American Institute of Mathematical Sciences

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November  2021, 15(4): 737-755. doi: 10.3934/amc.2020094

$ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes

Habibul Islam 1, , Om Prakash 1,, and  Patrick Solé 2,

1. 

Department of Mathematics, Indian Institute of Technology Patna, Patna- 801 106, India
2. 

I2M, (CNRS, Aix-Marseille University, Centrale Marseille), Marseilles, France

* Corresponding author: Om Prakash

Received  January 2020 Revised  May 2020 Published  November 2021 Early access  July 2020

Fund Project: The research is supported by the University Grants Commission (UGC), Govt. of India

We study mixed alphabet cyclic and constacyclic codes over the two alphabets $ \mathbb{Z}_{4}, $ the ring of integers modulo $ 4 $, and its quadratic extension $ \mathbb{Z}_{4}[u] = \mathbb{Z}_{4}+u\mathbb{Z}_{4}, u^{2} = 0. $ Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we find cyclic, quasi-cyclic codes over $ \mathbb{Z}_{4} $ as the Gray images of both $ \lambda $-constacyclic and skew $ \lambda $-constacyclic codes over $ \mathbb{Z}_{4}[u] $. Moreover, it is proved that the Gray images of $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive constacyclic and skew $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive constacyclic codes are generalized quasi-cyclic codes over $ \mathbb{Z}_{4} $. Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.

Keywords: Additive code, Gray map, Quasi-cyclic code, Skew cyclic code, Generalized quasi-cyclic code.
Mathematics Subject Classification: 94B15, 94B05, 94B60.
Citation: Habibul Islam, Om Prakash, Patrick Solé. $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes. Advances in Mathematics of Communications, 2021, 15 (4) : 737-755. doi: 10.3934/amc.2020094
References:
[1]

T. Abualrub and I. Siap, Reversible cyclic codes over $\mathbb{Z}_{4}$, Australas. J. Comb., 38 (2007), 195-205.   Google Scholar

[2]

T. Abualrub and I. Siap, Cyclic codes over the rings $\mathbb{Z}_{2} +u\mathbb{Z}_{2}$ and $\mathbb{Z}_{2} +u\mathbb{Z}_{2}+u^{2}\mathbb{Z}_{2}$, Des. Codes Cryptogr., 42 (2007), 273-287.  doi: 10.1007/s10623-006-9034-5.  Google Scholar

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T. Abualrub, I. Siap and N. Aydin, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-Additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014), 1508-1514.  doi: 10.1109/TIT.2014.2299791.  Google Scholar

[4]

T. Asamov and N. Aydin, Table of Z4 codes, Online available at http://http://www.asamov.com/Z4Codes. Accessed on 2019-12-12. Google Scholar

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N. Aydin and H. Halilovic, A generalization of quasi-twisted codes: Multi-twisted codes, Finite Fields Appl., 45 (2017), 96-106.  doi: 10.1016/j.ffa.2016.12.002.  Google Scholar

[6]

I. Aydogdu, T. Abualrub and I. Siap, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814.  doi: 10.1080/00207160.2013.859854.  Google Scholar

[7]

I. Aydogdu, T. Abualrub and I. Siap, The $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-cyclic and constacyclic codes, IEEE Trans. Inform. Theory, 63 (2017), 4883-4893.  doi: 10.1109/TIT.2016.2632163.  Google Scholar

[8]

I. Aydogdu, T. Abualrub and I. Siap, On the structure of $\mathbb{Z}_{2}\mathbb{Z}_{2}[u^{3}]$-linear and cyclic codes, Finite Fields Appl., 48 (2017), 241-260.  doi: 10.1016/j.ffa.2017.03.001.  Google Scholar

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I. Aydogdu and I. Siap, The structure of $\mathbb{Z}_2\mathbb{Z}_{2^s}$-additive codes: Bounds on the minimum distance, Appl. Math. Inf. Sci., 7 (2013), 2271-2278.  doi: 10.12785/amis/070617.  Google Scholar

[10]

I. Aydogdu and I. Siap, On $\mathbb{Z}_{p^{r}}\mathbb{Z}_{p^{s}}$-additive codes, Linear Multilinear Algebra, 63 (2015), 2089-2102.  doi: 10.1080/03081087.2014.952728.  Google Scholar

