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doi: 10.3934/amc.2020135
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Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $

Raj Kumar and  Maheshanand Bhaintwal ,

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

* Corresponding author: Maheshanand Bhaintwal

Received  August 2020 Early access March 2021

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 $.

Keywords: Abelian codes, duadic codes, codes over rings, self-dual codes.
Mathematics Subject Classification: Primary: 94B05, 94B15.
Citation: Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, doi: 10.3934/amc.2020135
References:
[1]

N. Aydin and T. Asamov, The Database of $\mathbb{Z}_4$ Codes, Available from: http://www.z4codes.info.

[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.

[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.

[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]

http://magma.maths.usyd.edu.au/magma/.

show all references

References:
[1]

N. Aydin and T. Asamov, The Database of $\mathbb{Z}_4$ Codes, Available from: http://www.z4codes.info.

[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.

[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.

[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]

http://magma.maths.usyd.edu.au/magma/.

Table 1.  Duadic codes of $ R(\mathbb{Z}_3\times \mathbb{Z}_3) $
$ \text{Duadic code}\; C $ $ |C| $ $ \psi(C) $ $ \phi(C) $
$ 2-1-0-1-0 $ $ 2^{18} $ $ [18, 4^82^2, 4 ]^* $ $ [18, 4^42^5, 8]^{**} $
$ u-1-0-1-0 $ $ 2^{18} $ $ [18, 4^92^0, 4 ] $ $ [18, 4^42^5, 6] $
$ (2+u)-1-0-1-0 $ $ 2^{18} $ $ [18, 4^92^0, 4 ] $ $ [18, 4^42^5, 6] $
$ 2-2-2-2-2 $ $ 2^{18} $ $ [18,4^02^{18},2 ]^* $ $ [18, 4^02^9, 4]^\dagger $
$ (2+u)-2-2-2-2 $ $ 2^{18} $ $ [18, 4^12^{16}, 4]^* $ $ [18, 4^02^9, 8]^{\dagger **} $
$ u-2-2-2-2 $ $ 2^{18} $ $ [18, 4^12^{16}, 4] $ $ [18, 4^02^9, 8]^{\dagger} $
$ 2-2-u-u-2 $ $ 2^{18} $ $ [18, 4^42^{10},4 ]^{**} $ $ [18, 4^02^9,8] $
$ u-2-u-u-2 $ $ 2^{18} $ $ [18, 4^52^8, 4]^{**} $ $ [18, 4^02^9,6]^\dagger $
$ (2+u)-2-u-u-2 $ $ 2^{18} $ $ [18,4^5 2^8, 4 ] $ $ [18, 4^02^9,8] $
$ (2+u) - (2+u) -0-1- (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 4 ] $ $ [18, 4^22^5, 6]^{**} $
$ (2+u) - (2+u) -u-u- (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 4 ] $ $ [18, 4^02^9,6] $
$ (2+u) - (2+u) -2-2- (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 4 ] $ $ [18, 4^02^9,6] $
$ (2+u) - (2+u) -(2+u) - (2+u) - (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 2 ] $ $ [18, 4^02^9,2]^\dagger $
$ u - (2+u) -2-2- (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 4 ] $ $ [18, 4^02^9,8] $
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$ \text{Duadic code}\; C $ $ |C| $ $ \psi(C) $ $ \phi(C) $
$ 2-1-0-1-0 $ $ 2^{18} $ $ [18, 4^82^2, 4 ]^* $ $ [18, 4^42^5, 8]^{**} $
$ u-1-0-1-0 $ $ 2^{18} $ $ [18, 4^92^0, 4 ] $ $ [18, 4^42^5, 6] $
$ (2+u)-1-0-1-0 $ $ 2^{18} $ $ [18, 4^92^0, 4 ] $ $ [18, 4^42^5, 6] $
$ 2-2-2-2-2 $ $ 2^{18} $ $ [18,4^02^{18},2 ]^* $ $ [18, 4^02^9, 4]^\dagger $
$ (2+u)-2-2-2-2 $ $ 2^{18} $ $ [18, 4^12^{16}, 4]^* $ $ [18, 4^02^9, 8]^{\dagger **} $
$ u-2-2-2-2 $ $ 2^{18} $ $ [18, 4^12^{16}, 4] $ $ [18, 4^02^9, 8]^{\dagger} $
$ 2-2-u-u-2 $ $ 2^{18} $ $ [18, 4^42^{10},4 ]^{**} $ $ [18, 4^02^9,8] $
$ u-2-u-u-2 $ $ 2^{18} $ $ [18, 4^52^8, 4]^{**} $ $ [18, 4^02^9,6]^\dagger $
$ (2+u)-2-u-u-2 $ $ 2^{18} $ $ [18,4^5 2^8, 4 ] $ $ [18, 4^02^9,8] $
$ (2+u) - (2+u) -0-1- (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 4 ] $ $ [18, 4^22^5, 6]^{**} $
$ (2+u) - (2+u) -u-u- (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 4 ] $ $ [18, 4^02^9,6] $
$ (2+u) - (2+u) -2-2- (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 4 ] $ $ [18, 4^02^9,6] $
$ (2+u) - (2+u) -(2+u) - (2+u) - (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 2 ] $ $ [18, 4^02^9,2]^\dagger $
$ u - (2+u) -2-2- (2+u) $ $ 2^{18} $ $ [18, 4^9 2^0, 4 ] $ $ [18, 4^02^9,8] $
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