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

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

doi:10.3934/amc.2018011

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2018, Volume 12, Issue 1: 169-179. Doi: 10.3934/amc.2018011

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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: February 2018

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Abstract

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.

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

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References

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