[11]

I. Aydogdu and F. Gursoy, On $\mathbb{Z}_{2}\mathbb{Z}_{4}[\xi]$-Skew cyclic codes, preprint (2017), arXiv: 1711.01816v1. Google Scholar

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N. BenBelkacem, F. M. Ezerman, T. Abualrub and A. Batoul, Skew Cyclic Codes over $ \mathbb{F}_4R, $, preprint (2017), arXiv: 1812.10692. Google Scholar

[13]

J. Bierbrauer, The theory of cyclic codes and a generalization to additive codes, Des. Codes Cryptogr, 25 (2002), 189-206.  doi: 10.1023/A:1013808515797.  Google Scholar

[14]

J. Borges, C. Fernandez-Cordoba, J. Pujol and J. Rifa, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes: Geneartor matrices and duality, Des. Codes Cryptogr., 54 (2010), 167-179.  doi: 10.1007/s10623-009-9316-9.  Google Scholar

[15]

J. Borges, C. Fernandez-Cordoba and R. Ten-Valls, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-additive cyclic codes, generator polynomials and dual codes, IEEE Trans. Inform. Theory, 62 (2016), 6348-6354.  doi: 10.1109/TIT.2016.2611528.  Google Scholar

[16]

J. Borges and C. Fernandez-Cordoba, A characterization of $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes, Des. Codes Cryptogr., 86 (2018), 1377-1389.  doi: 10.1007/s10623-017-0401-1.  Google Scholar

[17]

D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.  Google Scholar

[18]

P. Delsarte, An Algebraic Approach to Association Schemes of Coding Theory, Philips Res. Rep., Supplement, 1973.  Google Scholar

[19]

P. Delsarte and V. I. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory, 44 (1998), 2477-2504.  doi: 10.1109/18.720545.  Google Scholar

[20]

M. Esmaeilia and S. Yari, Generalized quasi-cyclic codes: Structrural properties and code construction, Algebra Engrg. Comm. Commput., 20 (2009), 159-173.  doi: 10.1007/s00200-009-0095-3.  Google Scholar

[21]

C. Fernandez-Cordoba, J. Pujol and M. Villanueva, $\mathbb{Z}_2\mathbb{Z}_4$-linear codes: Rank and kernel, Des. Codes Cryptogr., 56 (2010), 43-59.  doi: 10.1007/s10623-009-9340-9.  Google Scholar

[22]

H. Islam and O. Prakash, On $\mathbb{Z}_{p}\mathbb{Z}_{p}[u, v]$-additive cyclic and constacyclic codes, preprint (2019), arXiv: 1905.06686v1. Google Scholar

[23]

P. Li, W. Dai and X. Kai, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic Google Scholar

[24]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc., New York, 1974.  Google Scholar

[25]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.  Google Scholar

[26]

H. Rifa-Pous, J. Rifa and L. Ronquillo, $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in steganography, Adv. Math. Commun., 5 (2011), 425-433.  doi: 10.3934/amc.2011.5.425.  Google Scholar

[27]

A. Sharma and M. Bhaintwal, A class of skew-constacyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, Int. J. Inf. Coding Theory, 4 (2017), 289-303.  doi: 10.1504/IJICOT.2017.086918.  Google Scholar

[28]

A. Sharma and M. Bhaintwal, $\mathbb{F}_3R$-skew cyclic codes, Int. J. Inf. Coding Theory, 3 (2016), 234-251.  doi: 10.1504/IJICOT.2016.076967.  Google Scholar

[29] M. Shi, A. Alahmadi and P. Solé, Codes and Rings: Theory and Practice, Academic Press, 2017.   Google Scholar
[30]

M. Shi, R. Wu and D. S. Krotov, On $\mathbb{Z}_p\mathbb{Z}_{p^k}$-additive codes and their duality, IEEE Trans. Inf. Theory, 65 (2018), 3842-3847.  doi: 10.1109/TIT.2018.2883759.  Google Scholar

[31]

