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Channel decomposition for multilevel codes over multilevel and partial erasure channels
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 |
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.
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.
|
| [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.
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| [3] |
I. Aydogdu and I. Siap,
On $\mathbb Z_{p^r}\mathbb Z_{p^s}$-additive codes, Lin. Multilin. Algebra, 63 (2014), 2089-2102.
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| [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.
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| [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.
|
| [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. |
| [7] |
A. R. Calderbank and N. J. A. Sloane,
Modular and $p$-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35.
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H. Q. Dinh and S. R. López-Permouth,
Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inf. Theory, 50 (2004), 1728-1744.
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| [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.
|
| [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. Abualrub, I. 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. 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.
|
| [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.
|
| [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. |
| [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. Gao, M. Shi, T. 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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