Skip to main content
Log in

New pseudo-planar binomials in characteristic two and related schemes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Planar functions in odd characteristic were introduced by Dembowski and Ostrom in order to construct finite projective planes in 1968. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and signal sets. Recently, a new notion of pseudo-planar functions in even characteristic was proposed by Zhou. These new pseudo-planar functions, as an analogue of planar functions in odd characteristic, also bring about finite projective planes. There are three known infinite families of pseudo-planar monomial functions constructed by Schmidt and Zhou, and Scherr and Zieve. In this paper, three new classes of pseudo-planar binomials are provided. Moreover, we find that each pseudo-planar function gives an association scheme which is defined on a Galois ring.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdukhalikov K.: Symplectic spreads, planar functions and mutually unbiased bases. arXiv:1306.3478.

  2. Abdukhalikov K., Bannai E., Suda S.: Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets. J. Comb. Theory Ser. A 116(2), 434–448 (2009).

    Google Scholar 

  3. Bannai E.: Subschemes of some association schemes. J. Algebra 144(1), 167–188 (1991).

    Google Scholar 

  4. Bannai E., Ito T.: Algebraic Combinatorics I. Association Schemes. The Benjamin/Cummings Publishing Co., Inc., Menlo Park (1984).

  5. Bonnecaze A., Duursma I.M.: Translates of linear codes over \(Z_4\). IEEE Trans. Inf. Theory 43(4), 1218–1230 (1997).

    Google Scholar 

  6. Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Degular Graphs, vol. 18. Springer, Berlin (1989).

  7. Carlet C., Ding C., Yuan J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005).

    Google Scholar 

  8. Dembowski P., Ostrom T.G.: Planes of order \(n\) with collineation groups of order \(n^{2}\). Math. Z. 103, 239–258 (1968).

  9. Ding C.: Cyclic codes from APN and planar functions. arxiv:1206.4687.

  10. Ding C., Niederreiter H.: Systematic authentication codes from highly nonlinear functions. IEEE Trans. Inf. Theory 50(10), 2421–2428 (2004).

    Google Scholar 

  11. Ding C., Yin J.: Signal sets from functions with optimum nonlinearity. IEEE Trans. Commun. 55(5), 936–940 (2007).

    Google Scholar 

  12. Ding C., Yuan J.: A family of optimal constant-composition codes. IEEE Trans. Inf. Theory 51(10), 3668–3671 (2005).

    Google Scholar 

  13. Hammons A. R., Jr., Vijay Kumar P., Calderbank A. R., Sloane N. J. A., Solé P.: The \({ Z}_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994).

    Google Scholar 

  14. LeCompte N., Martin W.J., Owens W.: On the equivalence between real mutually unbiased bases and a certain class of association schemes. Eur. J. Comb. 31(6), 1499—1512 (2010).

    Google Scholar 

  15. Lidl R., Niederreiter H.: Finite fields, volume 20 of Encyclopedia of Mathematics and Its Applications. Addison-Wesley Publishing Company Advanced Book Program, Reading (1983).

  16. Liebler R.A., Mena R.A.: Certain distance-regular digraphs and related rings of characteristic \(4\). J. Comb. Theory Ser. A 47(1), 111–123 (1988).

    Google Scholar 

  17. Muzychuk M.E.: V-rings of permutation groups with invariant metric. PhD thesis, Kiev State University (1987).

  18. Nyberg K., Knudsen, L.R.: Provable security against differential cryptanalysis. In: Advances in Cryptology-CRYPTO ’92, Santa Barbara, CA, 1992. Lecture Notes in Comput. Sci., vol. 740, pp. 566–574. Springer, Berlin (1992).

  19. Scherr Z., Zieve M.E.: Planar monomials in characteristic 2. arXiv:1302.1244.

  20. Schmidt K.-U., Zhou Y.: Planar functions over fields of characteristic two. J. Algebraic Comb. (2014). doi:10.1007/s10801-013-0496-z.

  21. Wan Z.X.: Lectures on finite fields and Galois rings. World Scientific Publishing Co., Inc., River Edge (2003).

  22. Weng G., Qiu W., Wang Z., Xiang Q.: Pseudo-Paley graphs and skew Hadamard difference sets from presemifields. Des. Codes Cryptogr. 44(1–3), 49–62 (2007).

    Google Scholar 

  23. Yuan J., Carlet C., Ding C.: The weight distribution of a class of linear codes from perfect nonlinear functions. IEEE Trans. Inf. Theory 52(2), 712–717 (2006).

    Google Scholar 

  24. Zhou Y.: \((2^n,2^n,2^n,1)\)-relative difference sets and their representations. J. Comb. Des. 21(12), 563–584 (2013).