B. Srinivasulu and B. Maheshanand, The $\mathbb{Z}_{2}(\mathbb{Z}_{2}+u\mathbb{Z}_{2})$-additive cyclic codes and their duals, Discrete Math. Algorithm. Appl., 8 (2016), 1650027, 19 pp. doi: 10.1142/S1793830916500270.  Google Scholar

[32]

T. Yao and S. Zhu, $\mathbb{Z}_p\mathbb{Z}_{p^s}$-additive cyclic codes are asymptotically good, Cryptogr. Commun., 12 (2020), 253-264.  doi: 10.1007/s12095-019-00397-z.  Google Scholar

[33]

T. Yao, S. Zhu and X. Kai, Asymptotically good $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes, Finite Fields Appl., 63 (2020), 101633, 15 pp. doi: 10.1016/j.ffa.2020.101633.  Google Scholar

[34]

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

[35]

Félix Ulmer's publication list, http://felixulmer.epizy.com/fu_papers.html. Google Scholar

show all references

References:
[1]

T. Abualrub and I. Siap, Reversible cyclic codes over $\mathbb{Z}_{4}$, Australas. J. Comb., 38 (2007), 195-205.   Google Scholar

[2]

T. Abualrub and I. Siap, Cyclic codes over the rings $\mathbb{Z}_{2} +u\mathbb{Z}_{2}$ and $\mathbb{Z}_{2} +u\mathbb{Z}_{2}+u^{2}\mathbb{Z}_{2}$, Des. Codes Cryptogr., 42 (2007), 273-287.  doi: 10.1007/s10623-006-9034-5.  Google Scholar

[3]

T. Abualrub, I. Siap and N. Aydin, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-Additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014), 1508-1514.  doi: 10.1109/TIT.2014.2299791.  Google Scholar

[4]

T. Asamov and N. Aydin, Table of Z4 codes, Online available at http://http://www.asamov.com/Z4Codes. Accessed on 2019-12-12. Google Scholar

[5]

N. Aydin and H. Halilovic, A generalization of quasi-twisted codes: Multi-twisted codes, Finite Fields Appl., 45 (2017), 96-106.  doi: 10.1016/j.ffa.2016.12.002.  Google Scholar

[6]

I. Aydogdu, T. Abualrub and I. Siap, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes, Int. J. Comput. Math., 92 (2015), 1806-1814.  doi: 10.1080/00207160.2013.859854.  Google Scholar

[7]

I. Aydogdu, T. Abualrub and I. Siap, The $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-cyclic and constacyclic codes, IEEE Trans. Inform. Theory, 63 (2017), 4883-4893.  doi: 10.1109/TIT.2016.2632163.  Google Scholar

[8]

I. Aydogdu, T. Abualrub and I. Siap, On the structure of $\mathbb{Z}_{2}\mathbb{Z}_{2}[u^{3}]$-linear and cyclic codes, Finite Fields Appl., 48 (2017), 241-260.  doi: 10.1016/j.ffa.2017.03.001.  Google Scholar

[9]

I. Aydogdu and I. Siap, The structure of $\mathbb{Z}_2\mathbb{Z}_{2^s}$-additive codes: Bounds on the minimum distance, Appl. Math. Inf. Sci., 7 (2013), 2271-2278.  doi: 10.12785/amis/070617.  Google Scholar

[10]

I. Aydogdu and I. Siap, On $\mathbb{Z}_{p^{r}}\mathbb{Z}_{p^{s}}$-additive codes, Linear Multilinear Algebra, 63 (2015), 2089-2102.  doi: 10.1080/03081087.2014.952728.  Google Scholar

[11]

I. Aydogdu and F. Gursoy, On $\mathbb{Z}_{2}\mathbb{Z}_{4}[\xi]$-Skew cyclic codes, preprint (2017), arXiv: 1711.01816v1. Google Scholar

[12]

N. BenBelkacem, F. M. Ezerman, T. Abualrub and A. Batoul, Skew Cyclic Codes over $ \mathbb{F}_4R, $, preprint (2017), arXiv: 1812.10692. Google Scholar

[13]

J. Bierbrauer, The theory of cyclic codes and a generalization to additive codes, Des. Codes Cryptogr, 25 (2002), 189-206.  doi: 10.1023/A:1013808515797.  Google Scholar

