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On the Subfield Codes of a Subclass of Optimal Cyclic Codes and Their Covering Structures

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LATIN 2022: Theoretical Informatics (LATIN 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13568))

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Abstract

A class of optimal three-weight \([q^k-1,k+1,q^{k-1}(q-1)-1]\) cyclic codes over , with \(k\ge 2\), achieving the Griesmer lower bound, was presented by Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501–4513]. In this paper we study some of the subfield codes of this class of optimal cyclic codes when \(k=2\). The weight distributions of the subfield codes are settled. It turns out that some of these codes are optimal and others have the best known parameters. The duals of the subfield codes are also investigated and found to be almost optimal with respect to the sphere-packing bound. In addition, the covering structure for the studied subfield codes is determined. Some of these codes are found to have the important property that any nonzero codeword is minimal. This is a desirable property which is useful in the design of a secret sharing scheme based on a linear code. Moreover, we present a specific example of a secret sharing scheme based on one of these subfield codes.

F. Hernández - PhD student, manuscript partially supported by CONACyT, México.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped to improve the quality of the paper.

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Authors and Affiliations

  1. Posgrado en Ciencia e Ingeniería de la Computación, Universidad Nacional Autónoma de México, 04510, Ciudad de México, Mexico

    Félix Hernández

  2. Dirección General de Cómputo y de Tecnologías de Información y Comunicación, Universidad Nacional Autónoma de México, 04510, Ciudad de México, Mexico

    Gerardo Vega

Authors
  1. Félix Hernández
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  2. Gerardo Vega
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Correspondence to Félix Hernández .

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Editors and Affiliations

  1. Universidad Nacional Autónoma de México, Mexico City, Mexico

    Armando Castañeda

  2. Centro de investigación y de Estudios Avanzados, Mexico City, Mexico

    Francisco Rodríguez-Henríquez

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Hernández, F., Vega, G. (2022). On the Subfield Codes of a Subclass of Optimal Cyclic Codes and Their Covering Structures. In: Castañeda, A., Rodríguez-Henríquez, F. (eds) LATIN 2022: Theoretical Informatics. LATIN 2022. Lecture Notes in Computer Science, vol 13568. Springer, Cham. https://doi.org/10.1007/978-3-031-20624-5_16

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  • DOI: https://doi.org/10.1007/978-3-031-20624-5_16

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