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Structural properties of binary propelinear codes
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 |
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. |
[2] |
C. Baker, A. Bonato and P. Kergin, Skolem arrays and Skolem labellings of ladder graphs,, Ars Combin., 63 (2002), 97.
|
[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. |
[5] |
G. Caire, G. Taricco and E. Biglieri, Bit-interleaved coded modulation,, IEEE Trans. Inform. Theory, 44 (1998), 927.
doi: 10.1109/18.669123. |
[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. |
[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. |
[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. |
[9] |
C. J. Colbourn and J. H. Dinitz, "Handbook of Combinatorial Designs,'' 2nd edition,, Chapman & Hall/CRC, (2006).
doi: 10.1201/9781420010541. |
[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.
|
[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. |
[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. |
[17] |
R. Lidl and G. L. Mullen, Cycle structure of Dickson permutation polynomials,, Math. J. Okayama Univ., 33 (1991), 1.
|
[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. |
[19] |
R. Lidl and H. Niederreiter, "Finite Fields,'', Cambridge Univ. Press, (1997).
|
[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. |
[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.
|
[24] |
I. Rubio and C. Corrada, Cyclic decomposition of permutations of finite fields obtained using monomials,, in, (2004), 254.
|
[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.
|
[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. |
[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. |
[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. |
[31] |
O. Y. Takeshita, Permutation polynomials interleavers: an algebraic geometric perspective,, IEEE Trans. Inform. Theory, 53 (2007), 2116.
doi: 10.1109/TIT.2007.896870. |
[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. |
[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. |
[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. |
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. |
[2] |
C. Baker, A. Bonato and P. Kergin, Skolem arrays and Skolem labellings of ladder graphs,, Ars Combin., 63 (2002), 97.
|
[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. |
[5] |
G. Caire, G. Taricco and E. Biglieri, Bit-interleaved coded modulation,, IEEE Trans. Inform. Theory, 44 (1998), 927.
doi: 10.1109/18.669123. |
[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. |
[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. |
[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. |
[9] |
C. J. Colbourn and J. H. Dinitz, "Handbook of Combinatorial Designs,'' 2nd edition,, Chapman & Hall/CRC, (2006).
doi: 10.1201/9781420010541. |
[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.
|
[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. |
[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. |
[17] |
R. Lidl and G. L. Mullen, Cycle structure of Dickson permutation polynomials,, Math. J. Okayama Univ., 33 (1991), 1.
|
[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. |
[19] |
R. Lidl and H. Niederreiter, "Finite Fields,'', Cambridge Univ. Press, (1997).
|
[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. |
[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.
|
[24] |
I. Rubio and C. Corrada, Cyclic decomposition of permutations of finite fields obtained using monomials,, in, (2004), 254.
|
[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.
|
[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. |
[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. |
[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. |
[31] |
O. Y. Takeshita, Permutation polynomials interleavers: an algebraic geometric perspective,, IEEE Trans. Inform. Theory, 53 (2007), 2116.
doi: 10.1109/TIT.2007.896870. |
[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. |
[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. |
[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. |
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