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Introduction to coding theory and algebraic geometry. I: Coding theory. (English) Zbl 0648.94011
Coding theory and algebraic geometry, Lect. Semin., Düsseldorf/FRG 1987, DMV Semin. 12, 9-33 (1988).
[For the entire collection see Zbl 0639.00048.]
An exciting new development in algebraic coding theory is concerned with the application of methods from algebraic geometry to the construction of linear block codes. The principal achievement so far of this line of research is the construction of families of linear block codes that meet or even go beyond the classical Gilbert-Varshamov bound. In 1987 the Deutsche Mathematiker-Vereinigung organized a seminar on this topic which featured two series of lectures, one on the coding theory background and the other on the algebraic geometry background. The paper under review is based on the first series of lectures. It contains a masterful exposition of the theory of linear block codes which requires only minimal algebraic prerequisites. Besides basic concepts, the article covers BCH codes, classical Goppa codes, bounds on codes such as the Gilbert-Varshamov bound and the Plotkin bound, self-dual codes, and some examples of codes obtained from algebraic curves. The elegant, succinct style allows the presentation of a lot of material in a relatively short space.
Reviewer: H.Niederreiter

94B05 Linear codes, general
94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory
14H25 Arithmetic ground fields for curves