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Constructions of Codes from Number Fields

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2227))

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Abstract

We define number-theoretic error-correcting codes based on algebraic number fields,thereby providing a generalization of Chinese Remainder Codes akin to the generalization of Reed-Solomon codes to Algebraic-geometric codes. Our construction is very similar to (and in fact less general than) the one given by Lenstra [8],b ut the parallel with the function field case is more apparent,since we only use the non-archimedean places for the encoding. We prove that over an alphabet size as small as 19,there even exist asymptotically good number field codes of the type we consider. This result is based on the existence of certain number fields that have an infinite class field tower in which some primes of small norm split completely.

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Author information

Authors and Affiliations

  1. MIT Laboratory for Computer Science200 Technology Square, 01239, Cambridge, MA

    Venkatesan Guruswami

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  1. Venkatesan Guruswami
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Editor information

Editors and Affiliations

  1. Department of Mathematics, RMIT University, GPO Box 2476V, 3001, Melbourne, Australia

    Serdar Boztaş

  2. Department of Computing, Macquarie University, 2109, NSW, Australia

    Igor E. Shparlinski

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Guruswami, V. (2001). Constructions of Codes from Number Fields. In: Boztaş, S., Shparlinski, I.E. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2001. Lecture Notes in Computer Science, vol 2227. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45624-4_14

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  • DOI: https://doi.org/10.1007/3-540-45624-4_14

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42911-1

  • Online ISBN: 978-3-540-45624-7

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