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$ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes
1. | Department of Mathematics, Indian Institute of Technology Patna, Patna- 801 106, India |
2. | I2M, (CNRS, Aix-Marseille University, Centrale Marseille), Marseilles, France |
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.
References:
[1] |
T. Abualrub and I. Siap,
Reversible cyclic codes over $\mathbb{Z}_{4}$, Australas. J. Comb., 38 (2007), 195-205.
|
[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. |
[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. |
[4] |
T. Asamov and N. Aydin, Table of Z4 codes, Online available at http://http://www.asamov.com/Z4Codes. Accessed on 2019-12-12. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
I. Aydogdu and F. Gursoy, On $\mathbb{Z}_{2}\mathbb{Z}_{4}[\xi]$-Skew cyclic codes, preprint (2017), arXiv: 1711.01816v1. |
[12] |
N. BenBelkacem, F. M. Ezerman, T. Abualrub and A. Batoul, Skew Cyclic Codes over $ \mathbb{F}_4R, $, preprint (2017), arXiv: 1812.10692. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
P. Delsarte, An Algebraic Approach to Association Schemes of Coding Theory, Philips Res. Rep., Supplement, 1973. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. Li, W. Dai and X. Kai, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic |
[24] |
B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc., New York, 1974. |
[25] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. |
[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. |
[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. |
[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. |
[29] |
M. Shi, A. Alahmadi and P. Solé, Codes and Rings: Theory and Practice, Academic Press, 2017.
![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[35] |
Félix Ulmer's publication list, http://felixulmer.epizy.com/fu_papers.html. |
show all references
References:
[1] |
T. Abualrub and I. Siap,
Reversible cyclic codes over $\mathbb{Z}_{4}$, Australas. J. Comb., 38 (2007), 195-205.
|
[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. |
[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. |
[4] |
T. Asamov and N. Aydin, Table of Z4 codes, Online available at http://http://www.asamov.com/Z4Codes. Accessed on 2019-12-12. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
I. Aydogdu and F. Gursoy, On $\mathbb{Z}_{2}\mathbb{Z}_{4}[\xi]$-Skew cyclic codes, preprint (2017), arXiv: 1711.01816v1. |
[12] |
N. BenBelkacem, F. M. Ezerman, T. Abualrub and A. Batoul, Skew Cyclic Codes over $ \mathbb{F}_4R, $, preprint (2017), arXiv: 1812.10692. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
P. Delsarte, An Algebraic Approach to Association Schemes of Coding Theory, Philips Res. Rep., Supplement, 1973. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. Li, W. Dai and X. Kai, On $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic |
[24] |
B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc., New York, 1974. |
[25] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. |
[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. |
[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. |
