doi: 10.3934/amc.2022017
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On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes

1. 

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

2. 

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

* Corresponding author: Patrick Solé

Received  August 2021 Revised  January 2022 Early access March 2022

Let $ \mathbb{Z}_4 $ be the ring of integers modulo $ 4 $. This paper studies mixed alphabets $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive cyclic and $ \lambda $-constacyclic codes for units $ \lambda = 1+2u^2,3+2u^2 $. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the structure of additive constacyclic codes. Further, we define some Gray maps and obtain $ \mathbb{Z}_4 $-images of such codes. Finally, we present numerical examples that include six new and two best-known quaternary linear codes.

Citation: Om Prakash, Shikha Yadav, Habibul Islam, Patrick Solé. On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022017
References:
[1]

T. AbualrubI. 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.

[2]

T. Asamov and N. Aydin, Table of $\mathbb{Z}_4$ codes, available online at http://www.asamov.com/Z4Codes, Accessed on 2021-08-20.

[3]

I. AydogduT. 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.

[4]

I. AydogduT. Abualrub and I. Siap, The $\mathbb{Z}_2\mathbb{Z}_2[u]$-cyclic and constacyclic codes, IEEE Trans. Inform. Theory, 63 (2016), 4883-4893.  doi: 10.1109/TIT.2016.2632163.

[5]

I. AydogduI. Siap and R. Ten-Valls, 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.

[6]

I. Aydogdu, Codes over $\mathbb{Z}_p[u]/\langle u^r\rangle\times \mathbb{Z}_p[u]/\langle u^s\rangle$, J. Algebra Comb. Discrete Struct. Appl., 6 (2019), 39-51.  doi: 10.13069/jacodesmath.514339.

[7]

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.

[8]

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.

[9]

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.

[10]

J. BorgesC. Fernández-CórdobaJ. Pujol and J. Rifà, $\mathbb{Z}_2\mathbb{Z}_4$-linear codes: Geneartor matrices and duality, Des. Codes Cryptogr., 54 (2010), 167-179.  doi: 10.1007/s10623-009-9316-9.

[11]

J. BorgesC. Fernández-Córdoba 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.

[12]

J. Borges and C. Fernández-Córdoba, 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.

[13]

W. Bosma and J. Cannon, Handbook of Magma Functions., Univ. of Sydney, (1995).

[14]

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

[15]

M. Esmaeili and S. Yari, Generalised quasi-cyclic codes: Structural properties and code construction, Appl. Algebra Engrg. Comm. Comput., 20 (2009), 159-173.  doi: 10.1007/s00200-009-0095-3.

[16]

C. Fernández-CórdobaJ. 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.

[17]

H. IslamO. Prakash and P. Solé, $\mathbb{Z}_4\mathbb{Z}_4[u]$-Additive cyclic and constacyclic codes, Adv. Math. Commun., 15 (2020), 737-755. 

[18]

H. IslamT. Bag and O. Prakash, A class of constacyclic codes over $\mathbb{Z}_4[u]/\langle u^k \rangle$, J. Appl. Math. Comput., 60 (2019), 237-251.  doi: 10.1007/s12190-018-1211-y.

[19]

H. Islam and O. Prakash, On $\mathbb{Z}_p\mathbb{Z}_p[u, v]$-additive cyclic and constacyclic codes, (preprint), (2019), http://arXiv.org/abs/1905.06686v1.

[20]

P. Li, W. Dai and X. Kai, On $\mathbb{Z}_2\mathbb{Z}_2[u]$-$(1 + u)$-additive constacyclic, (preprint), (2016), arXiv: 1611.03169v1[cs.IT].

[21]

E. Martínez-MoroA. Pinera-Nicolas and I. F. Rua, Additive semisimple multivariable codes over $\mathbb{F}_4$, Des. Codes Cryptogr., 69 (2013), 161-180.  doi: 10.1007/s10623-012-9641-2.

[22]

E. Martínez-MoroK. Otal and F. Özbudak, Additive cyclic codes over finite commutative chain rings, Discrete Math., 341 (2018), 1873-1884.  doi: 10.1016/j.disc.2018.03.016.

[23]

H. Rifà-PousJ. 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.

[24]

M. ShiL. QianL. Sok and P. Solé, On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1 \rangle$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95.  doi: 10.1016/j.ffa.2016.11.016.

[25]

M. ShiR. Wu and D. S. Krotov, On $\mathbb{Z}_p\mathbb{Z}_{p^k}$-additive codes and their duality, IEEE Trans. Inform. Theory, 65 (2018), 3841-3847. 

