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Gröbner Bases, Coding, and Cryptography pp 107–120Cite as

Automorphisms and Encoding of AG and Order Domain Codes

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Abstract

We survey some encoding methods for AG codes, focusing primarily on one approach utilizing code automorphisms. If a linear code C over\(\mathbb{F}_{q}\)has a finite Abelian group H as a group of automorphisms, then C has the structure of a module over a polynomial ring ℘. This structure can be used to develop systematic encoding algorithms using Gröbner bases for modules. We illustrate these observations with several examples including geometric Goppa codes and codes from order domains.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, USA

    John B. Little

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  1. John B. Little
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Correspondence to John B. Little .

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Editors and Affiliations

  1. Boole Centre for Research in Informatics, University College Cork, Crosses Green 17 South Bank, Cork, Ireland

    Massimiliano Sala

  2. Faculty of Engineering Dept. Production Systems Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka Toyohashi, Aichi, Tempaku-cho, 441-8580, Japan

    Shojiro Sakata

  3. Dipto. Matematica, Università Genova, Via Dodecaneso, 35, 16146, Genova, Italy

    Teo Mora

  4. Dipto. Matematica, Università Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy

    Carlo Traverso

  5. LIP6, University of Paris VI, 104 avenue du Président Kennedy, 75016, Paris, France

    Ludovic Perret

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Little, J.B. (2009). Automorphisms and Encoding of AG and Order Domain Codes. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_7

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