Minimum Distance of Turbo Codes with Permutation Polynomial-Based Interleavers

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


In this chapter we present upper bounds on the minimum distance of turbo codes using QPP interleavers obtained by Eirik Rosnes in (2012), as well as an upper bound on the minimum distance of turbo codes using PP-based interleaver of any degree given in Ryu et al. (2015). We note that most of these upper bounds are partial, meaning that they are valid only for certain interleaver lengths complying with certain constraints on their prime decomposition. A single upper bound is general (see Theorem 5.10, Eq. (5.112)), meaning that it holds for any interleaver length, but there is a constraint on the minimum degree of the inverse polynomials to be equal to two (i.e. the QPP interleaver must have an inverse QPP (Ryu and Takeshita 2006)). Because some of the above mentioned upper bounds use inverse polynomials of QPPs, in the first section of this chapter we present the necessary and sufficient condition for a QPP to admit at least one quadratic inverse, derived by Jonghoon Ryu and Oscar Y. Takeshita in Ryu and Takeshita (2006). At the end of the first section we give, without proof, a theorem from Ryu (2007), Ryu and Takeshita (2011) which allows us to determine all inverse permutation polynomials of a QPP of the smallest degree.


Turbo Codes Interleaver Length Quadratic Inverse Inverse Polynomial Interleaver Pattern 
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  1. 3GPP TS 36.212 V8.3.0, 3rd Generation partnership project, multiplexing and channel coding (Release 8) (2008),
  2. S. Crozier, P. Guinand, A. Hunt, Computing the minimum distance of turbo-codes using iterative decoding techniques, in 22th Biennial Symposium on Communication, Kingston, Ontario, Canada, 31 May–3 June 2004, pp. 306–308Google Scholar
  3. R. Garello, P. Pierleoni, S. Benedetto, Computing the free distance of turbo codes and serially concatenated codes with interleavers: algorithms and applications. IEEE J. Sel. Areas Commun. 19(5), 800–812 (2001)CrossRefGoogle Scholar
  4. P. Guinand, J. Lodge, Trellis termination for turbo encoders, in 17th Biennial Symposium Communication, Queen’s University, Kingston, Canada, 30 May–1 June 1994, pp. 389–392Google Scholar
  5. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 4th edn. (Oxford University Press, Clarendon, 1975)Google Scholar
  6. E. Rosnes, On the minimum distance of turbo codes with quadratic permutation polynomial interleavers. IEEE Trans. Inf. Theory 58(7), 4781–4795 (2012)MathSciNetCrossRefGoogle Scholar
  7. E. Rosnes, O.Y. Takeshita, Optimum distance quadratic permutation polynomial-based interleavers for turbo codes, in IEEE International Symposium Information Theory (ISIT), Seattle, USA, 9–14 July 2006, pp. 1988–1992Google Scholar
  8. E. Rosnes, Ø. Ytrehus, Improved algorithms for the determination of turbo-code weight distributions. IEEE Trans. Commun. 53(1), 20–26 (2005)CrossRefGoogle Scholar
  9. J. Ryu, Permutation polynomial based interleavers for turbo codes over integer rings. Ph.D. thesis. Ohio State University (2007),
  10. J. Ryu, Efficient address generation for permutation polynomial based interleavers over integer rings. IEICE Trans. Fundam. E95–A(1), 421–424 (2012a)CrossRefGoogle Scholar
  11. J. Ryu, Permutation polynomials of higher degrees for turbo code interleavers. IEICE Trans. Commun. E95–B(12), 3760–3762 (2012b)CrossRefGoogle Scholar
  12. 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
  13. J. Ryu, O.Y. Takeshita, On inverses for quadratic permutation polynomials over integers rings, 10 February 2011,
  14. J. Ryu, L. Trifina, H. Balta, The limitation of permutation polynomial interleavers for turbo codes and a scheme for dithering permutation polynomials. AEÜ Int. J. Electron. Commun. 69(10), 1550–1556 (2015)CrossRefGoogle Scholar
  15. Telemetry channel coding, consultative committee for space data systems (CCSDS), CCSDS 101.0-B-6, Blue Book (2002)Google Scholar
  16. L. Trifina, J. Ryu, D. Tarniceriu, Up to five degree permutation polynomial interleavers for short length LTE turbo codes with optimum minimum distance, in IEEE International Symposium on Signals Circuits System (ISSCS), Iasi, Romania, 13–14 July 2017Google Scholar
  17. L. Trifina, D. Tarniceriu, Analysis of cubic permutation polynomials for turbo codes. Wirel. Pers. Commun. 69(1), 1–22 (2013)CrossRefGoogle Scholar
  18. L. Trifina, D. Tarniceriu, Improved method for searching interleavers from a certain set using Garello’s method with applications for the LTE standard. Ann. Telecommun. 69(5–6), 251–272 (2014)CrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

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

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