directed and cyclic designs. Ars Combin. 19A, 179–190 (1985)MathSciNetGoogle Scholar
  13. 13.
    Seberry J.R., Skillicorn D.: All directed BIBDs with k = 3 exist. J. Combin. Theory Ser. A 29, 244–248 (1980)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Seiden E.: A method of construction of resolvable BIBD. Sankhyã. 25A, 293–294 (1963)MathSciNetGoogle Scholar
  15. 15.
    Street D.J., Seberry J.: All DBIBDs with block size four exist. Util. Math. 18, 27–34 (1980)MATHMathSciNetGoogle Scholar
  16. 16.
    Street D.J., Wilson W.H.: On directed balanced incomplete block designs with block size five. Util. Math. 18, 161–174 (1980)MATHMathSciNetGoogle Scholar
  17. 17.
    Wang J., Yin J.: Constructions for perfect 5-deletion-correcting codes of length 7. IEEE Trans. Inform. Theory. 52, 3676–3685 (2006)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Wilson R.M.: Constructions and uses of pairwise balanced designs. Math. Centre Tracts. 55, 18–41 (1974)Google Scholar
  19. 19.
    Yin J.: Existence of directed GDDs with block size five and index λ ≥ 2. J. Statist. Plann. Inference. 86, 619–627 (2000)MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Department of MathematicsMount Saint Vincent UniversityHalifaxCanada

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