Some Notes on d-Form Functions with Difference-Balanced Property

  • Tongjiang Yan
  • Xiaoni Du
  • Enjian Bai
  • Guozhen Xiao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4547)


The relation between a cyclic relative difference set and a cyclic difference set is considered. Both the sets are with Singer parameters and can be constructed from a difference-balanced d-form function. Although neither of the inversions of Klapper A.′s and No J. S.′s main theorems is true, we prove that a difference-balanced d-form function can be obtained by the cyclic relative difference set and the cyclic difference set introduced by these two main theorems respectively.


Cyclic difference sets cyclic relative difference sets d-form functions difference-balanced 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baumert, L.D.: Cyclic Difference Sets. Lecture Notes in Mathematics, vol. 182. Springer-Verlag, Heidelberg (1971)zbMATHGoogle Scholar
  2. Butson, A.T.: Relations among generalized Hadamard matrices, relative difference sets and maximal length linear recurring sequences. Canad. J. Math. 15, 42–48 (1963)zbMATHMathSciNetGoogle Scholar
  3. Chandler, D., Xiang, Q.: Cyclic relative difference sets and their p-ranks. Des., Codes, Cryptogr. 30, 325–343 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. Dillon, J.F., Dobbertin, H.: Cyclic difference sets with singer parameters. Finite Fields Their Appl. 10, 342–389 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. Jungnickel, D., Pott, A.: Difference sets: An introduction, in Difference Sets, Sequences and their Correlation Properties. In: Pott, A., Kumar, P., Helleseth, T., and Jungnickel, D., (eds.) Kulwer Amsterdam, The Netherlands (1999) 259–295Google Scholar
  6. Chung, F.R.K., Salehi, J.A., Wei, V.K.: Optical orthogonal codes: Design, analysis, and applications. IEEE Trans. Inf. Theory 35(3), 595–604 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  7. Elliott, J.E.H., Butson, A.T.: Relative difference sets. Illinois J. Math. 10, 517–531 (1966)zbMATHMathSciNetGoogle Scholar
  8. Helleseth, T., Gong, G.: New nonbinary sequences with ideal two-level autocorrelation function. IEEE Trans. Inf. Theory 48(11), 2868–2872 (2002)CrossRefMathSciNetGoogle Scholar
  9. Helleseth, T., Kumar, P.V., Martinsen, H.M.: A new family of ternary sequences with ideal two-level autocorrelation. Des., Codes, Cryptogr. 23, 157–166 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  10. Jungnickel, D.: Difference sets. In: Dinitz, J., Stinson, D. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 241–324. Wiley, New York (1992)Google Scholar
  11. Klapper, A.: d-form sequence: Families of sequences with low correlation values and large linear spans. IEEE Trans. Inf. Theory 41(2), 423–431 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  12. No, J.S.: p-ary unified sequences: p-ary extended d-form sequences with ideal autocorrelation property. IEEE Trans. Inf. Theory 48(9), 2540–2546 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. No, J.S.: New cyclic difference sets with Singer parameters constructed from d −homogeneous function. Des., Codes, Cryptogr. 33, 199–213 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. Kim, S.H., No, J.S., Chung, H.: New cyclic relative difference sets constructed from d −homogeneous functions with difference-balanced properties. IEEE Transactions on Information Theory 51(3), 1155–1163 (2005)CrossRefMathSciNetGoogle Scholar
  15. Spence, E.: Hadamard matrices from relative difference sets. J. Combin. Theory 19, 287–300 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  16. Yamada, M.: On a relation between a cyclic relative difference sets associated with the quadratic extensions of a finite field and the szekeres difference sets. Combinatorica 8, 207–216 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  17. Singer, J.: A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 43, 377–385 (1938)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Tongjiang Yan
    • 1
    • 2
  • Xiaoni Du
    • 2
    • 4
  • Enjian Bai
    • 3
  • Guozhen Xiao
    • 2
  1. 1.Math. and Comp. Sci., China Univ. of Petro., Dongying 257061China
  2. 2.P.O.Box 119, Key Lab.on ISN, Xidian Univ., Xi’an 710071China
  3. 3.Inform. Sci. and Tech., Donghua Univ., Shanghai 201620China
  4. 4.Math. and Inform. Sci, Northwest Normal Univ., Lanzhou 730070China

Personalised recommendations