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Quantum Codes from Repeated-Root Cyclic and Negacyclic Codes of Length 4ps Over \(\mathbb {F}_{p^{m}}\)

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Abstract

Let \(\mathbb {F}_{p^{m}}\) be a finite field with pm elements for an odd prime p and a positive integer m. In this paper, we aim to construct non-binary quantum codes from repeated root cyclic and negacyclic codes of length 4ps over \(\mathbb {F}_{p^{m}}\). To achieve this, first we discuss the structure of all maximum distance separable (MDS) cyclic and negacyclic codes of length 4ps over \(\mathbb {F}_{p^{m}}\) and their Euclidean and Hermitian duals. Further, we establish a necessary and sufficient condition for these codes to contain their duals. Finally, many new and quantum MDS codes are obtained with the help of CSS and Hermitian constructions from dual containing cyclic and negacyclic codes.

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Acknowledgements

The first author would like to thanks the Department of Science and Technology (DST), India for financial support under the grant no. MTR/2018/001250. The second author would like to thanks to the Council of Scientific & Industrial Research (CSIR), Govt. of India for financial support under grant no. 09/1023(0014)/2015-EMR-I. The authors would also like to thank the Editor and anonymous referee(s) for careful reading and constructive suggestions to improve the presentation of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, S. A. Jain P. G. College, Ambala, 134003, India

    Saroj Rani

  2. Department of Mathematics, Indian Institute of Technology, Patna, 801106, India

    Ram Krishna Verma & Om Prakash

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  1. Saroj Rani
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  2. Ram Krishna Verma
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  3. Om Prakash
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Correspondence to Saroj Rani.

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Rani, S., Verma, R.K. & Prakash, O. Quantum Codes from Repeated-Root Cyclic and Negacyclic Codes of Length 4ps Over \(\mathbb {F}_{p^{m}}\). Int J Theor Phys 60, 1299–1327 (2021). https://doi.org/10.1007/s10773-021-04757-5

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