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Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$
Department of Mathematics, Bu Ali Sina University, Hamedan, Iran |
The purpose of this paper is to study the structure of quadratic residue codes over the ring $R=\mathbb{F}_{p^r}+u_1\mathbb{F}_{p^r}+u_2 \mathbb{F}_{p^r}+...+u_t \mathbb{F}_{p^r}$, where $r, t ≥ 1$ and $p$ is a prime number. First, we survey known results on quadratic residue codes over the field $\mathbb{F}_{p^r}$ and give general properties with quadratic residue codes over $R$. We introduce the Gray map from $R$ to $\mathbb{F}^{t+1}_{p^r}$ and study more details about the quadratic residue codes over the ring $R$ for $p=2, 3$. Finally, we obtain a number of Hermitian self-dual codes over $R$ in the following two cases, where $t$ is an odd number; the first case, when $p=2$ and $r$ is an even number or $r=1$, the second case, when $p=3$ and $r$ is an even number.
References:
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M. H. Chiu, S. T. Yau and Y. Yu,
$\mathbb{Z}_8$-cyclic codes and quadratic residue codes, Adv. Appl. Math, 25 (2000), 12-33.
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[2] |
S. T. Dougherty, J. L. Kim and H. Kulosman,
MDS codes over finite principal ideal rings, Des. Codes Cryptogr, 50 (2009), 77-92.
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W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes Cambrigde University press, 2003. |
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Quadratic residue codes over $\mathbb{F}_p+v \mathbb{F}_p$ and their Gray images, J. Pure App. Algebra, 218 (2014), 1999-2011.
doi: 10.1016/j.jpaa.2014.03.002. |
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A. Kaya, B. Yildiz and I. Siap,
New extremal binary self-dual codes of length 68 from quadratic residue codes over $\mathbb{F}_2+u \mathbb{F}_2+u^2 \mathbb{F}_2$, Finite Fields Appl, 29 (2014), 160-177.
doi: 10.1016/j.ffa.2014.04.009. |
[7] |
V. Pless and Z. Qian,
Cyclic codes and quadratic residue codes over $Z_4$, IEEE Trans. Inform. Theory, 42 (1996), 1594-1600.
doi: 10.1109/18.532906. |
[8] |
K. Samei and A. Soufi, Constacyclic codes over finite principal ideal rings, Submitted. |
[9] |
K. Samei and A. Soufi, Cyclic codes over $ \mathbb{F}_{2^r} + { u_1} \mathbb{F}_{2^r} +{u_2} \mathbb{F}_{2^r} + . . . +{u_t}\mathbb{F}_ {2^r} $, Submitted. |
[10] |
M. Shi, Q. Liqin, L. Sok, N. Aydin and P. Solé,
On constacyclic codes over $ \frac{\mathbb{Z}_{4}[u]}{<u^2-1>}$, Finite Fields Appl, 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[11] |
M. Shi, P. Solé and B. Wu,
Cyclic codes and the weight enumerators over $ \mathbb{F}_{2} + { v} \mathbb{F}_{2} +{v^2} \mathbb{F}_{2}$, Appl. Comput. Math, 12 (2013), 247-255.
|
[12] |
M. Shi, L. Xu and G. Yang,
A note on one weight and two weight projective $ \mathbb{Z}_{4}$-codes, IEEE Trans. Inform. Theory, 63 (2017), 177-182.
doi: 10.1109/TIT.2016.2628408. |
[13] |
M. Shi, S. Zhu and S. Yang,
A class of optimal $p$-ary codes from one-weight codes over $ \frac{\mathbb{F}_{p}[u]}{<u^m>}$, J. Franklin Inst, 350 (2013), 929-937.
doi: 10.1016/j.jfranklin.2012.05.014. |
[14] |
B. Taeri,
Quadratic residue codes over $Z_9$, J. Korean Math. Soc., 46 (2009), 13-30.
