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Quasi-perfect linear codes from planar and APN functions

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Abstract

Let p be a prime and m be a positive integer with m ≥ 3. Let f be a mapping from \(\mathbb {F}_{p^{m}}\) to itself and \(\mathcal {C}_{f}\) be the linear code of length p m − 1, whose parity-check matrix has its j-th column \({\left [\begin {array}{c} \pi ^{j}\\ f(\pi ^{j}) \end {array} \right ]}\), where π is a primitive element in \(\mathbb {F}_{p^{m}}\) and j = 0, 1, ⋯ , p m − 2. In the case of p = 2, it is proved that \(\mathcal {C}_{f}\) has covering radius 3 when f(x) is a quadratic APN function. This gives a number of binary quasi-perfect codes with minimum distance 5. In the case that p is an odd prime, we show that for all known planar functions f(x), the covering radius of \(\mathcal {C}_{f}\) is equal to 2 if m is odd and 3 if m is even. Consequently, several classes of p-ary quasi-perfect codes are derived.

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Authors and Affiliations

  1. Department of Electrical Engineering & Computer Science, University of Stavanger, 4036, Stavanger, Norway

    Chunlei Li

  2. Department of Informatics, University of Bergen, 5008, Bergen, Norway

    Tor Helleseth

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  1. Chunlei Li
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  2. Tor Helleseth
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Correspondence to Chunlei Li.

Additional information

Part of the results in this paper have been presented at the 7th International Workshop on Optimal Codes and Related Topics, OCRT 2013.

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This study was funded by the Norwegian Research Council.

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The work of both authors is supported by the Norwegian Research Council and the authors declare that they have no conflict of interest.

1.1 Research involving human participants and/or animals and informed consent

This is not applicable for the research in this manuscript.

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Li, C., Helleseth, T. Quasi-perfect linear codes from planar and APN functions. Cryptogr. Commun. 8, 215–227 (2016). https://doi.org/10.1007/s12095-015-0132-y

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  • DOI: https://doi.org/10.1007/s12095-015-0132-y

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