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An elementary approach to algebraic geometry codes. (English) Zbl 0978.94046

Authors’ introduction: In this paper (talk given by the second author at the 29th Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, 1998) we give a short exposition of a recent elementary approach to algebraic geometry codes. The theory is treated extensively in the chapter on algebraic geometry codes in the Handbook of Coding Theory, V. S. Pless, W. C. Huffman and R. A. Brualdi (eds.), 871-961 (1998; Zbl 0922.94015). For a long list of references and the historical background, we refer the reader to that chapter. To introduce the topic, we first give a brief summary of what we call the classical approach to these codes. An intermediate step uses Bézout’s theorem, which does not require all the machinery of algebraic geometry. The main part of the paper is based on the concept of a weight function on an algebra.
Chapter headings: Reed-Solomon codes, Algebraic geometry codes; Bézout’s theorem; Order, degree and weight functions; Evaluation codes; The order bounds.

MSC:

94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
94-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory

Citations:

Zbl 0922.94015
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