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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), 370-374.
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), 97-107. |
[3] |
C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: turbo codes, in "Proc. International Conference on Communications,'' (1993), 1064-1070. |
[4] |
J. Boutros and G. Zemor, On quasi-cyclic interleavers for parallel turbo codes, IEEE Trans. Inform. Theory, 52 (2006), 1732-1739.
doi: 10.1109/TIT.2006.871061. |
[5] |
G. Caire, G. Taricco and E. Biglieri, Bit-interleaved coded modulation, IEEE Trans. Inform. Theory, 44 (1998), 927-946.
doi: 10.1109/18.669123. |
[6] |
L. Carlitz, A note on permutation functions over a finite field, Duke Math. J., 29 (1962), 325-332.
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), 593-614.
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), 174-185.
doi: 10.1117/12.591043. |
[9] |
C. J. Colbourn and J. H. Dinitz, "Handbook of Combinatorial Designs,'' 2nd edition, Chapman & Hall/CRC, Boca Raton, FL, 2006.
doi: 10.1201/9781420010541. |
[10] |
C. Corrada and I. Rubio, Deterministic interleavers for turbo codes with random-like performance and simple implementation, in "Proc. of the 3rd Int. Symp. on Turbo Codes and Related Topics,'' (2003), 555-558. |
[11] |
C. Corrada and I. Rubio, Algebraic construction of interleavers using permutation monomials, in "Proc. of the 2004 IEEE Int. Conf. on Communications,'' (2004), 911-915. |
[12] |
S. Crozier, New high-spread high-distance interleavers for turbo codes, in "Proc. 20th Biennial Symp. Communications,'' (2000), 3-7. |
[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), 10-28. |
[14] |
A. R. Eckler, The construction of missile guidance codes resistant to random interference, Bell System Tech. J., 39 (1960), 973-994. |
[15] |
S. A. Eldin, N. Shalaby and F. Al-Thukair, Construction of Skolem sequences, Int. J. Comp. Math., 70 (1998), 333-345.
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), 39-76.
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), 1-11. |
[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), 71-74.
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, New Jersey, 2003. |
[21] |
V. Linek and Z. Jiang, Hooked $k$-extended Skolem sequences, Discrete Math., 196 (1999), 229-238.
doi: 10.1016/S0012-365X(98)00202-7. |
[22] |
B. Moision and M. Klimesh, Some observations on permutation polynomials,, JPL, ().
|
[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 "7th Int. Conf. on Finite Fields and their Applications,'' Springer-Verlag, (2004), 254-261. |
[25] |
I. Rubio, G. L. Mullen, C. Corrada and F. Castro, Dickson permutation polynomials that decompose in cycles of the same length, in "8th Int. Conf. on Finite Fields and their Applications,'' (2008), 229-239. |
[26] |
J. Ryu and O. Y. Takeshita, On quadratic inverses for quadratic permutation polynomials over integer rings, IEEE Trans. Inform. Theory, 52 (2006), 1254-1260.
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 "Proc. of 48th Ann. Allerton Conf. Commun. Control, and Computing,'' (2010), 22-28. |
[28] |
A. Sakzad and M.-R. Sadeghi, On cycle-free lattices with high rate label codes, Adv. Math. Commun., 4 (2010), 441-452.
doi: 10.3934/amc.2010.4.441. |
[29] |
A. Sakzad, M-R. Sadeghi and D. Panario, Construction of turbo lattices, in "Proc. of 48th Ann. Allerton Conf. Commun. Control, and Computing,'' (2010), 14-21. |
[30] |
J. Sun and O. Y. Takeshita, Interleavers for turbo codes using permutation polynomials over integer rings, IEEE Trans. Inform. Theory, 51 (2005), 101-119.
doi: 10.1109/TIT.2004.839478. |
[31] |
O. Y. Takeshita, Permutation polynomials interleavers: an algebraic geometric perspective, IEEE Trans. Inform. Theory, 53 (2007), 2116-2132.
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), 1988-2006.
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), 190-205.
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), 1323-1344.
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), 370-374.
