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Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India |
In this paper, we study the structure of duadic codes of an odd length $ n $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $, $ u^2 = 0 $, (more generally over $ \mathbb{Z}_{q}+u\mathbb{Z}_{q} $, $ u^2 = 0 $, where $ q = p^r $, $ p $ a prime and $ (n, p) = 1 $) using the discrete Fourier transform approach. We study these codes by considering them as a class of abelian codes. Some results related to self-duality and self-orthogonality of duadic codes are presented. Some conditions on the existence of self-dual augmented and extended duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ are determined. We present a sufficient condition for abelian codes of the same length over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ to have the same minimum Hamming distance. A new Gray map over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ is defined, and it is shown that the Gray image of an abelian code over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ is an abelian code over $ \mathbb{Z}_4 $. We have obtained five new linear codes of length $ 18 $ over $ \mathbb{Z}_4 $ from duadic codes of length $ 9 $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ through the Gray map and a new map from $ \mathbb{Z}_4+u\mathbb{Z}_4 $ to $ \mathbb{Z}_4^2 $.
References:
[1] |
N. Aydin and T. Asamov, The Database of $\mathbb{Z}_4$ Codes, Available from: http://www.z4codes.info. Google Scholar |
[2] |
R. K. Bandi and M. Bhaintwal, A note on cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, Discrete Mathematics, Algorithms and Applications, 8 (2016), 1650017.
doi: 10.1142/S1793830916500178. |
[3] |
R. K. Bandi and M. Bhaintwal, Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, In the Proceedings of IWSDA'15, (2015), 47-52. Google Scholar |
[4] |
S. D. Berman,
Semisimple cyclic and abelian codes II, Kibernetika, 3 (1967), 17-23.
doi: 10.1007/BF01119999. |
[5] |
Y. Cao and Q. Li,
Cyclic codes of odd length over $\frac{\mathbb{Z}_u[u]}{\langle u^k \rangle}$, Cryptogr. Comm., 9 (2017), 599-624.
doi: 10.1007/s12095-016-0204-7. |
[6] |
J. Gao, F. W. Fu, L. Xiao and R. K. Bandi, Some results on cyclic codes over $\mathbb{Z}_q+u\mathbb{Z}_q$, Discrete Mathematics, Algorithms and Applications, 7 (2015), 1550058.
doi: 10.1142/S1793830915500585. |
[7] |
A. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé,
The $\mathbb{Z}_4$ linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[8] |
P. Langevin and P. Solé,
Duadic $\mathbb{Z}_4$-codes, Finite Fields Appl., 6 (2000), 309-326.
doi: 10.1006/ffta.2000.0285. |
[9] |
S. Ling and P. Solé,
Duadic codes over $\mathbb{F}_2+u\mathbb{F}_2$, Appl. Algebra Engrg. Comm. Comput., 12 (2000), 365-389.
doi: 10.1007/s002000100079. |
[10] |
S. Ling and P. Solé,
Duadic codes over $\mathbb{Z}_2k$, IEEE Trans. Inform. Theory, 47 (2000), 1581-1588.
doi: 10.1109/18.923740. |
[11] |
R. Luo and U. Parampalli, Self-dual cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, In the Proceedings of IWSDA'15, (2015), 57-61. Google Scholar |
[12] |
F. J. MacWilliams,
Binary codes which are ideals in the group algebra of an Abelian group, Bell Syst. Tech. J., 49 (1970), 987-1011.
doi: 10.1002/j.1538-7305.1970.tb01812.x. |
[13] |
B. S. Rajan and M. U. Siddiqui,
Transform domain characterzation of cyclic codes over $\mathbb{Z}_m$, Appl. Algebra Engrg. Comm. Comput., 5 (1994), 261-275.
doi: 10.1007/BF01225641. |
[14] |
B. S. Rajan and M. U. Siddiqui,
A generalized DFT for Abelian codes over $\mathbb{Z}_m$, IEEE Trans. Inform. Theory, 40 (1994), 2082-2090.
doi: 10.1109/18.340486. |
[15] |
M. Shi, L. Qian, L. Sok and P. Solé,
On constacyclic codes over $\frac{\mathbb{Z}_4[u]}{\langle u^2-1 \rangle}$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[16] |
M. Shi, D. Wang, J. Gao and B. Wu,
Construction of one-Gray weight codes and two-Gray weight codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, J. Syst. Sci. Complex, 29 (2016), 1472-1484.
doi: 10.1007/s11424-016-5286-y. |
[17] |
P. Solé, Codes over Rings, Proceedings of the Cimpa Summer School, Ankara, Turkey, 2008, 18-29.
doi: 10.1142/7140. |
[18] |
E. Speigel,
Codes over $\mathbb{Z}_m$, Inform. Control, 35 (1977), 48-51.
