# American Institute of Mathematical Sciences

August  2012, 6(3): 347-361. doi: 10.3934/amc.2012.6.347

## Cycle structure of permutation functions over finite fields and their applications

 1 Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran, Iran 2 School of Mathematics and Statistics, Carleton University, Ottawa, Canada

Received  November 2011 Revised  May 2012 Published  August 2012

In this work we establish some new interleavers based on permutation functions. The inverses of these interleavers are known over a finite field $\mathbb F_q$. For the first time Möbius and Rédei functions are used to give new deterministic interleavers. Furthermore we employ Skolem sequences in order to find new interleavers with known cycle structure. In the case of Rédei functions an exact formula for the inverse function is derived. The cycle structure of Rédei functions is also investigated. The self-inverse and non-self-inverse versions of these permutation functions can be used to construct new interleavers.
Citation: Amin Sakzad, Mohammad-Reza Sadeghi, Daniel Panario. Cycle structure of permutation functions over finite fields and their applications. Advances in Mathematics of Communications, 2012, 6 (3) : 347-361. doi: 10.3934/amc.2012.6.347
##### References:
 [1] S. Ahmad, Cycle structure of automorphisms of finite cyclic groups,, J. Comb. Theory, 6 (1969), 370. doi: 10.1016/S0021-9800(69)80032-3. Google Scholar [2] C. Baker, A. Bonato and P. Kergin, Skolem arrays and Skolem labellings of ladder graphs,, Ars Combin., 63 (2002), 97. Google Scholar [3] C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: turbo codes,, in, (1993), 1064. Google Scholar [4] J. Boutros and G. Zemor, On quasi-cyclic interleavers for parallel turbo codes,, IEEE Trans. Inform. Theory, 52 (2006), 1732. doi: 10.1109/TIT.2006.871061. Google Scholar [5] G. Caire, G. Taricco and E. Biglieri, Bit-interleaved coded modulation,, IEEE Trans. Inform. Theory, 44 (1998), 927. doi: 10.1109/18.669123. Google Scholar [6] L. Carlitz, A note on permutation functions over a finite field,, Duke Math. J., 29 (1962), 325. doi: 10.1215/S0012-7094-62-02931-9. Google Scholar [7] A. Cesmelioglu, W. Meidl and A. Topuzoglu, On the cycle structure of permutation polynomials,, Finite Fields Appl., 14 (2008), 593. doi: 10.1016/j.ffa.2007.08.003. Google Scholar [8] M. Cheng, M. Nakashima, J. Hamkins, B. Moision and M. Barsoum, A decoder architecture for high-speed free-space laser communications,, Proc. SPIE, 5712 (2005), 174. doi: 10.1117/12.591043. Google Scholar [9] C. J. Colbourn and J. H. Dinitz, "Handbook of Combinatorial Designs,'' 2nd edition,, Chapman & Hall/CRC, (2006). doi: 10.1201/9781420010541. Google Scholar [10] C. Corrada and I. Rubio, Deterministic interleavers for turbo codes with random-like performance and simple implementation,, in, (2003), 555. Google Scholar [11] C. Corrada and I. Rubio, Algebraic construction of interleavers using permutation monomials,, in, (2004), 911. Google Scholar [12] S. Crozier, New high-spread high-distance interleavers for turbo codes,, in, (2000), 3. Google Scholar [13] S. Crozier and P. Guinand, Distance upper bounds and true minimum distance results for turbo-codes designed with DRP interleavers,, Ann. Telecommun., 60 (2005), 10. Google Scholar [14] A. R. Eckler, The construction of missile guidance codes resistant to random interference,, Bell System Tech. J., 39 (1960), 973. Google Scholar [15] S. A. Eldin, N. Shalaby and F. Al-Thukair, Construction of Skolem sequences,, Int. J. Comp. Math., 70 (1998), 333. doi: 10.1080/00207169808804756. Google Scholar [16] N. Francetić and E. Mendelsohn, A survey of Skolem-type sequences and Rosa's use of them,, Math. Slovaca, 59 (2009), 39. doi: 10.2478/s12175-008-0110-3. Google Scholar [17] R. Lidl and G. L. Mullen, Cycle structure of Dickson permutation polynomials,, Math. J. Okayama Univ., 33 (1991), 1. Google Scholar [18] R. Lidl and G. L. Mullen, When does a polynomial over a finite field permute the elements of the field?,, Amer. Math. Monthly, 100 (1993), 71. doi: 10.2307/2324822. Google Scholar [19] R. Lidl and H. Niederreiter, "Finite Fields,'', Cambridge Univ. Press, (1997). Google Scholar [20] S. Lin and D. J. Costello, "Error Control Coding Fundamentals and Application,'' 2nd edition,, Pearson Prentice Hall, (2003). Google Scholar [21] V. Linek and Z. Jiang, Hooked $k$-extended Skolem sequences,, Discrete Math., 196 (1999), 229. doi: 10.1016/S0012-365X(98)00202-7. Google Scholar [22] B. Moision and M. Klimesh, Some observations on permutation polynomials,, JPL, (). Google Scholar [23] L. Rédei, Uber eindeuting umkehrbare polynome in endlichen kopern,, Acta Sci. Math., 11 (): 1946. Google Scholar [24] I. Rubio and C. Corrada, Cyclic decomposition of permutations of finite fields obtained using monomials,, in, (2004), 254. Google Scholar [25] I. Rubio, G. L. Mullen, C. Corrada and F. Castro, Dickson permutation polynomials that decompose in cycles of the same length,, in, (2008), 229. Google Scholar [26] J. Ryu and O. Y. Takeshita, On quadratic inverses for quadratic permutation polynomials over integer rings,, IEEE Trans. Inform. Theory, 52 (2006), 1254. doi: 10.1109/TIT.2005.864442. Google Scholar [27] A. Sakzad, D. Panario, M-R. Sadeghi and N. Eshghi, Self-inverse interleavers based on permutation functions for turbo codes,, in, (2010), 22. Google Scholar [28] A. Sakzad and M.-R. Sadeghi, On cycle-free lattices with high rate label codes,, Adv. Math. Commun., 4 (2010), 441. doi: 10.3934/amc.2010.4.441. Google Scholar [29] A. Sakzad, M-R. Sadeghi and D. Panario, Construction of turbo lattices,, in, (2010), 14. Google Scholar [30] J. Sun and O. Y. Takeshita, Interleavers for turbo codes using permutation polynomials over integer rings,, IEEE Trans. Inform. Theory, 51 (2005), 101. doi: 10.1109/TIT.2004.839478. Google Scholar [31] O. Y. Takeshita, Permutation polynomials interleavers: an algebraic geometric perspective,, IEEE Trans. Inform. Theory, 53 (2007), 2116. doi: 10.1109/TIT.2007.896870. Google Scholar [32] O. Y. Takeshita and D. J. Costello, New deterministic interleaver designs for turbo codes,, IEEE Trans. Inform. Theory, 46 (2000), 1988. doi: 10.1109/18.868474. Google Scholar [33] D. V. Truhachev, M. Lentmaier and K. S. Zigangirov, Some results concerning design and decoding of turbo-codes,, Probl. Inform. Trans., 37 (2001), 190. doi: 10.1023/A:1013821922527. Google Scholar [34] B. Vucetic, Y. Li, L. C. Perez and F. Jiang, Recent advances in turbo code design and theory,, Proc. IEEE, 95 (2007), 1323. doi: 10.1109/JPROC.2007.897975. Google Scholar