[14]

J. Borges, C. Fernandez-Cordoba, J. Pujol and J. Rifa, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear codes: Geneartor matrices and duality, Des. Codes Cryptogr., 54 (2010), 167-179.  doi: 10.1007/s10623-009-9316-9.  Google Scholar

[15]

J. Borges, C. Fernandez-Cordoba and R. Ten-Valls, $\mathbb{Z}_{2}\mathbb{Z}_{4}$-additive cyclic codes, generator polynomials and dual codes, IEEE Trans. Inform. Theory, 62 (2016), 6348-6354.  doi: 10.1109/TIT.2016.2611528.  Google Scholar

[16]

J. Borges and C. Fernandez-Cordoba, A characterization of $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes, Des. Codes Cryptogr., 86 (2018), 1377-1389.  doi: 10.1007/s10623-017-0401-1.  Google Scholar

[17]

D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.  Google Scholar

[18]

P. Delsarte, An Algebraic Approach to Association Schemes of Coding Theory, Philips Res. Rep., Supplement, 1973.  Google Scholar

[19]

P. Delsarte and V. I. Levenshtein, Association schemes and coding theory, IEEE Trans. Inform. Theory, 44 (1998), 2477-2504.  doi: 10.1109/18.720545.  Google Scholar

[20]

M. Esmaeilia and S. Yari, Generalized quasi-cyclic codes: Structrural properties and code construction, Algebra Engrg. Comm. Commput., 20 (2009), 159-173.  doi: 10.1007/s00200-009-0095-3.  Google Scholar

[21]

C. Fernandez-Cordoba, J. Pujol and M. Villanueva, $\mathbb{Z}_2\mathbb{Z}_4$-linear codes: Rank and kernel, Des. Codes Cryptogr., 56 (2010), 43-59.  doi: 10.1007/s10623-009-9340-9.  Google Scholar

[22]

H. Islam and O. Prakash, On $\mathbb{Z}_{p}\mathbb{Z}_{p}[u, v]$-additive cyclic and constacyclic codes, preprint (2019), arXiv: 1905.06686v1. Google Scholar

[23]

P. Li, W. Dai and X. Kai, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic Google Scholar

[24]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc., New York, 1974.  Google Scholar

[25]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.  Google Scholar

[26]

H. Rifa-Pous, J. Rifa and L. Ronquillo, $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in steganography, Adv. Math. Commun., 5 (2011), 425-433.  doi: 10.3934/amc.2011.5.425.  Google Scholar

[27]

A. Sharma and M. Bhaintwal, A class of skew-constacyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, Int. J. Inf. Coding Theory, 4 (2017), 289-303.  doi: 10.1504/IJICOT.2017.086918.  Google Scholar

[28]

A. Sharma and M. Bhaintwal, $\mathbb{F}_3R$-skew cyclic codes, Int. J. Inf. Coding Theory, 3 (2016), 234-251.  doi: 10.1504/IJICOT.2016.076967.  Google Scholar

[29] M. Shi, A. Alahmadi and P. Solé, Codes and Rings: Theory and Practice, Academic Press, 2017.   Google Scholar
[30]

M. Shi, R. Wu and D. S. Krotov, On $\mathbb{Z}_p\mathbb{Z}_{p^k}$-additive codes and their duality, IEEE Trans. Inf. Theory, 65 (2018), 3842-3847.  doi: 10.1109/TIT.2018.2883759.  Google Scholar

[31]

B. Srinivasulu and B. Maheshanand, The $\mathbb{Z}_{2}(\mathbb{Z}_{2}+u\mathbb{Z}_{2})$-additive cyclic codes and their duals, Discrete Math. Algorithm. Appl., 8 (2016), 1650027, 19 pp. doi: 10.1142/S1793830916500270.  Google Scholar

[32]

T. Yao and S. Zhu, $\mathbb{Z}_p\mathbb{Z}_{p^s}$-additive cyclic codes are asymptotically good, Cryptogr. Commun., 12 (2020), 253-264.  doi: 10.1007/s12095-019-00397-z.  Google Scholar

[33]

