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November  2017, 11(4): 791-804. doi: 10.3934/amc.2017058

Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$

Karim Samei and  Arezoo Soufi

Department of Mathematics, Bu Ali Sina University, Hamedan, Iran

Received  July 2016 Published  November 2017

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.

Keywords: Quadratic residue codes, Euclidean self-dual codes, Hermitian self-dual codes, Gray map.
Mathematics Subject Classification: Primary: 94B05, 94B15.
Citation: Karim Samei, Arezoo Soufi. Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$. Advances in Mathematics of Communications, 2017, 11 (4) : 791-804. doi: 10.3934/amc.2017058
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.  Google Scholar

[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.  Google Scholar

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M. Grassl, http://codetables.de, accessed on 04.11.2012. Google Scholar

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W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes Cambrigde University press, 2003. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[8]

K. Samei and A. Soufi, Constacyclic codes over finite principal ideal rings, Submitted. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. Grassl, http://codetables.de, accessed on 04.11.2012. Google Scholar

[4]

W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes Cambrigde University press, 2003. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

K. Samei and A. Soufi, Constacyclic codes over finite principal ideal rings, Submitted. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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