Algebraic Geometric Codes
The most important development in the theory of error-correcting codes in recent years is the introduction of methods from algebraic geometry to construct good codes. The ideas are based on generalizations of so-called Goppa codes. The (by now) “classical” Goppa codes (1970, cf.) were already a great improvement on codes known at that time. The algebraic geometric codes were also inspired by ideas of Goppa but the most sensational development was a paper by Tsfasman, Vlădut and Zink (1982,cf.). In this paper the idea of codes from algebraic curves was combined with certain recent deep results from algebraic geometry to produce a sequence of error-correcting codes that led to a new lower bound on the information rate of good codes that is better than the Gilbert-Varshamov bound. The novice reader should realize that the G-V-bound (1952) was never improved (until 1982) and believed by many to be best possible. Actually, the improvement is only achieved for alphabets of size at least 49 and several binary coding experts still have hope that no improvement of the G-V-bound for F2 will be possible; (this author is not one of them).
KeywordsRational Point Linear Code Algebraic Curf Cyclic Code Projective Curve
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