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Reversible DNA codes over $F_{16}+uF_{16}+vF_{16}+uvF_{16}$
Arrays composed from the extended rational cycle
1. | Universidad de Cantabria, Avd. Los Castros, s/n. Facultad de Ciencias, Santander, Spain |
2. | Universidad de Cantabria, Avd. Los Castros, s/n. EIIT, Santander, Spain |
3. | Scientific Technology, 8 Cecil St., East Brighton, Vic, 3187, Australia |
We present a 3D array construction with application to video watermarking. This new construction uses two main ingredients: an extended rational cycle (ERC) as a shift sequence and a Legendre array as a base. This produces a family of 3D arrays with good auto and cross-correlation. We calculate exactly the values of the auto correlation and the cross-correlation function and their frequency. We present a unified method of obtaining multivariate recursion polynomials and their footprints for all finite multidimensional arrays. Also, we describe new results for arbitrary arrays and enunciate a result for arrays constructed using the method of composition. We also show that the size of the footprint is invariant under dimensional transformations based on the Chinese Remainder Theorem.
References:
[1] |
E. Berlekamp and O. Moreno,
Extended double-error-correcting binary Goppa codes are cyclic (Corresp.), IEEE Trans. Inf. Theory, 19 (1973), 817-818.
|
[2] |
S. T. Blake, O. Moreno and A. Z. Tirkel, Families of 3d arrays for video watermarking, in Sequences and Their Applications, 2014,134-145.
doi: 10.1007/978-3-319-12325-7_12. |
[3] |
S. T. Blake and A. Z. Tirkel, A construction for perfect periodic autocorrelation sequences, in Int. Conf. Seq. Their Appl. -SETA 2014, Springer, 2014, 104-108.
doi: 10.1007/978-3-319-12325-7_9. |
[4] |
L. Bömer and M. Antweiler,
Construction of a new class of higher-dimensional Legendre-and pseudonoise-arrays, IEEE Symposium on IT, 90 (1990), p.76.
|
[5] |
I. J. Cox, J. Kilian, F. T. Leighton and T. Shamoon,
Secure spread spectrum watermarking for multimedia, IEEE Trans. Image Proc., 6 (1997), 1673-1687.
|
[6] |
J.-C. Faugere, P. Gianni, D. Lazard and T. Mora,
Efficient computation of zero-dimensional Gröbner bases by change of ordering, J. Symb. Comput., 16 (1993), 329-344.
doi: 10.1006/jsco.1993.1051. |
[7] |
D. Gomez-Perez, T. Høholdt, O. Moreno and I. Rubio, Linear complexity for multidimensional
arrays-a numerical invariant, in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2015,2697-2701. |
[8] |
J. K. Holmes, Coherent spread spectrum systems, New York Wiley-Interscience, 1 (1982), p. 636. |
[9] |
S. Katzenbeisser and F. Petitcolas, Information Hiding Techniques for Steganography and Digital Watermarking, Artech house, 2000.
![]() |
[10] |
E. I. Krengel, A. Z. Tirkel and T. E. Hall, New sets of binary and ternary sequences with low
correlation, in Int. Conf. Seq. Their Appl., Springer, 2004,220-235.
doi: 10.1109/TIT.2003.818399. |
[11] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, Elsevier, 1977.
![]() ![]() |
[12] |
J. Massey,
Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, 15 (1969), 122-127.
|
[13] |
O. Moreno and S. Maric,
A New Family of Frequency-Hop Codes, IEEE Trans. Commun., 48 (2000), 1241-1244.
|
[14] |
O. Moreno and A. Z. Tirkel, Multi-dimensional arrays for watermarking, in Proc. IEEE Int.
Symp. Inf. Theory (ISIT), 2011,2691-2695. |
[15] |
O. Moreno and A. Z. Tirkel, New optimal low correlation sequences for wireless communications, in International Conference on Sequences and Their Applications, Springer, 2012,
212-223.
doi: 10.1007/978-3-642-30615-0_20. |
[16] |
H. Niederreiter and J. Rivat,
On the correlation of pseudorandom numbers generated by inversive methods, Monatsh. Math., 153 (2008), 251-264.
doi: 10.1007/s00605-007-0503-3. |
[17] |
H. Niederreiter and I. E. Shparlinski, Recent advances in the theory of nonlinear pseudorandom number generators, in Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer,
2002, 86-102. |
[18] |
H. Qi, Stream Ciphers and Linear Complexity, Ph. D thesis, National Univ. Singapore, 2008. |
[19] |
I. M. Rubio, M. Sweedler and C. Heegard, Finding a Gröbner basis for the ideal of recurrence relations on m-dimensional periodic arrays, in Contemporary Developments in Finite Fields and Applications, World Scientific, 2016,296-320. |
[20] |
S. Sakata,
Extension of the Berlekamp-Massey algorithm to N dimensions, Inf. Comput., 84 (1990), 207-239.
doi: 10.1016/0890-5401(90)90039-K. |
[21] |
W. M. Schmidt, Linear recurrence sequences, in Diophantine Approximation, Springer, 2003, 171-247.
doi: 10.1007/3-540-44979-5_4. |
[22] |
A. Z. Tirkel and T. Hall,
New matrices with good auto and cross-correlation, IEICE Trans. Fundam. Electr. Commun. Comp. Sci., 89 (2006), 2315-2321.
|
[23] |
A. Z. Tirkel, T. E. Hall and C. F. Osborne, Steganography applications of coding theory, in IEEE Information Theory Workshop, Svalbard, 1997, 57-59. |
[24] |
A. Z. Tirkel, T. E. Hall, C. F. Osborne, N. Meinhold and O. Moreno, Collusion resistant fingerprinting of digital audio, in Proc. 4th Int. Conf. Sec. Inf. Networks, 2011, 5-12. |
[25] |
A. Z. Tirkel, R. G. van Schyndel and C. F. Osborne,
A two-dimensional digital watermark, DICTA, 95 (1995), 5-8.
