Nonexistence of a Kind of Generalized Perfect Binary Array

  • Zhang Xiyong
  • Guo Hua
  • Han Wenbao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4086)


Generalized perfect binary array(GPBA) is a useful tool in the construction of perfect binary arrays. By investigating the character values of corresponding relative difference sets, we obtain some nonexistence results of GPBAs. In particular, we show that no GPBA(2,2,p n ) of any type z exists for n=1 and any odd prime p, or for any n and any odd prime \(p\not\equiv 1 (mod{8})\). For the case p=2, there exists a GPBA(2,2,2 n ) of type z=(z 1,z 2,z 3) if and only if z=(0,0,0) and n=0,2,4, or z≠(0,0,0) with z 3=0 and 0≤n≤5, with z 3=1 and 0≤n≤3.


Abelian Group Cyclic Subgroup Type Vector Binary Array Coset Representation 
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  1. 1.
    Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Hughes, G.: Cocyclic Theory of Generalized Perfect Binary Arrays. Royal Melbourne Institute of Technology, Department of Mathematics, Research Report No. 6 (1998)Google Scholar
  3. 3.
    Jedwab, J.: Generalized Perfect Arrays and Menon Difference Sets. Designs Codes and Cryptography 2, 19–68 (1992)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Jedwab, J., Mitchell, C.: Constructing new perfect binary arrays. Electronics Letters 24, 650–652 (1988)CrossRefGoogle Scholar
  5. 5.
    Kraemer, R.G.: Proof of a conjecture on Hadamard 2-groups. Journal of Combinatorial Theory(A) 63, 1–10 (1993)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ma, S.L.: Polynomial addition sets. Ph.D. thesis. University of Hong Kong (1985)Google Scholar
  7. 7.
    Ma, S.L.: Planar Functions, Relative Difference Sets, and Character Theory. Journal of Algebra 185, 342–356 (1996)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ma, S.L., Schmidt, B.: On (p a,p,p a,p a − 1)-relative difference sets. Designs, Codes and Cryptography 6, 57–71 (1995)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Pott, A.: A survey on relative difference sets. In: Arasu, K.T., et al. (eds.) Groups, Difference sets and the Monster, pp. 195–232. deGruyter, Berlag-New York (1996)Google Scholar
  10. 10.
    Schmidt, B.: Cyclotomic Integers of Prescribed Absolute Value and the Class Group. J.Number Theory 72, 269–281 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Turyn, R.J.: Character sums and difference sets. Pacific J. Math. 15, 319–346 (1965)MATHMathSciNetGoogle Scholar
  12. 12.
    Yang, Y.X.: Quasi-perfect binary arrays. Acta Electronica Sinica 20(4), 37–44 (1992) (in Chinese)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zhang Xiyong
    • 1
    • 2
  • Guo Hua
    • 3
  • Han Wenbao
    • 2
  1. 1.Department of MathematicsZhengzhou UniversityZhengzhouChina
  2. 2.Department of Applied MathematicsInformation Engineering UniversityZhengzhouChina
  3. 3.School of Computer Science EngineeringBeihang UniversityBeijingChina

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