Good integers and some applications in coding theory

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Abstract

A class of good integers has been introduced by P. Moree in 1997 together with the characterization of good odd integers. Such integers have shown to have nice number theoretical properties and wide applications. In this paper, a complete characterization of all good integers is given. Two subclasses of good integers are introduced, namely, oddly-good and evenly-good integers. The characterization and properties of good integers in these two subclasses are determined. As applications, good integers and oddly-good integers are applied in the study of the hulls of abelian codes. The average dimension of the hulls of abelian codes is given together with some upper and lower bounds.

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  • 18 May 2018

    Proposition 2.1 and Proposition 2.3 in the original publication are incorrectly worded and they should be as follows.

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Acknowledgements

The author would like to thank the anonymous referees for their helpful comments and suggestions.

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Correspondence to Somphong Jitman.

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This research was supported by the Thailand Research Fund and the Office of Higher Education Commission of Thailand under Research Grant MRG6080012.

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Jitman, S. Good integers and some applications in coding theory. Cryptogr. Commun. 10, 685–704 (2018). https://doi.org/10.1007/s12095-017-0255-4

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Keywords

  • Good integers
  • Abelian codes
  • Hulls
  • Euclidean inner product
  • Hermitian inner product

Mathematics Subject Classification (2010)

  • 94B15
  • 94B60
  • 11N25