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An asymmetric ZCZ sequence set with inter-subset uncorrelated property and flexible ZCZ length
A first step towards the skew duadic codes
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France |
This text gives a first definition of the $θ$-duadic codes where $θ$ is an automorphism of $\mathbb{F}_q$. A link with the self-orthogonal $θ$-cyclic codes is established. A construction and an enumeration are provided when $q$ is the square of a prime number $p$. In addition, new self-dual binary codes $ [72, 36, 12] $ are obtained from extended $θ$-duadic codes defined on $\mathbb{F}_4$.
References:
[1] |
D. Boucher, W. Geiselmann and F. Ulmer,
Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389.
doi: 10.1007/s00200-007-0043-z. |
[2] |
D. Boucher and F. Ulmer,
Coding with skew polynomial rings, J. Symb. Comp., 44 (2009), 1644-1656.
doi: 10.1016/j.jsc.2007.11.008. |
[3] |
D. Boucher and F. Ulmer,
A note on the dual codes of module skew codes, Cryptography and coding, Lecture Notes in Comput. Sci., 7089 (2011), 230-243.
doi: 10.1007/978-3-642-25516-8_14. |
[4] |
D. Boucher and F. Ulmer,
Self-dual skew codes and factorization of skew polynomials, J. Symb. Comp., 60 (2014), 47-61.
doi: 10.1016/j.jsc.2013.10.003. |
[5] |
D. Boucher,
Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$, Adv. Math. Commun., 10 (2016), 765-795.
doi: 10.3934/amc.2016040. |
[6] |
X. Caruso and J. Le Borgne,
A new faster algorithm for factoring skew polynomials over finite fields, J. Symb. Comp., 79 (2017), 411-443.
doi: 10.1016/j.jsc.2016.02.016. |
[7] |
S. T. Dougherty, T. A. Gulliver and H. Masaaki,
Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.
|
[8] |
M. Giesbrecht,
Factoring in skew-polynomial rings over finite fields, J. Symb. Comput., 26 (1998), 463-486.
doi: 10.1006/jsco.1998.0224. |
[9] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077. |
[10] | |
[11] |
A. Kaya, B. Yildiz and I. Siap,
New extremal binary self-dual codes of length 68 from quadratic residue codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$, Finite Fields and their Applications, 29 (2014), 160-177.
doi: 10.1016/j.ffa.2014.04.009. |
[12] |
R. W. K. Odoni,
On additive polynomials over a finite field, Proc. Edinburgh Math. Soc., 42 (1999), 1-16.
doi: 10.1017/S0013091500019970. |
[13] |
O. Ore,
Theory of Non-Commutative Polynomials, Ann. Math., 34 (1933), 480-508.
doi: 10.2307/1968173. |
[14] |
J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2013.
doi: 10.1017/CBO9781139856065. |
[15] |
A. Zhdanov, New self-dual codes of length $72$, preprint, arXiv: 1705.05779. |
show all references
References:
[1] |
D. Boucher, W. Geiselmann and F. Ulmer,
Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389.
doi: 10.1007/s00200-007-0043-z. |
[2] |
D. Boucher and F. Ulmer,
Coding with skew polynomial rings, J. Symb. Comp., 44 (2009), 1644-1656.
doi: 10.1016/j.jsc.2007.11.008. |
[3] |
D. Boucher and F. Ulmer,
A note on the dual codes of module skew codes, Cryptography and coding, Lecture Notes in Comput. Sci., 7089 (2011), 230-243.
doi: 10.1007/978-3-642-25516-8_14. |
[4] |
D. Boucher and F. Ulmer,
Self-dual skew codes and factorization of skew polynomials, J. Symb. Comp., 60 (2014), 47-61.
doi: 10.1016/j.jsc.2013.10.003. |
[5] |
D. Boucher,
Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$, Adv. Math. Commun., 10 (2016), 765-795.
doi: 10.3934/amc.2016040. |
[6] |
X. Caruso and J. Le Borgne,
A new faster algorithm for factoring skew polynomials over finite fields, J. Symb. Comp., 79 (2017), 411-443.
