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Structure of \({\mathbb {F}}_q{\mathcal {R}}\)-linear \((\varTheta ,\varDelta _\varTheta )\)-cyclic codes

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Abstract

Let \({\mathbb {F}}_q\) be the finite field of order \(q=p^m\) where p is a prime, and m is a positive integer. This work introduces \({\mathbb {F}}_q{\mathcal {R}}\)-linear \((\varTheta ,\varDelta _\varTheta )\)-cyclic codes where \({\mathcal {R}}={\mathbb {F}}_q+u{\mathbb {F}}_q\) with \(u^2=u\), i.e., \((\varTheta ,\varDelta _\varTheta )\)-cyclic codes over \({\mathbb {F}}_q{\mathcal {R}}\). With the help of the decomposition method, we study the structural properties and determine the generator polynomials of \({\mathbb {F}}_q{\mathcal {R}}\)-linear \((\varTheta ,\varDelta _\varTheta )\)-cyclic codes. Further, we define the Gray map over \({\mathbb {F}}_q{\mathcal {R}}\) and find the Gray images of \({\mathbb {F}}_q{\mathcal {R}}\)-linear \((\varTheta ,\varDelta _\varTheta )\)-cyclic codes over \({\mathbb {F}}_q\). Finally, with the help of our established results, we have constructed some new codes corresponding to \({\mathbb {F}}_q{\mathcal {R}}\)-linear \((\varTheta ,\varDelta _\varTheta )\)-cyclic codes.

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The authors declare that [the/all other] data supporting the findings of this study are available within the article. Any clarification may be requested from the corresponding author.

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Acknowledgements

The first and second authors are thankful to the Department of Science and Technology (DST) and the Council of Scientific and Industrial Research (CSIR), Govt. of India for financial support under Ref No. DST/INSPIRE/03/2016/ 001445 and Grant No. 09/1023(0027)/2019-EMR-I, respectively. The authors are thankful to the Indian Institute of Technology Patna for providing the research facilities.

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Authors and Affiliations

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

    Shikha Patel, Ashutosh Singh & Om Prakash

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  1. Shikha Patel
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  2. Ashutosh Singh
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  3. Om Prakash
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Correspondence to Om Prakash.

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Communicated by Gaojun Luo.

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Patel, S., Singh, A. & Prakash, O. Structure of \({\mathbb {F}}_q{\mathcal {R}}\)-linear \((\varTheta ,\varDelta _\varTheta )\)-cyclic codes. Comp. Appl. Math. 43, 151 (2024). https://doi.org/10.1007/s40314-024-02631-8

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  • DOI: https://doi.org/10.1007/s40314-024-02631-8

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