show all references

##### References:
 [1] S. Ahmad, Cycle structure of automorphisms of finite cyclic groups,, J. Comb. Theory, 6 (1969), 370. doi: 10.1016/S0021-9800(69)80032-3. Google Scholar [2] C. Baker, A. Bonato and P. Kergin, Skolem arrays and Skolem labellings of ladder graphs,, Ars Combin., 63 (2002), 97. Google Scholar [3] C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: turbo codes,, in, (1993), 1064. Google Scholar [4] J. Boutros and G. Zemor, On quasi-cyclic interleavers for parallel turbo codes,, IEEE Trans. Inform. Theory, 52 (2006), 1732. doi: 10.1109/TIT.2006.871061. Google Scholar [5] G. Caire, G. Taricco and E. Biglieri, Bit-interleaved coded modulation,, IEEE Trans. Inform. Theory, 44 (1998), 927. doi: 10.1109/18.669123. Google Scholar [6] L. Carlitz, A note on permutation functions over a finite field,, Duke Math. J., 29 (1962), 325. doi: 10.1215/S0012-7094-62-02931-9. Google Scholar [7] A. Cesmelioglu, W. Meidl and A. Topuzoglu, On the cycle structure of permutation polynomials,, Finite Fields Appl., 14 (2008), 593. doi: 10.1016/j.ffa.2007.08.003. Google Scholar [8] M. Cheng, M. Nakashima, J. Hamkins, B. Moision and M. Barsoum, A decoder architecture for high-speed free-space laser communications,, Proc. SPIE, 5712 (2005), 174. doi: 10.1117/12.591043. Google Scholar [9] C. J. Colbourn and J. H. Dinitz, "Handbook of Combinatorial Designs,'' 2nd edition,, Chapman & Hall/CRC, (2006). doi: 10.1201/9781420010541. Google Scholar [10] C. Corrada and I. Rubio, Deterministic interleavers for turbo codes with random-like performance and simple implementation,, in, (2003), 555. Google Scholar [11] C. Corrada and I. Rubio, Algebraic construction of interleavers using permutation monomials,, in, (2004), 911. Google Scholar [12] S. Crozier, New high-spread high-distance interleavers for turbo codes,, in, (2000), 3. Google Scholar [13] S. Crozier and P. Guinand, Distance upper bounds and true minimum distance results for turbo-codes designed with DRP interleavers,, Ann. Telecommun., 60 (2005), 10. Google Scholar [14] A. R. Eckler, The construction of missile guidance codes resistant to random interference,, Bell System Tech. J., 39 (1960), 973. Google Scholar [15] S. A. Eldin, N. Shalaby and F. Al-Thukair, Construction of Skolem sequences,, Int. J. Comp. Math., 70 (1998), 333. doi: 10.1080/00207169808804756. Google Scholar [16] N. Francetić and E. Mendelsohn, A survey of Skolem-type sequences and Rosa's use of them,, Math. Slovaca, 59 (2009), 39. doi: 10.2478/s12175-008-0110-3. Google Scholar [17] R. Lidl and G. L. Mullen, Cycle structure of Dickson permutation polynomials,, Math. J. Okayama Univ., 33 (1991), 1. Google Scholar [18] R. Lidl and G. L. Mullen, When does a polynomial over a finite field permute the elements of the field?,, Amer. Math. Monthly, 100 (1993), 71. doi: 10.2307/2324822. Google Scholar [19] R. Lidl and H. Niederreiter, "Finite Fields,'', Cambridge Univ. Press, (1997). Google Scholar [20] S. Lin and D. J. Costello, "Error Control Coding Fundamentals and Application,'' 2nd edition,, Pearson Prentice Hall, (2003). Google Scholar [21] V. Linek and Z. Jiang, Hooked $k$-extended Skolem sequences,, Discrete Math., 196 (1999), 229. doi: 10.1016/S0012-365X(98)00202-7. Google Scholar [22] B. Moision and M. Klimesh, Some observations on permutation polynomials,, JPL, (). Google Scholar [23] L. Rédei, Uber eindeuting umkehrbare polynome in endlichen kopern,, Acta Sci. Math., 11 (): 1946. Google Scholar [24] I. Rubio and C. Corrada, Cyclic decomposition of permutations of finite fields obtained using monomials,, in, (2004), 254. Google Scholar [25] I. Rubio, G. L. Mullen, C. Corrada and F. Castro, Dickson permutation polynomials that decompose in cycles of the same length,, in, (2008), 229. Google Scholar [26] J. Ryu and O. Y. Takeshita, On quadratic inverses for quadratic permutation polynomials over integer rings,, IEEE Trans. Inform. Theory, 52 (2006), 1254. doi: 10.1109/TIT.2005.864442. Google Scholar [27] A. Sakzad, D. Panario, M-R. Sadeghi and N. Eshghi, Self-inverse interleavers based on permutation functions for turbo codes,, in, (2010), 22. Google Scholar [28] A. Sakzad and M.-R. Sadeghi, On cycle-free lattices with high rate label codes,, Adv. Math. Commun., 4 (2010), 441. doi: 10.3934/amc.2010.4.441. Google Scholar [29] A. Sakzad, M-R. Sadeghi and D. Panario, Construction of turbo lattices,, in, (2010), 14. Google Scholar [30] J. Sun and O. Y. Takeshita, Interleavers for turbo codes using permutation polynomials over integer rings,, IEEE Trans. Inform. Theory, 51 (2005), 101. doi: 10.1109/TIT.2004.839478. Google Scholar [31] O. Y. Takeshita, Permutation polynomials interleavers: an algebraic geometric perspective,, IEEE Trans. Inform. Theory, 53 (2007), 2116. doi: 10.1109/TIT.2007.896870. Google Scholar [32] O. Y. Takeshita and D. J. Costello, New deterministic interleaver designs for turbo codes,, IEEE Trans. Inform. Theory, 46 (2000), 1988. doi: 10.1109/18.868474. Google Scholar [33] D. V. Truhachev, M. Lentmaier and K. S. Zigangirov, Some results concerning design and decoding of turbo-codes,, Probl. Inform. Trans., 37 (2001), 190. doi: 10.1023/A:1013821922527. Google Scholar [34] B. Vucetic, Y. Li, L. C. Perez and F. Jiang, Recent advances in turbo code design and theory,, Proc. IEEE, 95 (2007), 1323. doi: 10.1109/JPROC.2007.897975. Google Scholar
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