doi: 10.1016/j.jsc.2016.02.016. |
[7] |
S. T. Dougherty, T. A. Gulliver and H. Masaaki,
Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.
|
[8] |
M. Giesbrecht,
Factoring in skew-polynomial rings over finite fields, J. Symb. Comput., 26 (1998), 463-486.
doi: 10.1006/jsco.1998.0224. |
[9] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077. |
[10] | |
[11] |
A. Kaya, B. Yildiz and I. Siap,
New extremal binary self-dual codes of length 68 from quadratic residue codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$, Finite Fields and their Applications, 29 (2014), 160-177.
doi: 10.1016/j.ffa.2014.04.009. |
[12] |
R. W. K. Odoni,
On additive polynomials over a finite field, Proc. Edinburgh Math. Soc., 42 (1999), 1-16.
doi: 10.1017/S0013091500019970. |
[13] |
O. Ore,
Theory of Non-Commutative Polynomials, Ann. Math., 34 (1933), 480-508.
doi: 10.2307/1968173. |
[14] |
J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2013.
doi: 10.1017/CBO9781139856065. |
[15] |
A. Zhdanov, New self-dual codes of length $72$, preprint, arXiv: 1705.05779. |
Coefficients of |
||
-3072 | ||
-3276 | ||
-3480 | ||
-3582 | ||
-3684 | ||
-3990 | ||
-4092 |
Coefficients of |
||
-3072 | ||
-3276 | ||
-3480 | ||
-3582 | ||
-3684 | ||
-3990 | ||
-4092 |
Require: |
Ensure: All irreducible skew polynomials with bound |
1. |
2. |
3. if |
4. |
5. |
6. for |
7. |
8. |
9. add |
10. endfor |
11. else |
12. |
13. |
14. for |
15. |
16. |
17. add |
18. endfor |
19. endif |
20. return |
Require: |
Ensure: All irreducible skew polynomials with bound |
1. |
2. |
3. if |
4. |
5. |
6. for |
7. |
8. |
9. add |
10. endfor |
11. else |
12. |
13. |
14. for |
15. |
16. |
17. add |
18. endfor |
19. endif |
20. return |
Coefficients of |
|
-2820 | |
-3204 | |
-3276 | |
-3312 | |
-3336 | |
-3372 | |
-3408 | |
-3420 | |
-3456 | |
-3504 | |
-3540 | |
-3564 | |
-3576 | |
-3600 | |
-3612 | |
-3636 | |
-3660 | |
-3696 | |
-3732 | |
-3744 | |
-3768 | |
-3816 | |
-3828 | |
-3924 |
Coefficients of |
|
-2820 | |
-3204 | |
-3276 | |
-3312 | |
-3336 | |
-3372 | |
-3408 | |
-3420 | |
-3456 | |
-3504 | |
-3540 | |
-3564 | |
-3576 | |
-3600 | |
-3612 | |
-3636 | |
-3660 | |
-3696 | |
-3732 | |
-3744 | |
-3768 | |
-3816 | |
-3828 | |
-3924 |
Coefficients of |
||
|
201 | 0 |
237 | 0 | |
249 | 0 | |
273 | 0 | |
273 | 36 | |
309 | 0 | |
345 | 0 | |
381 | 0 | |
393 | 36 | |
489 | 36 |
Coefficients of |
||
|
201 | 0 |
237 | 0 | |
249 | 0 | |
273 | 0 | |
273 | 36 | |
309 | 0 | |
345 | 0 | |
381 | 0 | |
393 | 36 | |
489 | 36 |
Coefficients of |
|||
|
221 | 0 | |
323 | 0 | ||
238 | 0 | ||
391 | 0 | ||
289 | 0 | ||
102 | 0 | ||
255 | 0 | ||
153 | 0 |
Coefficients of |
|||
|
221 | 0 | |
323 | 0 | ||
238 | 0 | ||
391 | 0 | ||
289 | 0 | ||
102 | 0 | ||
255 | 0 | ||
153 | 0 |
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