[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. |
[29] |
M. Shi, A. Alahmadi and P. Solé, Codes and Rings: Theory and Practice, Academic Press, 2017.
![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[35] |
Félix Ulmer's publication list, http://felixulmer.epizy.com/fu_papers.html. |
$\beta$ | $h(x)$ | $k(x)$ | $\phi_{1}(\mathcal{S})$ | $\phi_{2}(\mathcal{S})$ |
$3$ | $[1, 3+2u, 3+u]$ | $[3u, u]$ | $[6, 4^0 2^4, 2]^*$ | $[6, 4^4 2^0, 2]^{\#}$ |
$7$ | $[1, 3+2u, 3, 0, 1+u]$ | $[3u, 3u, 2u, u]$ | $[14, 4^32^7, 2]$ | $[14, 4^72^3, 4]^{\#}$ |
$7$ | $[3+2u, 2, 3+2u, 1+u]$ | $[3u, 3u, 2u, u]$ | $[14, 4^0 2^{11}, 2]$ | $[14, 4^7 2^4, 2]$ |
$9$ | $[3+2u, 0, 2, 1+u]$ | $[u, 0, 0, 3u, 0, 0, 3u]$ | $[18, 4^02^{15}, 2]$ | $[18, 4^9 2^6, 2]$ |
$9$ | $[3+2u, 3, 0, 1, 1+2u, 0, 1+2u, 1+u]$ | $[u, 3u, 3u]$ | $[18, 4^0 2^{11}, 2]^*$ | $[18, 4^9 2^2, 2]$ |
$15$ | $[3+2u, 3, 2, 1, 2+u]$ | $[u, 2u, 2u, 3u, u, u, 3u]$ | $[30, 4^0 2^{26}, 2]$ | $[30, 4^{15} 2^{11}, 2]^*$ |
$15$ | $[1, 0, 2, 2, 1, 3+2u, 1, 3+2u, 3+u]$ | $[u, 3u, 3u]$ | $[30, 4^0 2^{20}, 4]$ | $[30, 4^{15} 2^5, 4]^*$ |
$17$ | $[3+2u, 3, 2, 1, 2, 2, 1+2u, 2, 1+2u, 2, 1+2u, 1+u]$ | $[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$ | $[34, 4^0 2^{25}, 2]^*$ | $[34, 4^{17} 2^8, 2]^*$ |
$17$ | $[1, 0, 2, 1+2u, 1, 1+2u, 2, 0, 3+u]$ | $[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$ | $[34, 4^0 2^{26}, 2]^*$ | $[34, 4^{17} 2^9, 2]^*$ |
$21$ | $[3+2u, 0, 0, 3, 0, 0, 0, 0, 2, 1+u]$ | $[u, 3u, 3u, 0, 3u, 2u, 3u]$ | $[42, 4^0 2^{33}, 2]^*$ | $[42, 4^{21} 2^{18}, 2]^*$ |
$21$ | $[3+2u, 3, 2, 2, 3+2u, 1, 2, 1, 3+2u, 3+u]$ | $[3u, 0, 2u, u]$ | $[42, 4^0 2^{33}, 2]^*$ | $[42, 4^{21} 2^{12}, 2]^*$ |
$\beta$ | $h(x)$ | $k(x)$ | $\phi_{1}(\mathcal{S})$ | $\phi_{2}(\mathcal{S})$ |
$3$ | $[1, 3+2u, 3+u]$ | $[3u, u]$ | $[6, 4^0 2^4, 2]^*$ | $[6, 4^4 2^0, 2]^{\#}$ |
$7$ | $[1, 3+2u, 3, 0, 1+u]$ | $[3u, 3u, 2u, u]$ | $[14, 4^32^7, 2]$ | $[14, 4^72^3, 4]^{\#}$ |
$7$ | $[3+2u, 2, 3+2u, 1+u]$ | $[3u, 3u, 2u, u]$ | $[14, 4^0 2^{11}, 2]$ | $[14, 4^7 2^4, 2]$ |
$9$ | $[3+2u, 0, 2, 1+u]$ | $[u, 0, 0, 3u, 0, 0, 3u]$ | $[18, 4^02^{15}, 2]$ | $[18, 4^9 2^6, 2]$ |
$9$ | $[3+2u, 3, 0, 1, 1+2u, 0, 1+2u, 1+u]$ | $[u, 3u, 3u]$ | $[18, 4^0 2^{11}, 2]^*$ | $[18, 4^9 2^2, 2]$ |
$15$ | $[3+2u, 3, 2, 1, 2+u]$ | $[u, 2u, 2u, 3u, u, u, 3u]$ | $[30, 4^0 2^{26}, 2]$ | $[30, 4^{15} 2^{11}, 2]^*$ |
$15$ | $[1, 0, 2, 2, 1, 3+2u, 1, 3+2u, 3+u]$ | $[u, 3u, 3u]$ | $[30, 4^0 2^{20}, 4]$ | $[30, 4^{15} 2^5, 4]^*$ |
$17$ | $[3+2u, 3, 2, 1, 2, 2, 1+2u, 2, 1+2u, 2, 1+2u, 1+u]$ | $[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$ | $[34, 4^0 2^{25}, 2]^*$ | $[34, 4^{17} 2^8, 2]^*$ |
$17$ | $[1, 0, 2, 1+2u, 1, 1+2u, 2, 0, 3+u]$ | $[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$ | $[34, 4^0 2^{26}, 2]^*$ | $[34, 4^{17} 2^9, 2]^*$ |
$21$ | $[3+2u, 0, 0, 3, 0, 0, 0, 0, 2, 1+u]$ | $[u, 3u, 3u, 0, 3u, 2u, 3u]$ | $[42, 4^0 2^{33}, 2]^*$ | $[42, 4^{21} 2^{18}, 2]^*$ |
$21$ | $[3+2u, 3, 2, 2, 3+2u, 1, 2, 1, 3+2u, 3+u]$ | $[3u, 0, 2u, u]$ | $[42, 4^0 2^{33}, 2]^*$ | $[42, 4^{21} 2^{12}, 2]^*$ |
Generators | |||
Generators | |||
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