Let be a monic polynomial over the finite field and let be a set of elements of such that for . The classical Goppa code with Goppa polynomial is the set of all words in for which . Here is to be interpreted as where is the unique polynomial of degree degree such that .
The basic idea of this construction can be generalized as follows. Let be a non-singular projective curve (in the sense of algebraic geometry) defined over the finite field . Let be a set of rational points of . Let be the divisor . Let be another divisor with support disjoint from . Let be the linear system associated to the divisor , i.e. , where is the set of non-zero rational functions on and is the divisor of zeros and poles defined by an . The geometric Goppa code of length associated to the pair is the image of the linear mapping defined by . If the supports of and are not disjoint, there is still a code associated to the pair , but not a canonical one; however, all the codes so obtained are equivalent in a suitable sense. A second code associated to the pair is the following. Let be the vector space of rational differential forms on with , with the zero form added. The second linear code associated to the pair is the image of the linear mapping defined by . This is the construction which more directly generalizes the classical Goppa codes mentioned above. The codes and are dual to each other. For fixed and varying the codes and yield the same family.
At present (1989) one approach to finding good codes is to find curves with large numbers of rational points compared to their genus. This brings in advanced algebraic geometry. Using Shimura curves (modular curves) one now can find a sequence of codes which beats the Gilbert–Varshamov bound for . For more details cf. [a4], [a7]–[a9].
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|[a3]||R.J. McEliece, E.R. Rodemich, H. Rumsey, L.R. Welch, "New upper bounds on the rate of a code via the Delsarte–MacWilliams inequalities" IEEE Trans. Inform. Theory , 23 (1977) pp. 157–166 MR439403|
|[a4]||M.A. Tsfasman, S.G. Vladuts, T. Zink, "Modular curves, Shimura curves and Goppa codes, better than Varshamov–Gilbert bound" Math. Nachr. , 109 (1982) pp. 21–28 MR0705893 Zbl 0574.94013|
|[a5]||P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint) MR0893813 Zbl 0611.10017|
|[a6]||R. Hill, "A first course in coding theory" , Clarendon Press (1986) MR0853914 Zbl 0616.94006|
|[a7]||J.H. van Lint, G. van der Geer, "Introduction to coding theory and algebraic geometry" , Birkhäuser (1988) Zbl 0648.94011 Zbl 0639.00048|
|[a8]||V.D. Goppa, "Geometry and codes" , Kluwer (1988) MR1029027 Zbl 1097.14502|
|[a9]||M.A. Tsfasman, S.G. Vlăduts, "Algebraic geometric codes" , Kluwer (1989) MR1003354 Zbl 0674.94012|
Goppa code. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Goppa_code&oldid=34180