Algebraic-geometry codes

The theory of error-correcting codes derived from curves in an algebraic geometry was initiated by the work of Goppa as generalizations of Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon (RS), and Goppa codes. The development of the theory has received intense consideration since that time and the pu...

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Bibliographic Details
Published in:IEEE transactions on information theory 1998-10, Vol.44 (6), p.2596-2618
Main Authors: Blake, I., Heegard, C., Hoholdt, T., Wei, V.
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Codes
Decoding
Error correction codes
Galois fields
Geometry
Laboratories
Modular construction
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Summary:The theory of error-correcting codes derived from curves in an algebraic geometry was initiated by the work of Goppa as generalizations of Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon (RS), and Goppa codes. The development of the theory has received intense consideration since that time and the purpose of the paper is to review this work. Elements of the theory of algebraic curves, at a level sufficient to understand the code constructions and decoding algorithms, are introduced. Code constructions from particular classes of curves, including the Klein quartic, elliptic, and hyperelliptic curves, and Hermitian curves, are presented. Decoding algorithms for these classes of codes, and others, are considered. The construction of classes of asymptotically good codes using modular curves is also discussed.
ISSN:0018-9448
1557-9654
DOI:10.1109/18.720550