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), 97-107. |
[3] |
C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: turbo codes, in "Proc. International Conference on Communications,'' (1993), 1064-1070. |
[4] |
J. Boutros and G. Zemor, On quasi-cyclic interleavers for parallel turbo codes, IEEE Trans. Inform. Theory, 52 (2006), 1732-1739.
doi: 10.1109/TIT.2006.871061. |
[5] |
G. Caire, G. Taricco and E. Biglieri, Bit-interleaved coded modulation, IEEE Trans. Inform. Theory, 44 (1998), 927-946.
doi: 10.1109/18.669123. |
[6] |
L. Carlitz, A note on permutation functions over a finite field, Duke Math. J., 29 (1962), 325-332.
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), 593-614.
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), 174-185.
doi: 10.1117/12.591043. |
[9] |
C. J. Colbourn and J. H. Dinitz, "Handbook of Combinatorial Designs,'' 2nd edition, Chapman & Hall/CRC, Boca Raton, FL, 2006.
doi: 10.1201/9781420010541. |
[10] |
C. Corrada and I. Rubio, Deterministic interleavers for turbo codes with random-like performance and simple implementation, in "Proc. of the 3rd Int. Symp. on Turbo Codes and Related Topics,'' (2003), 555-558. |
[11] |
C. Corrada and I. Rubio, Algebraic construction of interleavers using permutation monomials, in "Proc. of the 2004 IEEE Int. Conf. on Communications,'' (2004), 911-915. |
[12] |
S. Crozier, New high-spread high-distance interleavers for turbo codes, in "Proc. 20th Biennial Symp. Communications,'' (2000), 3-7. |
[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), 10-28. |
[14] |
A. R. Eckler, The construction of missile guidance codes resistant to random interference, Bell System Tech. J., 39 (1960), 973-994. |
[15] |
S. A. Eldin, N. Shalaby and F. Al-Thukair, Construction of Skolem sequences, Int. J. Comp. Math., 70 (1998), 333-345.
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), 39-76.
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), 1-11. |
[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), 71-74.
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, New Jersey, 2003. |
[21] |
V. Linek and Z. Jiang, Hooked $k$-extended Skolem sequences, Discrete Math., 196 (1999), 229-238.
doi: 10.1016/S0012-365X(98)00202-7. |
[22] |
B. Moision and M. Klimesh, Some observations on permutation polynomials,, JPL, ().
|
[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 "7th Int. Conf. on Finite Fields and their Applications,'' Springer-Verlag, (2004), 254-261. |
[25] |
I. Rubio, G. L. Mullen, C. Corrada and F. Castro, Dickson permutation polynomials that decompose in cycles of the same length, in "8th Int. Conf. on Finite Fields and their Applications,'' (2008), 229-239. |
[26] |
J. Ryu and O. Y. Takeshita, On quadratic inverses for quadratic permutation polynomials over integer rings, IEEE Trans. Inform. Theory, 52 (2006), 1254-1260.
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 "Proc. of 48th Ann. Allerton Conf. Commun. Control, and Computing,'' (2010), 22-28. |
[28] |
A. Sakzad and M.-R. Sadeghi, On cycle-free lattices with high rate label codes, Adv. Math. Commun., 4 (2010), 441-452.
doi: 10.3934/amc.2010.4.441. |
[29] |
A. Sakzad, M-R. Sadeghi and D. Panario, Construction of turbo lattices, in "Proc. of 48th Ann. Allerton Conf. Commun. Control, and Computing,'' (2010), 14-21. |
[30] |
J. Sun and O. Y. Takeshita, Interleavers for turbo codes using permutation polynomials over integer rings, IEEE Trans. Inform. Theory, 51 (2005), 101-119.
doi: 10.1109/TIT.2004.839478. |
[31] |
O. Y. Takeshita, Permutation polynomials interleavers: an algebraic geometric perspective, IEEE Trans. Inform. Theory, 53 (2007), 2116-2132.
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), 1988-2006.
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), 190-205.
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), 1323-1344.
doi: 10.1109/JPROC.2007.897975. |
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