Advertisement

Permutation Polynomial Based Interleavers. Conditions on Coefficients

  • Lucian TrifinaEmail author
  • Daniela Tarniceriu
Chapter
  • 158 Downloads
Part of the Signals and Communication Technology book series (SCT)

Abstract

In this chapter we obtain the conditions on the coefficients of a polynomial so that it is a PP modulo a positive integer. Definition of a PP-based interleaver is given in Sect. 3.1. Conditions on the coefficients of a polynomial of arbitrary degree so that it is a PP modulo a power of 2 are given in Sect. 3.2. Necessary and sufficient conditions so that a polynomial is a PP modulo a power of an arbitrary prime are given in Sect. 3.3. Simplified necessary and sufficient conditions on the coefficients of a polynomial so that it is a PP modulo an arbitrary positive integer are given in Sect. 3.4. Conditions on the coefficients of a polynomial of first, second, third, fourth and fifth degree so that it is a LPP, QPP, CPP, permutation polynomial of fourth degree (4-PP), and permutation polynomial of fifth degree (5-PP), are obtained in Sects. 3.5, 3.6, 3.7, 3.8, and 3.9, respectively. Zhao and Fan sufficient conditions (Zhao and Fan 2007) on the coefficients of a polynomial of arbitrary degree so that is a PP are given in Sect. 3.10. Finally, an algorithm obtained by Weng and Dong (2008) to get all PPs up to five degree is presented in Sect. 3.11.

References

  1. Y.-L. Chen, J. Ryu, O.Y. Takeshita, A simple coefficient test for cubic permutation polynomials over integer rings. IEEE Commun. Lett. 10(7), 549–551 (2006)CrossRefGoogle Scholar
  2. L.E. Dickson, The analytic representation of substitutions on a power of a prime number of letters with a discussion of the liner group. Ann. Math. 11(1–6), 65–120 (1896)MathSciNetCrossRefGoogle Scholar
  3. L.E. Dickson, Linear Groups: With an Exposition of the Galois Field Theory, Dover Phoenix edn. (Dover, New York, 1901), https://ia801406.us.archive.org/22/items/lineargroupswith00dickuoft/lineargroupswith00dickuoft.pdf
  4. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 4th edn. (Oxford University Press, Clarendon, 1975)Google Scholar
  5. R. Lidl, G.L. Mullen, When does a polynomial over a finite field permute the elements of the field? Am. Math. Mon. 95(3), 243–246 (1988)MathSciNetCrossRefGoogle Scholar
  6. G. Mullen, H. Stevens, Polynomial functions (mod m). Acta Math. Hung. 44(3–4), 237–241 (1984)MathSciNetCrossRefGoogle Scholar
  7. W. Nöbauer, Über permutationspolynome und permutationsfunktionen für primzahlpotenzen. Mon. Math. 69(3), 230–238 (1965)Google Scholar
  8. R.L. Rivest, Permutation polynomials modulo 2w. Finite Fields Appl. 7(2), 287–292 (2001)MathSciNetCrossRefGoogle Scholar
  9. J. Ryu, O.Y. Takeshita, On quadratic inverses for quadratic permutation polynomials over integers rings. IEEE Trans. Inf. Theory 52(3), 1254–1260 (2006)Google Scholar
  10. J. Sun, O.Y. Takeshita, Interleavers for turbo codes using permutation polynomial over integers rings. IEEE Trans. Inf. Theory 51(1), 101–119 (2005)Google Scholar
  11. J. Sun, O.Y. Takeshita, M.P. Fitz, Permutation polynomial based deterministic interleavers for turbo codes, in IEEE International Symposium on Information Theory (ISIT), Yokohama, Japan, 29 June–4 July 2003, p. 319Google Scholar
  12. O.Y. Takeshita, Maximum contention-free interleavers and permutation polynomials over integers rings. IEEE Trans. Inf. Theory 52(3), 1249–1253 (2006)Google Scholar
  13. O.Y. Takeshita, Permutation polynomial interleavers: an algebraic-geometric perspective. IEEE Trans. Inf. Theory 53(6), 2116–2132 (2007)MathSciNetCrossRefGoogle Scholar
  14. L. Trifina, D. Tarniceriu, A coefficient test for fourth degree permutation polynomials over integer rings. AEÜ Int. J. Electron. Commun. 70(11), 1565–1568 (2016)CrossRefGoogle Scholar
  15. L. Trifina, D. Tarniceriu, A coefficient test for quintic permutation polynomials over integer rings. IEEE Access 6(2), 37893–37909 (2018).  https://doi.org/10.1109/ACCESS.2018.2854373CrossRefGoogle Scholar
  16. G. Weng, C. Dong, A note on permutation polynomial over Zn. IEEE Trans. Inf. Theory 54(9), 4388–4390 (2008)Google Scholar
  17. H. Zhao, P. Fan, Simple method for generating mth-order permutation polynomials over integer rings. Electron. Lett. 43(8), 449–451 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Faculty of Electronics, Telecommunications and Information TechnologyGheorghe Asachi Technical UniversityIașiRomania

Personalised recommendations