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Codes from unit groups of division algebras over number fields

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Abstract

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. We prove that the noncommutative unit group construction yields asymptotically good families of codes for the sum-rank metric from division algebras of any degree, and we estimate the size of the alphabet in terms of the degree.

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Notes

  1. A factor \(2^{-r_2d^2}\) is missing in [11, Proposition 9], but this does not affect the results of the paper.

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Authors and Affiliations

  1. Institut FEMTO-ST, Univ. Bourgogne Franche-Comté, 15B Av. des Montboucons, 25030, Besançon cédex, France

    Christian Maire

  2. INRIA, Univ. Bordeaux, CNRS, IMB, UMR 5251, 33400, Talence, France

    Aurel Page

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  1. Christian Maire
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  2. Aurel Page
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Correspondence to Christian Maire.

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The authors would like to thank the Warwick Mathematics Department and the Department of Mathematics at Cornell University for providing a stimulating research atmosphere. We also thank Xavier Caruso for suggesting the use of the sum-rank distance, and an anonymous referee for their valuable feedback. We also thank the EPSRC for financial support via the EPSRC Programme Grant EP/K034383/1 LMF: L-functions and modular forms. CM was also partially supported by the Region Bourgogne Franche-Comté, the ANR project FLAIR (ANR-17-CE40-0012) and the EIPHI Graduate School (ANR-17-EURE-0002).

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Maire, C., Page, A. Codes from unit groups of division algebras over number fields. Math. Z. 298, 327–348 (2021). https://doi.org/10.1007/s00209-020-02614-5

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  • DOI: https://doi.org/10.1007/s00209-020-02614-5

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