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Two and three weight codes over \(\mathbb {F}_{p}+u\mathbb {F}_{p}\)

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Abstract

We construct an infinite family of three-Lee-weight codes of dimension 2m, where m is singly-even, over the ring \(\mathbb {F}_{p}+u\mathbb {F}_{p}\) with u 2=0. These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain an infinite family of abelian p-ary three-weight codes. When m is odd, and p≡3 (mod 4), we obtain an infinite family of two-weight codes which meets the Griesmer bound with equality. An application to secret sharing schemes is given.

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Acknowledgments

The authors are grateful to the reviewers and the editor for their helpful comments that improved the presentation and quality of this paper. This research is supported by NNSF of China (61672036), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133), the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11) and Key Projects of Support Program for outstanding young talents in Colleges and Universities (gxyqZD2016008).

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Authors and Affiliations

  1. Key Laboratory of Intelligent Computing, Signal Processing, Ministry of Education, Anhui University No. 3 Feixi Road, Hefei, Anhui Province, 230039, People’s Republic of China

    Minjia Shi

  2. National Mobile Communications Research Laboratory, Southeast University, Nanjing, 210096, People’s Republic of China

    Minjia Shi

  3. School of Mathematical Sciences, Anhui University, Hefei, Anhui Province, 230601, People’s Republic of China

    Minjia Shi, Rongsheng Wu & Yan Liu

  4. CNRS/ LTCI, University Paris-Saclay, 75013, Paris, France

    Patrick Solé

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  1. Minjia Shi
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  2. Rongsheng Wu
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  3. Yan Liu
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  4. Patrick Solé
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Correspondence to Minjia Shi.

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Shi, M., Wu, R., Liu, Y. et al. Two and three weight codes over \(\mathbb {F}_{p}+u\mathbb {F}_{p}\) . Cryptogr. Commun. 9, 637–646 (2017). https://doi.org/10.1007/s12095-016-0206-5

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  • DOI: https://doi.org/10.1007/s12095-016-0206-5

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Mathematics Subject Classification (2010)