# American Institue of Mathematical Sciences

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

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

 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.

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, 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, 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, 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, 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, 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, P. Solé, On constacyclic codes over $\frac{\mathbb{Z}_{4}[u]}{}$, Finite Fields Appl, 45 (2017), 86-95. doi: 10.1016/j.ffa.2016.11.016. [11] M. Shi, P. Solé, 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, 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, S. Yang, A class of optimal $p$-ary codes from one-weight codes over $\frac{\mathbb{F}_{p}[u]}{}$, 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, 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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##### References:
 [1] M. H. Chiu, S. T. Yau, 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, 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, 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, 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, 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, P. Solé, On constacyclic codes over $\frac{\mathbb{Z}_{4}[u]}{}$, Finite Fields Appl, 45 (2017), 86-95. doi: 10.1016/j.ffa.2016.11.016. [11] M. Shi, P. Solé, 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, 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, S. Yang, A class of optimal $p$-ary codes from one-weight codes over $\frac{\mathbb{F}_{p}[u]}{}$, 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, 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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