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(f, σ, δ)-skew Polycyclic Codes and Their Applications to Quantum Codes

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Abstract

Let \(\mathbb {F}_{q}\) be the finite field of q elements where q = pm and p is a prime. This work presents quantum codes from (f, σ, δ)-skew polycyclic codes (SPC) over the ring \(\mathcal {R}=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,\) where σ is an automorphism and δ is a σ-derivation of \(\mathcal {R}\). First, the Euclidean dual containing properties for (f, σ, δ)-skew polycyclic codes of length n over \(\mathbb {F}_{q}\) are established. Then we discuss the necessary and sufficient conditions of (f, σ, δ)-skew polycyclic codes for containing their duals over \(\mathcal {R}\). To obtain quantum codes, a Gray map φ, which preserves orthogonality of linear codes from \(\mathcal {R}\) to \(\mathbb {F}_{q},\) is defined. Finally, we apply the CSS (Calderbank-Shor-Steane) construction on the Gray images of such (f, σ, δ)-skew polycyclic codes and obtain many better quantum codes compared to the best-known codes in the literature.

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Acknowledgements

The first and third authors are thankful to the Department of Science and Technology (DST), Govt. of India for financial support under Ref No. DST/INSPIRE/03/2016/001445 and CRG/2020 /005927, vide Diary No. SERB/F/6780/ 2020-2021 dated 31 December, 2020, respectively. Also, the authors would like to thank the anonymous referee(s) and the Editor for their valuable comments to improve the presentation of the paper.

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  1. Department of Mathematics, Indian Institute of Technology Patna, Patna, 801 106, India

    Shikha Patel, Habibul Islam & Om Prakash

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  1. Shikha Patel
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Correspondence to Om Prakash.

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Patel, S., Islam, H. & Prakash, O. (f, σ, δ)-skew Polycyclic Codes and Their Applications to Quantum Codes. Int J Theor Phys 61, 47 (2022). https://doi.org/10.1007/s10773-022-05035-8

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