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Suzuki-invariant codes from the Suzuki curve

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Abstract

In this paper we consider the Suzuki curve \(y^q + y = x^{q_0}(x^q + x)\) over the field with \(q = 2^{2m+1}\) elements. The automorphism group of this curve is known to be the Suzuki group \(\mathrm{{Sz}}(q)\) with \(q^2(q-1)(q^2+1)\) elements. We construct AG codes over \(\mathbb {F}_{q^4}\) from an \(\mathrm{{Sz}}(q)\)-invariant divisor D, giving an explicit basis for the Riemann–Roch space \(L(\ell D)\) for \(0 < \ell \le q^2-1\). The full Suzuki group \(\mathrm{{Sz}}(q)\) acts faithfully on each code. These families of codes have very good parameters and information rate close to 1. In addition, they are explicitly constructed. The dual codes of these families are of the same kind if \(2g-1 \le \ell \le q^2-1\).

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Acknowledgments

The authors would like to thank Rachel Pries for organizing the workshop on rational points on Suzuki Curves in which this paper was conceived. We also would like to thank the two anonymous reviewers for the very useful comments and suggestions which helped us improve the quality of our paper. This work was conducted at the Mathematics Department, Colorado State University, Summer 2011, funded by NSF grant DMS-11-01712.

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Authors and Affiliations

  1. Department of Mathematics, University of Bahrain, Sakhir Campus, Bahrain

    Abdulla Eid

  2. Department of Mathematics, Stanford University, Palo Alto, CA, 94305, USA

    Hilaf Hasson

  3. Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402, USA

    Amy Ksir

  4. Arlington, VA, USA

    Justin Peachey

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  1. Abdulla Eid
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  2. Hilaf Hasson
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  3. Amy Ksir
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Correspondence to Hilaf Hasson.

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Communicated by P. Charpin.

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Eid, A., Hasson, H., Ksir, A. et al. Suzuki-invariant codes from the Suzuki curve. Des. Codes Cryptogr. 81, 413–425 (2016). https://doi.org/10.1007/s10623-015-0164-5

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  • DOI: https://doi.org/10.1007/s10623-015-0164-5

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