doi: 10.1016/S0019-9958(77)90526-5. |
[19] |
Z. X. Wan, Finite Fields and Galois Rings, World Scientific Pub. Co. Inc., Singapore, 2012. |
[20] |
B. Yildiz and S. Karadeniz, Linear codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, MacWilliams identities, projections, and formally self-dual codes, Finite Fields Appl., 27, 24-40, (2014)
doi: 10.1016/j.ffa.2013.12.007. |
[21] |
B. Yildiz and N. Aydin,
On cyclic codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$ and their $\mathbb{Z}_4$-images, Int. J. Inf. Coding Theory, 2 (2014), 226-237.
doi: 10.1504/IJICOT.2014.066107. |
[22] |
show all references
References:
[1] |
N. Aydin and T. Asamov, The Database of $\mathbb{Z}_4$ Codes, Available from: http://www.z4codes.info. Google Scholar |
[2] |
R. K. Bandi and M. Bhaintwal, A note on cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, Discrete Mathematics, Algorithms and Applications, 8 (2016), 1650017.
doi: 10.1142/S1793830916500178. |
[3] |
R. K. Bandi and M. Bhaintwal, Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, In the Proceedings of IWSDA'15, (2015), 47-52. Google Scholar |
[4] |
S. D. Berman,
Semisimple cyclic and abelian codes II, Kibernetika, 3 (1967), 17-23.
doi: 10.1007/BF01119999. |
[5] |
Y. Cao and Q. Li,
Cyclic codes of odd length over $\frac{\mathbb{Z}_u[u]}{\langle u^k \rangle}$, Cryptogr. Comm., 9 (2017), 599-624.
doi: 10.1007/s12095-016-0204-7. |
[6] |
J. Gao, F. W. Fu, L. Xiao and R. K. Bandi, Some results on cyclic codes over $\mathbb{Z}_q+u\mathbb{Z}_q$, Discrete Mathematics, Algorithms and Applications, 7 (2015), 1550058.
doi: 10.1142/S1793830915500585. |
[7] |
A. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé,
The $\mathbb{Z}_4$ linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.
doi: 10.1109/18.312154. |
[8] |
P. Langevin and P. Solé,
Duadic $\mathbb{Z}_4$-codes, Finite Fields Appl., 6 (2000), 309-326.
doi: 10.1006/ffta.2000.0285. |
[9] |
S. Ling and P. Solé,
Duadic codes over $\mathbb{F}_2+u\mathbb{F}_2$, Appl. Algebra Engrg. Comm. Comput., 12 (2000), 365-389.
doi: 10.1007/s002000100079. |
[10] |
S. Ling and P. Solé,
Duadic codes over $\mathbb{Z}_2k$, IEEE Trans. Inform. Theory, 47 (2000), 1581-1588.
doi: 10.1109/18.923740. |
[11] |
R. Luo and U. Parampalli, Self-dual cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, In the Proceedings of IWSDA'15, (2015), 57-61. Google Scholar |
[12] |
F. J. MacWilliams,
Binary codes which are ideals in the group algebra of an Abelian group, Bell Syst. Tech. J., 49 (1970), 987-1011.
doi: 10.1002/j.1538-7305.1970.tb01812.x. |
[13] |
B. S. Rajan and M. U. Siddiqui,
Transform domain characterzation of cyclic codes over $\mathbb{Z}_m$, Appl. Algebra Engrg. Comm. Comput., 5 (1994), 261-275.
doi: 10.1007/BF01225641. |
[14] |
B. S. Rajan and M. U. Siddiqui,
A generalized DFT for Abelian codes over $\mathbb{Z}_m$, IEEE Trans. Inform. Theory, 40 (1994), 2082-2090.
doi: 10.1109/18.340486. |
[15] |
M. Shi, L. Qian, L. Sok and P. Solé,
On constacyclic codes over $\frac{\mathbb{Z}_4[u]}{\langle u^2-1 \rangle}$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[16] |
M. Shi, D. Wang, J. Gao and B. Wu,
Construction of one-Gray weight codes and two-Gray weight codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, J. Syst. Sci. Complex, 29 (2016), 1472-1484.
doi: 10.1007/s11424-016-5286-y. |
[17] |
P. Solé, Codes over Rings, Proceedings of the Cimpa Summer School, Ankara, Turkey, 2008, 18-29.
doi: 10.1142/7140. |
[18] |
E. Speigel,
Codes over $\mathbb{Z}_m$, Inform. Control, 35 (1977), 48-51.
doi: 10.1016/S0019-9958(77)90526-5. |
[19] |
Z. X. Wan, Finite Fields and Galois Rings, World Scientific Pub. Co. Inc., Singapore, 2012. |
[20] |
B. Yildiz and S. Karadeniz, Linear codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, MacWilliams identities, projections, and formally self-dual codes, Finite Fields Appl., 27, 24-40, (2014)
doi: 10.1016/j.ffa.2013.12.007. |
[21] |
B. Yildiz and N. Aydin,
On cyclic codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$ and their $\mathbb{Z}_4$-images, Int. J. Inf. Coding Theory, 2 (2014), 226-237.
doi: 10.1504/IJICOT.2014.066107. |
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