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

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  • 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.

    Mathematics Subject Classification: Primary: 94B05, 94B15.


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  • [1] M. H. ChiuS. 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. DoughertyJ. 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. KayaB. 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. KayaB. 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. ShiQ. LiqinL. SokN. 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. ShiP. 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. ShiL. 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. ShiS. 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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