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February 2018, 12(1): 169-179. doi: 10.3934/amc.2018011

On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes

Department of Information and Communications Engineering, Universitat Autónoma de Barcelona, 08193-Bellaterra, Spain

Received  October 2016 Published  March 2018

A ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive code, $r≤ s$, is a${\mathbb{Z}}_{p^s}$-submodule of ${{\mathbb{Z}}_{p^r}^α× {\mathbb{Z}}_{p^s}^β}$. We introduce ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes. These codes can be seen as ${\mathbb{Z}}_{p^s}[x]$-submodules of ${\mathcal{R}^{α,β}_{r,s}}= \frac{{\mathbb{Z}}_{p^r}[x]}{\langle x^α-1\rangle}×\frac{{\mathbb{Z}}_{p^s}[x]}{\langle x^β-1\rangle}$. We determine the generator polynomials of a code over ${\mathcal{R}^{α,β}_{r,s}}$ and a minimal spanning set over ${{\mathbb{Z}}_{p^r}^α× {\mathbb{Z}}_{p^s}^β}$ in terms of the generator polynomials. We also study the duality in the module ${\mathcal{R}^{α,β}_{r,s}}$.Our results generalise those for ${\mathbb{Z}}_{2}{\mathbb{Z}}_{4}$-additive cyclic codes.

Citation: Joaquim Borges, Cristina Fernández-Córdoba, Roger Ten-Valls. On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 169-179. doi: 10.3934/amc.2018011
References:
[1]

T. AbualrubI. Siap and N. Aydin, $\mathbb Z_2\mathbb Z_4$-additive cyclic codes, IEEE Trans. Inf. Theory, 60 (2014), 1508-1514.

[2]

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.

[3]

I. Aydogdu and I. Siap, On $\mathbb Z_{p^r}\mathbb Z_{p^s}$-additive codes, Lin. Multilin. Algebra, 63 (2014), 2089-2102.

[4]

J. BorgesC. Fernández-CórdobaJ. PujolJ. Rifá and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: generator matrices and duality, Des. Codes Crypt., 54 (2010), 167-179.

[5]

J. BorgesC. Fernández-Córdoba and R. Ten-Valls, $\mathbb Z_2\mathbb Z_4$-additive cyclic codes, generator polynomials and dual codes, IEEE Trans. Inf. Theory, 62 (2016), 6348-6354.

[6]

J. BorgesC. Fernández-Córdoba and R. Ten-Valls, $\mathbb Z_2$-double cyclic codes, Des. Codes Crypt., 86 (2018), 463-479. doi: 10.1007/s10623-017-0334-8.

[7]

A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35.

[8]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inf. Theory, 50 (2004), 1728-1744.

[9]

J. GaoM. ShiT. Wu and F. Fu, On double cyclic codes over $\mathbb Z_4$, Finite Fields Appl., 39 (2016), 233-250.

[10]

P. Kanwar and S. R. López-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl., 3 (1997), 334-352.

show all references

References:
[1]

T. AbualrubI. Siap and N. Aydin, $\mathbb Z_2\mathbb Z_4$-additive cyclic codes, IEEE Trans. Inf. Theory, 60 (2014), 1508-1514.

[2]

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.

[3]

I. Aydogdu and I. Siap, On $\mathbb Z_{p^r}\mathbb Z_{p^s}$-additive codes, Lin. Multilin. Algebra, 63 (2014), 2089-2102.

[4]

J. BorgesC. Fernández-CórdobaJ. PujolJ. Rifá and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: generator matrices and duality, Des. Codes Crypt., 54 (2010), 167-179.

[5]

J. BorgesC. Fernández-Córdoba and R. Ten-Valls, $\mathbb Z_2\mathbb Z_4$-additive cyclic codes, generator polynomials and dual codes, IEEE Trans. Inf. Theory, 62 (2016), 6348-6354.

[6]

J. BorgesC. Fernández-Córdoba and R. Ten-Valls, $\mathbb Z_2$-double cyclic codes, Des. Codes Crypt., 86 (2018), 463-479. doi: 10.1007/s10623-017-0334-8.

[7]

A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35.

[8]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inf. Theory, 50 (2004), 1728-1744.

[9]

J. GaoM. ShiT. Wu and F. Fu, On double cyclic codes over $\mathbb Z_4$, Finite Fields Appl., 39 (2016), 233-250.

[10]

P. Kanwar and S. R. López-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl., 3 (1997), 334-352.

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