T. Yao, S. Zhu and X. Kai, Asymptotically good $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes, Finite Fields Appl., 63 (2020), 101633, 15 pp. doi: 10.1016/j.ffa.2020.101633.  Google Scholar

[34]

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

[35]

Félix Ulmer's publication list, http://felixulmer.epizy.com/fu_papers.html. Google Scholar

Table 1.  Gray images of $(3+2u)$-constacyclic codes over $\mathbb Z_4[u]$
$\beta$ $h(x)$ $k(x)$ $\phi_{1}(\mathcal{S})$ $\phi_{2}(\mathcal{S})$
$3$ $[1, 3+2u, 3+u]$ $[3u, u]$ $[6, 4^0 2^4, 2]^*$ $[6, 4^4 2^0, 2]^{\#}$
$7$ $[1, 3+2u, 3, 0, 1+u]$ $[3u, 3u, 2u, u]$ $[14, 4^32^7, 2]$ $[14, 4^72^3, 4]^{\#}$
$7$ $[3+2u, 2, 3+2u, 1+u]$ $[3u, 3u, 2u, u]$ $[14, 4^0 2^{11}, 2]$ $[14, 4^7 2^4, 2]$
$9$ $[3+2u, 0, 2, 1+u]$ $[u, 0, 0, 3u, 0, 0, 3u]$ $[18, 4^02^{15}, 2]$ $[18, 4^9 2^6, 2]$
$9$ $[3+2u, 3, 0, 1, 1+2u, 0, 1+2u, 1+u]$ $[u, 3u, 3u]$ $[18, 4^0 2^{11}, 2]^*$ $[18, 4^9 2^2, 2]$
$15$ $[3+2u, 3, 2, 1, 2+u]$ $[u, 2u, 2u, 3u, u, u, 3u]$ $[30, 4^0 2^{26}, 2]$ $[30, 4^{15} 2^{11}, 2]^*$
$15$ $[1, 0, 2, 2, 1, 3+2u, 1, 3+2u, 3+u]$ $[u, 3u, 3u]$ $[30, 4^0 2^{20}, 4]$ $[30, 4^{15} 2^5, 4]^*$
$17$ $[3+2u, 3, 2, 1, 2, 2, 1+2u, 2, 1+2u, 2, 1+2u, 1+u]$ $[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$ $[34, 4^0 2^{25}, 2]^*$ $[34, 4^{17} 2^8, 2]^*$
$17$ $[1, 0, 2, 1+2u, 1, 1+2u, 2, 0, 3+u]$ $[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$ $[34, 4^0 2^{26}, 2]^*$ $[34, 4^{17} 2^9, 2]^*$
$21$ $[3+2u, 0, 0, 3, 0, 0, 0, 0, 2, 1+u]$ $[u, 3u, 3u, 0, 3u, 2u, 3u]$ $[42, 4^0 2^{33}, 2]^*$ $[42, 4^{21} 2^{18}, 2]^*$
$21$ $[3+2u, 3, 2, 2, 3+2u, 1, 2, 1, 3+2u, 3+u]$ $[3u, 0, 2u, u]$ $[42, 4^0 2^{33}, 2]^*$ $[42, 4^{21} 2^{12}, 2]^*$
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$\beta$ $h(x)$ $k(x)$ $\phi_{1}(\mathcal{S})$ $\phi_{2}(\mathcal{S})$
$3$ $[1, 3+2u, 3+u]$ $[3u, u]$ $[6, 4^0 2^4, 2]^*$ $[6, 4^4 2^0, 2]^{\#}$
$7$ $[1, 3+2u, 3, 0, 1+u]$ $[3u, 3u, 2u, u]$ $[14, 4^32^7, 2]$ $[14, 4^72^3, 4]^{\#}$
$7$ $[3+2u, 2, 3+2u, 1+u]$ $[3u, 3u, 2u, u]$ $[14, 4^0 2^{11}, 2]$ $[14, 4^7 2^4, 2]$
$9$ $[3+2u, 0, 2, 1+u]$ $[u, 0, 0, 3u, 0, 0, 3u]$ $[18, 4^02^{15}, 2]$ $[18, 4^9 2^6, 2]$