|
[26] |
L.-J. Weng,
Decomposition of M-sequences and its applications, IEEE Trans. Inf. Theory, 17 (1971), 457-463.
|
show all references
References:
[1] |
E. Berlekamp and O. Moreno,
Extended double-error-correcting binary Goppa codes are cyclic (Corresp.), IEEE Trans. Inf. Theory, 19 (1973), 817-818.
|
[2] |
S. T. Blake, O. Moreno and A. Z. Tirkel, Families of 3d arrays for video watermarking, in Sequences and Their Applications, 2014,134-145.
doi: 10.1007/978-3-319-12325-7_12. |
[3] |
S. T. Blake and A. Z. Tirkel, A construction for perfect periodic autocorrelation sequences, in Int. Conf. Seq. Their Appl. -SETA 2014, Springer, 2014, 104-108.
doi: 10.1007/978-3-319-12325-7_9. |
[4] |
L. Bömer and M. Antweiler,
Construction of a new class of higher-dimensional Legendre-and pseudonoise-arrays, IEEE Symposium on IT, 90 (1990), p.76.
|
[5] |
I. J. Cox, J. Kilian, F. T. Leighton and T. Shamoon,
Secure spread spectrum watermarking for multimedia, IEEE Trans. Image Proc., 6 (1997), 1673-1687.
|
[6] |
J.-C. Faugere, P. Gianni, D. Lazard and T. Mora,
Efficient computation of zero-dimensional Gröbner bases by change of ordering, J. Symb. Comput., 16 (1993), 329-344.
doi: 10.1006/jsco.1993.1051. |
[7] |
D. Gomez-Perez, T. Høholdt, O. Moreno and I. Rubio, Linear complexity for multidimensional
arrays-a numerical invariant, in Proc. IEEE Int. Symp. Inf. Theory (ISIT), 2015,2697-2701. |
[8] |
J. K. Holmes, Coherent spread spectrum systems, New York Wiley-Interscience, 1 (1982), p. 636. |
[9] |
S. Katzenbeisser and F. Petitcolas, Information Hiding Techniques for Steganography and Digital Watermarking, Artech house, 2000.
![]() |
[10] |
E. I. Krengel, A. Z. Tirkel and T. E. Hall, New sets of binary and ternary sequences with low
correlation, in Int. Conf. Seq. Their Appl., Springer, 2004,220-235.
doi: 10.1109/TIT.2003.818399. |
[11] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, Elsevier, 1977.
![]() ![]() |
[12] |
J. Massey,
Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, 15 (1969), 122-127.
|
[13] |
O. Moreno and S. Maric,
A New Family of Frequency-Hop Codes, IEEE Trans. Commun., 48 (2000), 1241-1244.
|
[14] |
O. Moreno and A. Z. Tirkel, Multi-dimensional arrays for watermarking, in Proc. IEEE Int.
Symp. Inf. Theory (ISIT), 2011,2691-2695. |
[15] |
O. Moreno and A. Z. Tirkel, New optimal low correlation sequences for wireless communications, in International Conference on Sequences and Their Applications, Springer, 2012,
212-223.
doi: 10.1007/978-3-642-30615-0_20. |
[16] |
H. Niederreiter and J. Rivat,
On the correlation of pseudorandom numbers generated by inversive methods, Monatsh. Math., 153 (2008), 251-264.
doi: 10.1007/s00605-007-0503-3. |
[17] |
H. Niederreiter and I. E. Shparlinski, Recent advances in the theory of nonlinear pseudorandom number generators, in Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer,
2002, 86-102. |
[18] |
H. Qi, Stream Ciphers and Linear Complexity, Ph. D thesis, National Univ. Singapore, 2008. |
[19] |
I. M. Rubio, M. Sweedler and C. Heegard, Finding a Gröbner basis for the ideal of recurrence relations on m-dimensional periodic arrays, in Contemporary Developments in Finite Fields and Applications, World Scientific, 2016,296-320. |
[20] |
S. Sakata,
Extension of the Berlekamp-Massey algorithm to N dimensions, Inf. Comput., 84 (1990), 207-239.
doi: 10.1016/0890-5401(90)90039-K. |
[21] |
W. M. Schmidt, Linear recurrence sequences, in Diophantine Approximation, Springer, 2003, 171-247.
doi: 10.1007/3-540-44979-5_4. |
[22] |
A. Z. Tirkel and T. Hall,
New matrices with good auto and cross-correlation, IEICE Trans. Fundam. Electr. Commun. Comp. Sci., 89 (2006), 2315-2321.
|
[23] |
A. Z. Tirkel, T. E. Hall and C. F. Osborne, Steganography applications of coding theory, in IEEE Information Theory Workshop, Svalbard, 1997, 57-59. |
[24] |
A. Z. Tirkel, T. E. Hall, C. F. Osborne, N. Meinhold and O. Moreno, Collusion resistant fingerprinting of digital audio, in Proc. 4th Int. Conf. Sec. Inf. Networks, 2011, 5-12. |
[25] |
A. Z. Tirkel, R. G. van Schyndel and C. F. Osborne,
A two-dimensional digital watermark, DICTA, 95 (1995), 5-8.
|
[26] |
L.-J. Weng,
Decomposition of M-sequences and its applications, IEEE Trans. Inf. Theory, 17 (1971), 457-463.
|




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(0, 0) | (0, 1) | (1, 2) | (2, 1) | (1, 0) | (0, 2) | (1, 1) | (2, 2) | (2, 0) |
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