[26]

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.

[27]

W. C. Huffman, Additive cyclic codes over $\mathbb{F}_4$, Adv. Math. Commun., 1 (2007), 427-459.  doi: 10.3934/amc.2007.1.427.

[28]

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.

[29]

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.

[30]

, Magma Online Calculator, http://magma.maths.usyd.edu.au/calc/.

show all references

References:
[1]

T. AbualrubI. 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.

[2]

T. Asamov and N. Aydin, Table of $\mathbb{Z}_4$ codes, available online at http://www.asamov.com/Z4Codes, Accessed on 2021-08-20.

[3]

I. AydogduT. 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.

[4]

I. AydogduT. Abualrub and I. Siap, The $\mathbb{Z}_2\mathbb{Z}_2[u]$-cyclic and constacyclic codes, IEEE Trans. Inform. Theory, 63 (2016), 4883-4893.  doi: 10.1109/TIT.2016.2632163.

[5]

I. AydogduI. Siap and R. Ten-Valls, 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.

[6]

I. Aydogdu, Codes over $\mathbb{Z}_p[u]/\langle u^r\rangle\times \mathbb{Z}_p[u]/\langle u^s\rangle$, J. Algebra Comb. Discrete Struct. Appl., 6 (2019), 39-51.  doi: 10.13069/jacodesmath.514339.

[7]

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.

[8]

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.

[9]

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.

[10]

J. BorgesC. Fernández-CórdobaJ. Pujol and J. Rifà, $\mathbb{Z}_2\mathbb{Z}_4$-linear codes: Geneartor matrices and duality, Des. Codes Cryptogr., 54 (2010), 167-179.  doi: 10.1007/s10623-009-9316-9.

[11]

J. BorgesC. Fernández-Córdoba 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.

[12]

J. Borges and C. Fernández-Córdoba, 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.

[13]

W. Bosma and J. Cannon, Handbook of Magma Functions., Univ. of Sydney, (1995).

[14]

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

[15]

M. Esmaeili and S. Yari, Generalised quasi-cyclic codes: Structural properties and code construction, Appl. Algebra Engrg. Comm. Comput., 20 (2009), 159-173.  doi: 10.1007/s00200-009-0095-3.

[16]

C. Fernández-CórdobaJ. 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.

[17]

H. IslamO. Prakash and P. Solé, $\mathbb{Z}_4\mathbb{Z}_4[u]$-Additive cyclic and constacyclic codes, Adv. Math. Commun., 15 (2020), 737-755. 

[18]

H. IslamT. Bag and O. Prakash, A class of constacyclic codes over $\mathbb{Z}_4[u]/\langle u^k \rangle$, J. Appl. Math. Comput., 60 (2019), 237-251.  doi: 10.1007/s12190-018-1211-y.

[19]

H. Islam and O. Prakash, On $\mathbb{Z}_p\mathbb{Z}_p[u, v]$-additive cyclic and constacyclic codes, (preprint), (2019), http://arXiv.org/abs/1905.06686v1.

[20]

P. Li, W. Dai and X. Kai, On $\mathbb{Z}_2\mathbb{Z}_2[u]$-$(1 + u)$-additive constacyclic, (preprint), (2016), arXiv: 1611.03169v1[cs.IT].

[21]

E. Martínez-MoroA. Pinera-Nicolas and I. F. Rua, Additive semisimple multivariable codes over $\mathbb{F}_4$, Des. Codes Cryptogr., 69 (2013), 161-180.  doi: 10.1007/s10623-012-9641-2.

[22]

E. Martínez-MoroK. Otal and F. Özbudak, Additive cyclic codes over finite commutative chain rings, Discrete Math., 341 (2018), 1873-1884.  doi: 10.1016/j.disc.2018.03.016.

[23]

H. Rifà-PousJ. 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.

[24]

M. ShiL. QianL. Sok and P. Solé, On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1 \rangle$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95.  doi: 10.1016/j.ffa.2016.11.016.

[25]

M. ShiR. Wu and D. S. Krotov, On $\mathbb{Z}_p\mathbb{Z}_{p^k}$-additive codes and their duality, IEEE Trans. Inform. Theory, 65 (2018), 3841-3847. 

[26]

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.

[27]

W. C. Huffman, Additive cyclic codes over $\mathbb{F}_4$, Adv. Math. Commun., 1 (2007), 427-459.  doi: 10.3934/amc.2007.1.427.

[28]

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.

[29]

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.

[30]

, Magma Online Calculator, http://magma.maths.usyd.edu.au/calc/.

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