Download references

Acknowledgments

The authors express their gratitude to the anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of this paper, and to Prof. Claude Carlet, the Associate Editor, for his excellent editorial job. We would like to thank Professor Qing Xiang and the anonymous reviewer for suggestions on the proofs of Proposition 3.6 and Proposition 3.8, and Dr. Yue Zhou for valuable comments and suggestions. S. Hu was supported by the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education. T. Feng was supported in part by Fundamental Research Fund for the Central Universities of China, Zhejiang Provincial Natural Science Foundation under Grant No.  LQ12A01019, in part by the National Natural Science Foundation of China under Grant No. 11201418, and in part by the Research Fund for Doctoral Programs from the Ministry of Education of China under Grant No. 20120101120089. G. Ge was supported by the National Natural Science Foundation of China under Grant No. 61171198 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ13A010001.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gennian Ge.

Additional information

Communicated by C. Carlet.

Appendix

Appendix

When \(n\) is odd, the second eigenmatrix of the association scheme is

$$\begin{aligned} Q=\left[ \begin{array}{cccccc} 1&{}2\,{b}^{2}-1&{}\frac{b}{2} \left( 2\,{b}^{3}+2\,{b}^{2}-b-1 \right) &{}\frac{b}{2} \left( 2\,{b}^{3}+2\,{b}^{2}-b-1 \right) &{} \frac{b}{2} \left( 2\,{b}^{3}-2\,{b}^{2}-b+1 \right) &{}\frac{b}{2} \left( 2\,{b}^{3}-2\,{b}^{2}-b+1 \right) \\ 1&{}-1&{}\frac{b}{2} \left( {b}^{2}-1-({b}^{2}+b)\mathbf {i} \right) &{}\frac{b}{2} \left( {b}^{2}-1+({b}^{2}+b)\mathbf {i} \right) &{} \frac{b}{2} \left( 1-{b}^{2}-({b}^{2}-b)\mathbf {i} \right) &{}\frac{b}{2} \left( 1-{b}^{2}+({b}^{2}-b)\mathbf {i} \right) \\ 1&{}-1&{}\frac{b}{2} \left( {b}^{2}-1+({b}^{2}+b)\mathbf {i} \right) &{}\frac{b}{2} \left( {b}^{2}-1-({b}^{2}+b)\mathbf {i} \right) &{} \frac{b}{2} \left( 1-{b}^{2}+({b}^{2}-b)\mathbf {i} \right) &{}\frac{b}{2} \left( 1-{b}^{2}-({b}^{2}-b)\mathbf {i} \right) \\ 1&{}2\,{b}^{2}-1&{}-\frac{b}{2} \left( 1+b \right) &{}-\frac{b}{2} \left( 1+b \right) &{} \frac{b}{2} \left( 1-b \right) &{}\frac{b}{2} \left( 1-b \right) \\ 1&{}-1&{}-\frac{b}{2} \left( 1+b\mathbf {i} \right) &{}\frac{b}{2} \left( -1+b\mathbf {i} \right) &{} \frac{b}{2} \left( 1+b\mathbf {i} \right) &{}\frac{b}{2} \left( 1-b\mathbf {i} \right) \\ 1&{}-1&{}\frac{b}{2} \left( -1+b\mathbf {i} \right) &{}-\frac{b}{2} \left( 1+b\mathbf {i} \right) &{} \frac{b}{2} \left( 1-b\mathbf {i} \right) &{}{\frac{b \left( {b}^{2}+1 \right) }{2(1-b\mathbf {i})}} \end{array} \right] . \end{aligned}$$

When \(n\) is even, the second eigenmatrix of the association scheme is

$$\begin{aligned} Q=\left[ \begin{array}{cccccc} 1&{}4\,{b}^{2}-1&{}b \left( 4\,{b}^{3}-b+4\,{b}^{2}-1 \right) &{}b \left( 4\,{b}^{3}-4\,{b}^{2}-b+1 \right) &{} {b}^{2} \left( 4\,{b}^{2}-1 \right) &{}{b}^{2} \left( 4\,{b}^{2}-1 \right) \\ 1&{}-1&{}b \left( b+2\,{b}^{2}-1 \right) &{}- \left( 2\,{b}^{2}-b-1 \right) b &{} -{b}^{2} \left( 1+2\,b\mathbf {i} \right) &{}{b}^{2} \left( -1+2\,b\mathbf {i} \right) \\ 1&{}-1&{}b \left( b+2\,{b}^{2}-1 \right) &{}- \left( 2\,{b}^{2}-b-1 \right) b &{} {b}^{2} \left( -1+2\,b\mathbf {i} \right) &{}-{b}^{2} \left( 1+2\,b\mathbf {i} \right) \\ 1&{}4\,{b}^{2}-1&{}-b \left( 1+b \right) &{}-b \left( -1+b \right) &{} -{b}^{2}&{}-{b}^{2}\\ 1&{}-1&{}b \left( -1+b \right) &{}b \left( 1+b\right) &{} -{b}^{2}&{}-{b}^{2}\\ 1&{}-1&{}-b \left( 1+b\right) &{}-b \left( -1+b \right) &{} {b}^{2}&{}{b}^{2}\end{array} \right] . \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hu, S., Li, S., Zhang, T. et al. New pseudo-planar binomials in characteristic two and related schemes. Des. Codes Cryptogr. 76, 345–360 (2015). https://doi.org/10.1007/s10623-014-9958-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-014-9958-0

Keywords

Mathematics Subject Classification

Navigation