doi: 10.4134/JKMS.2009.46.1.013. |
[15] |
S. X. Zhu and L. Wang,
A class of constacyclic codes over $\mathbb{F}_p+v \mathbb{F}_p$ and its Gray image, Discrete Math., 311 (2011), 2677-2682.
doi: 10.1016/j.disc.2011.08.015. |
show all references
References:
[1] |
M. H. Chiu, S. T. Yau and Y. Yu,
$\mathbb{Z}_8$-cyclic codes and quadratic residue codes, Adv. Appl. Math, 25 (2000), 12-33.
doi: 10.1006/aama.2000.0687. |
[2] |
S. T. Dougherty, J. L. Kim and H. Kulosman,
MDS codes over finite principal ideal rings, Des. Codes Cryptogr, 50 (2009), 77-92.
doi: 10.1007/s10623-008-9215-5. |
[3] |
M. Grassl, http://codetables.de, accessed on 04.11.2012. |
[4] |
W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes Cambrigde University press, 2003. |
[5] |
A. Kaya, B. Yildiz and I. Siap,
Quadratic residue codes over $\mathbb{F}_p+v \mathbb{F}_p$ and their Gray images, J. Pure App. Algebra, 218 (2014), 1999-2011.
doi: 10.1016/j.jpaa.2014.03.002. |
[6] |
A. Kaya, B. Yildiz and I. Siap,
New extremal binary self-dual codes of length 68 from quadratic residue codes over $\mathbb{F}_2+u \mathbb{F}_2+u^2 \mathbb{F}_2$, Finite Fields Appl, 29 (2014), 160-177.
doi: 10.1016/j.ffa.2014.04.009. |
[7] |
V. Pless and Z. Qian,
Cyclic codes and quadratic residue codes over $Z_4$, IEEE Trans. Inform. Theory, 42 (1996), 1594-1600.
doi: 10.1109/18.532906. |
[8] |
K. Samei and A. Soufi, Constacyclic codes over finite principal ideal rings, Submitted. |
[9] |
K. Samei and A. Soufi, Cyclic codes over $ \mathbb{F}_{2^r} + { u_1} \mathbb{F}_{2^r} +{u_2} \mathbb{F}_{2^r} + . . . +{u_t}\mathbb{F}_ {2^r} $, Submitted. |
[10] |
M. Shi, Q. Liqin, L. Sok, N. Aydin and P. Solé,
On constacyclic codes over $ \frac{\mathbb{Z}_{4}[u]}{<u^2-1>}$, Finite Fields Appl, 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[11] |
M. Shi, P. Solé and B. Wu,
Cyclic codes and the weight enumerators over $ \mathbb{F}_{2} + { v} \mathbb{F}_{2} +{v^2} \mathbb{F}_{2}$, Appl. Comput. Math, 12 (2013), 247-255.
|
[12] |
M. Shi, L. Xu and G. Yang,
A note on one weight and two weight projective $ \mathbb{Z}_{4}$-codes, IEEE Trans. Inform. Theory, 63 (2017), 177-182.
doi: 10.1109/TIT.2016.2628408. |
[13] |
M. Shi, S. Zhu and S. Yang,
A class of optimal $p$-ary codes from one-weight codes over $ \frac{\mathbb{F}_{p}[u]}{<u^m>}$, J. Franklin Inst, 350 (2013), 929-937.
doi: 10.1016/j.jfranklin.2012.05.014. |
[14] |
B. Taeri,
Quadratic residue codes over $Z_9$, J. Korean Math. Soc., 46 (2009), 13-30.
doi: 10.4134/JKMS.2009.46.1.013. |
[15] |
S. X. Zhu and L. Wang,
A class of constacyclic codes over $\mathbb{F}_p+v \mathbb{F}_p$ and its Gray image, Discrete Math., 311 (2011), 2677-2682.
doi: 10.1016/j.disc.2011.08.015. |
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