Quadratic residue codes over <inline-formula><tex-math id="M1">\begin{document} \mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r} \end{document}</tex-math></inline-formula>

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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.
[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.

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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