On <inline-formula><tex-math id="M1">\begin{document}${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$\end{document}</tex-math></inline-formula>-additive cyclic codes

Joaquim Borges; Cristina Fernández-Córdoba; Roger Ten-Valls

doi:10.3934/amc.2018011

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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

Joaquim Borges , Cristina Fernández-Córdoba and Roger Ten-Valls

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.

Keywords: Additive codes, codes over rings, cyclic codes, duality.

Mathematics Subject Classification: 94B05, 94B15, 94B60.

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. Abualrub, I. Siap and N. Aydin, $\mathbb Z_2\mathbb Z_4$-additive cyclic codes, IEEE Trans. Inf. Theory, 60 (2014), 1508-1514. Google Scholar
[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. Google Scholar
[3]	I. Aydogdu and I. Siap, On $\mathbb Z_{p^r}\mathbb Z_{p^s}$-additive codes, Lin. Multilin. Algebra, 63 (2014), 2089-2102. Google Scholar
[4]	J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifá and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: generator matrices and duality, Des. Codes Crypt., 54 (2010), 167-179. Google Scholar
[5]	J. Borges, C. 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. Google Scholar
[6]	J. Borges, C. 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. Google Scholar
[7]	A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35. Google Scholar
[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. Google Scholar
[9]	J. Gao, M. Shi, T. Wu and F. Fu, On double cyclic codes over $\mathbb Z_4$, Finite Fields Appl., 39 (2016), 233-250. Google Scholar
[10]	P. Kanwar and S. R. López-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl., 3 (1997), 334-352. Google Scholar

show all references

References:

[1]	T. Abualrub, I. Siap and N. Aydin, $\mathbb Z_2\mathbb Z_4$-additive cyclic codes, IEEE Trans. Inf. Theory, 60 (2014), 1508-1514. Google Scholar
[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. Google Scholar
[3]	I. Aydogdu and I. Siap, On $\mathbb Z_{p^r}\mathbb Z_{p^s}$-additive codes, Lin. Multilin. Algebra, 63 (2014), 2089-2102. Google Scholar
[4]	J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rifá and M. Villanueva, $\mathbb Z_2\mathbb Z_4$-linear codes: generator matrices and duality, Des. Codes Crypt., 54 (2010), 167-179. Google Scholar
[5]	J. Borges, C. 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. Google Scholar
[6]	J. Borges, C. 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. Google Scholar
[7]	A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35. Google Scholar
[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. Google Scholar
[9]	J. Gao, M. Shi, T. Wu and F. Fu, On double cyclic codes over $\mathbb Z_4$, Finite Fields Appl., 39 (2016), 233-250. Google Scholar
[10]	P. Kanwar and S. R. López-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl., 3 (1997), 334-352. Google Scholar

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

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