Codes from unit groups of division algebras over number fields
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Mathematische Zeitschrift
Codes from unit groups of division algebras over number fields Christian Maire1 · Aurel Page2 Received: 12 December 2019 / Accepted: 27 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Number field codes · Division algebras over number fields · Asymptotically good codes Mathematics Subject Classification 94B40 · 11R52 · 11T71 · 94B75
1 Introduction Number field codes, introduced by Lenstra [9] and independently rediscovered by Guruswami [5], are number field analogues of the geometric codes of Goppa [4] built from curves over
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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Christian Maire [email protected] Aurel Page [email protected]
1
Institut FEMTO-ST, Univ. Bourgogne Franche-Comté, 15B Av. des Montboucons, 25030 Besançon cédex, France
2
INRIA, Univ. Bordeaux, CNRS, IMB, UMR 5251, 33400 Talence, France
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C. Maire, A. Page
finite fields. In these original constructions, the codes are constructed from the ring of integers of a number field. In [11], the first author and Oggier extended these ideas: they explained how to construct codes from any arithmetic group, and they analysed the parameters of the resulting codes in the case of the unit group of the ring of integers of a number field, and in the case of an order in a division algebra. In every case, it was possible to obtain asymptotically good families of codes using towers of number fields with bounded root discriminant. The multiplicative group of an order in a quaternion algebra was also considered with a partial analysis and the question of constructing asymptotically good codes from these groups was left open, short of having adapted techniques, as explained in Remark 12 of [11]. We completely analyse the noncommutative unit group codes in any degree, by using a metric that is adapted to those groups. We also measure the minimal distance
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