$9$ $[3+2u, 3, 0, 1, 1+2u, 0, 1+2u, 1+u]$ $[u, 3u, 3u]$ $[18, 4^0 2^{11}, 2]^*$ $[18, 4^9 2^2, 2]$
$15$ $[3+2u, 3, 2, 1, 2+u]$ $[u, 2u, 2u, 3u, u, u, 3u]$ $[30, 4^0 2^{26}, 2]$ $[30, 4^{15} 2^{11}, 2]^*$
$15$ $[1, 0, 2, 2, 1, 3+2u, 1, 3+2u, 3+u]$ $[u, 3u, 3u]$ $[30, 4^0 2^{20}, 4]$ $[30, 4^{15} 2^5, 4]^*$
$17$ $[3+2u, 3, 2, 1, 2, 2, 1+2u, 2, 1+2u, 2, 1+2u, 1+u]$ $[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$ $[34, 4^0 2^{25}, 2]^*$ $[34, 4^{17} 2^8, 2]^*$
$17$ $[1, 0, 2, 1+2u, 1, 1+2u, 2, 0, 3+u]$ $[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$ $[34, 4^0 2^{26}, 2]^*$ $[34, 4^{17} 2^9, 2]^*$
$21$ $[3+2u, 0, 0, 3, 0, 0, 0, 0, 2, 1+u]$ $[u, 3u, 3u, 0, 3u, 2u, 3u]$ $[42, 4^0 2^{33}, 2]^*$ $[42, 4^{21} 2^{18}, 2]^*$
$21$ $[3+2u, 3, 2, 2, 3+2u, 1, 2, 1, 3+2u, 3+u]$ $[3u, 0, 2u, u]$ $[42, 4^0 2^{33}, 2]^*$ $[42, 4^{21} 2^{12}, 2]^*$
Table 2.  $ \mathbb{Z}_4 $-Gray images of $ \mathbb{Z}_4\mathbb{Z}_4[u] $-additive cyclic codes of length $ (\alpha, \beta) $
$ (\alpha, \beta) $ Generators $ \Phi_{1}(\mathcal{C}) $ $ \Phi_{2}(\mathcal{C}) $
$ (3, 3) $ $ g_1=g_2=x^2+x+1, g_3=x+1 $, $ a_1=a_2=a_3=1, p=1, f_1=f_2=x+1 $ $ [9, 4^3 2^3, 1]^* $ $ [9, 4^6 2^0, 1] $
$ (3, 3) $ $ g_1=g_2=g_3=x^3-1, a_1=x^2+x+1 $, $ a_2=a_3=x+3, p=1, f_1=f_2=x^2+2 $ $ [9, 4^3 2^2, 1]^* $ $ [9, 4^5 2^1, 1]^{*} $
$ (3, 7) $ $ g_1=x^3-1, g_2=x^4+x^3+3x^2+2x+1, g_3=x^7-1 $,
$ a_1=x^2+x+1, a_2=x+3, a_3=x^3+3x^2+2x+3, p=1, f_1=f_2=x+1 $ $ [17, 4^2 2^8, 2]^* $ $ [17, 4^9 2^1, 2]^* $
$ (7, 3) $ $ g_1=x^7-1, g_2=x^3-1, g_3=x+3 $,
$ a_1=x^4+2x^3+3x^2+x+1, a_2=x^2+x+1, a_3=1, p=1, f_1=f_2=x+3 $ $ [13, 4^3 2^5, 2]^* $ $ [13, 4^6 2^2, 2]^* $
$ (7, 7) $ $ g_1=x^4+x^3+3x^2+2x+1, g_2=x^4+2x^3+3x^2+x+1, g_3=x+3 $,
$ a_1=x^3+2x^2+x+3, a_2=x^3+3x^2+2x+3, a_3=1, p=1, f_1=f_2=x+3 $ $ [21, 4^6 2^8, 2]^* $ $ [21, 4^{13} 2^2, 2]^* $
$ (9, 9) $ $ g_1=x^2+x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^6+x^3+1 $,
$ a_1=1, a_2=x^6+x^3+1, a_3=1, p=1, f_1=f_2=x+3 $ $ [27, 4^9 2^9, 2]^* $ $ [27, 4^{16} 2^4, 2]^* $
$ (3, 9) $ $ g_1=x^2+x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^3+3 $,
