Three-weight codes and near-bent functions from two-weight codes

Original Paper
  • 28 Downloads

Abstract

We introduce a construction of binary 3-weight codes and near-bent functions from 2-weight projective codes.

Keywords

2-weight codes 3-weight codes Near-bent functions 

References

  1. 1.
    Canteaut, A., Charpin, P.: Decomposing bent functions. IEEE Trans. Inf. Theory. 49(8), 2004–2019 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baumert, D., McEliece, R.J.: Weights of irreducible cyclic codes. Inf. Control 20, 158–175 (1972)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Delsarte, P.: Weights of linear codes and strongly regular normed spaces. Discrete Math. 3, 47–64 (1972)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dillon, J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974)Google Scholar
  5. 5.
    Ding, C., Li, C., Li, N., Zhou, Z.C.: Three-weight cyclic codes and their weight distributions. Discrete Math. 339, 415–427 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Leander, G., McGuire, G.: Construction of bent functions from near-bent functions. J. Comb. Theory Ser. A 116(4), 960–970 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Mac Williams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. North-Holland, Amsterdam (1977)Google Scholar
  8. 8.
    Pless, V.: Power moment identities on weight distribution in error correcting codes. Inf. Control 6, 147–152 (1963)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Pless, V., Huffman, W.C. (eds.): Handbook of Coding Theory. Elsevier, Amsterdam (1998)Google Scholar
  10. 10.
    Rothaus, O.S.: On bent functions. J. Comb. Theory Ser. A 20, 300–305 (1976)CrossRefMATHGoogle Scholar
  11. 11.
    Tang, C., Li, N., Qi, Y.F., Zhou, Z.C., Helleseth, T.: Linear codes with two or three weights from weakly regular bent functions. IEEE Trans. Inf. Theory 62(3), 1166–1176 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Wolfmann, J.: Bent functions and coding theory. In: Pott, A., Kumar, P.V., Helleseth, T., Jungnickel, D. (eds.) Difference Sets, Sequences and Their Correlation Properties, NATO Sciences Series, Series C, vol. 542, pp. 393–418. Kluwer, Dordrecht (1999)Google Scholar
  13. 13.
    Wolfmann, J.: Are 2-weight projective cyclic codes irreducible? IEEE Trans. Inf. Theory 51(2), 733–737 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Wolfmann, J.: Cyclic code aspects of bent functions. In: Finite Fields: Theory and Applications, AMS Series “Contemporary Mathematics”. vol. 518, pp. 363–384 (2010)Google Scholar
  15. 15.
    Wolfmann, J.: Special bent and near-bent functions. Adv. Math. Commun. 8(1), 21–33 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Wolfmann, J.: From near-bent to bent: a special case. In: Topics in Finite Fields, AMS Series “Contemporary Mathematics” vol. 632, pp. 359–371 (2015)Google Scholar
  17. 17.
    Wolfmann, J.: Codes Projectifs à deux poids, Caps complets et Ensembles de Différences. J. Comb. Theory Ser. A 23, 208–222 (1977)CrossRefMATHGoogle Scholar
  18. 18.
    Zhang, D., Fan, C., Peng, D., Tang, X.: Complete weight enumerators of some linear codes from quadratic forms. Cryptogr. Commun. 9(1), 151–163 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zhou, Z., Li, N., Fan, C., Helleseth, T.: Linear code with two or three weights from quadratic bent functions. Des. Codes Cryptogr. 81(2), 283–295 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IMATH (IAA)Université de ToulonToulon Cedex 9France

Personalised recommendations