# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020135

## Duadic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$

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

* Corresponding author: Maheshanand Bhaintwal

Received  August 2020 Published  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 $\mathbb{Z}_4^2$.

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. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] P. Langevin and P. Solé, Duadic $\mathbb{Z}_4$-codes, Finite Fields Appl., 6 (2000), 309-326.  doi: 10.1006/ffta.2000.0285.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] P. Solé, Codes over Rings, Proceedings of the Cimpa Summer School, Ankara, Turkey, 2008, 18-29. doi: 10.1142/7140.  Google Scholar [18] E. Speigel, Codes over $\mathbb{Z}_m$, Inform. Control, 35 (1977), 48-51.  doi: 10.1016/S0019-9958(77)90526-5.  Google Scholar [19] Z. X. Wan, Finite Fields and Galois Rings, World Scientific Pub. Co. Inc., Singapore, 2012.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22]

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] P. Langevin and P. Solé, Duadic $\mathbb{Z}_4$-codes, Finite Fields Appl., 6 (2000), 309-326.  doi: 10.1006/ffta.2000.0285.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] P. Solé, Codes over Rings, Proceedings of the Cimpa Summer School, Ankara, Turkey, 2008, 18-29. doi: 10.1142/7140.  Google Scholar [18] E. Speigel, Codes over $\mathbb{Z}_m$, Inform. Control, 35 (1977), 48-51.  doi: 10.1016/S0019-9958(77)90526-5.  Google Scholar [19] Z. X. Wan, Finite Fields and Galois Rings, World Scientific Pub. Co. Inc., Singapore, 2012.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22]
Duadic codes of $R(\mathbb{Z}_3\times \mathbb{Z}_3)$
 $\text{Duadic code}\; C$ $|C|$ $\psi(C)$ $\phi(C)$ $2-1-0-1-0$ $2^{18}$ $[18, 4^82^2, 4 ]^*$ $[18, 4^42^5, 8]^{**}$ $u-1-0-1-0$ $2^{18}$ $[18, 4^92^0, 4 ]$ $[18, 4^42^5, 6]$ $(2+u)-1-0-1-0$ $2^{18}$ $[18, 4^92^0, 4 ]$ $[18, 4^42^5, 6]$ $2-2-2-2-2$ $2^{18}$ $[18,4^02^{18},2 ]^*$ $[18, 4^02^9, 4]^\dagger$ $(2+u)-2-2-2-2$ $2^{18}$ $[18, 4^12^{16}, 4]^*$ $[18, 4^02^9, 8]^{\dagger **}$ $u-2-2-2-2$ $2^{18}$ $[18, 4^12^{16}, 4]$ $[18, 4^02^9, 8]^{\dagger}$ $2-2-u-u-2$ $2^{18}$ $[18, 4^42^{10},4 ]^{**}$ $[18, 4^02^9,8]$ $u-2-u-u-2$ $2^{18}$ $[18, 4^52^8, 4]^{**}$ $[18, 4^02^9,6]^\dagger$ $(2+u)-2-u-u-2$ $2^{18}$ $[18,4^5 2^8, 4 ]$ $[18, 4^02^9,8]$ $(2+u) - (2+u) -0-1- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^22^5, 6]^{**}$ $(2+u) - (2+u) -u-u- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,6]$ $(2+u) - (2+u) -2-2- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,6]$ $(2+u) - (2+u) -(2+u) - (2+u) - (2+u)$ $2^{18}$ $[18, 4^9 2^0, 2 ]$ $[18, 4^02^9,2]^\dagger$ $u - (2+u) -2-2- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,8]$
 $\text{Duadic code}\; C$ $|C|$ $\psi(C)$ $\phi(C)$ $2-1-0-1-0$ $2^{18}$ $[18, 4^82^2, 4 ]^*$ $[18, 4^42^5, 8]^{**}$ $u-1-0-1-0$ $2^{18}$ $[18, 4^92^0, 4 ]$ $[18, 4^42^5, 6]$ $(2+u)-1-0-1-0$ $2^{18}$ $[18, 4^92^0, 4 ]$ $[18, 4^42^5, 6]$ $2-2-2-2-2$ $2^{18}$ $[18,4^02^{18},2 ]^*$ $[18, 4^02^9, 4]^\dagger$ $(2+u)-2-2-2-2$ $2^{18}$ $[18, 4^12^{16}, 4]^*$ $[18, 4^02^9, 8]^{\dagger **}$ $u-2-2-2-2$ $2^{18}$ $[18, 4^12^{16}, 4]$ $[18, 4^02^9, 8]^{\dagger}$ $2-2-u-u-2$ $2^{18}$ $[18, 4^42^{10},4 ]^{**}$ $[18, 4^02^9,8]$ $u-2-u-u-2$ $2^{18}$ $[18, 4^52^8, 4]^{**}$ $[18, 4^02^9,6]^\dagger$ $(2+u)-2-u-u-2$ $2^{18}$ $[18,4^5 2^8, 4 ]$ $[18, 4^02^9,8]$ $(2+u) - (2+u) -0-1- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^22^5, 6]^{**}$ $(2+u) - (2+u) -u-u- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,6]$ $(2+u) - (2+u) -2-2- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,6]$ $(2+u) - (2+u) -(2+u) - (2+u) - (2+u)$ $2^{18}$ $[18, 4^9 2^0, 2 ]$ $[18, 4^02^9,2]^\dagger$ $u - (2+u) -2-2- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,8]$
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