$ a_1=1, a_2=a_3=x+3, p=1, f_1=f_2=x+3 $ $ [21, 4^3 2^9, 1]^* $ $ [21, 4^{12} 2^0, 1] $
$ (9, 3) $ $ g_1=x^3+3, g_2=x^3-1, g_3=x+3 $,
$ a_1=x^2+x+1, a_2=x+3, a_3=1, p=1, f_1=f_2=x+3 $ $ [15, 4^8 2^4, 2]^* $ $ [15, 4^{11} 2^1, 2]^* $
$ (7, 9) $ $ g_1=x^4+x^3+3x^2+2x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^2+x+1 $,
$ a_1=x^3+2x^2+x+3, a_2=x^6+x^3+1, a_3=1, p=1, f_1=f_2=x+1 $ $ [25, 4^6 2^{10}, 2]^* $ $ [25, 4^{15} 2^1, 2]^* $
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$ (\alpha, \beta) $ Generators $ \Phi_{1}(\mathcal{C}) $ $ \Phi_{2}(\mathcal{C}) $
$ (3, 3) $ $ g_1=g_2=x^2+x+1, g_3=x+1 $, $ a_1=a_2=a_3=1, p=1, f_1=f_2=x+1 $ $ [9, 4^3 2^3, 1]^* $ $ [9, 4^6 2^0, 1] $
$ (3, 3) $ $ g_1=g_2=g_3=x^3-1, a_1=x^2+x+1 $, $ a_2=a_3=x+3, p=1, f_1=f_2=x^2+2 $ $ [9, 4^3 2^2, 1]^* $ $ [9, 4^5 2^1, 1]^{*} $
$ (3, 7) $ $ g_1=x^3-1, g_2=x^4+x^3+3x^2+2x+1, g_3=x^7-1 $,
$ a_1=x^2+x+1, a_2=x+3, a_3=x^3+3x^2+2x+3, p=1, f_1=f_2=x+1 $ $ [17, 4^2 2^8, 2]^* $ $ [17, 4^9 2^1, 2]^* $
$ (7, 3) $ $ g_1=x^7-1, g_2=x^3-1, g_3=x+3 $,
$ a_1=x^4+2x^3+3x^2+x+1, a_2=x^2+x+1, a_3=1, p=1, f_1=f_2=x+3 $ $ [13, 4^3 2^5, 2]^* $ $ [13, 4^6 2^2, 2]^* $
$ (7, 7) $ $ g_1=x^4+x^3+3x^2+2x+1, g_2=x^4+2x^3+3x^2+x+1, g_3=x+3 $,
$ a_1=x^3+2x^2+x+3, a_2=x^3+3x^2+2x+3, a_3=1, p=1, f_1=f_2=x+3 $ $ [21, 4^6 2^8, 2]^* $ $ [21, 4^{13} 2^2, 2]^* $
$ (9, 9) $ $ g_1=x^2+x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^6+x^3+1 $,
$ a_1=1, a_2=x^6+x^3+1, a_3=1, p=1, f_1=f_2=x+3 $ $ [27, 4^9 2^9, 2]^* $ $ [27, 4^{16} 2^4, 2]^* $
$ (3, 9) $ $ g_1=x^2+x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^3+3 $,
$ a_1=1, a_2=a_3=x+3, p=1, f_1=f_2=x+3 $ $ [21, 4^3 2^9, 1]^* $ $ [21, 4^{12} 2^0, 1] $
$ (9, 3) $ $ g_1=x^3+3, g_2=x^3-1, g_3=x+3 $,
$ a_1=x^2+x+1, a_2=x+3, a_3=1, p=1, f_1=f_2=x+3 $ $ [15, 4^8 2^4, 2]^* $ $ [15, 4^{11} 2^1, 2]^* $
$ (7, 9) $ $ g_1=x^4+x^3+3x^2+2x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^2+x+1 $,
$ a_1=x^3+2x^2+x+3, a_2=x^6+x^3+1, a_3=1, p=1, f_1=f_2=x+1 $ $ [25, 4^6 2^{10}, 2]^* $ $ [25, 4^{15} 2^1, 2]^* $
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