By Henning Stichtenoth, Michael A. Tsfasman

About ten years in the past, V.D. Goppa came across a stunning connection among the idea of algebraic curves over a finite box and error-correcting codes. the purpose of the assembly "Algebraic Geometry and Coding thought" was once to provide a survey at the current country of study during this box and similar issues. The complaints comprise study papers on numerous points of the speculation, between them: Codes constituted of targeted curves and from higher-dimensional kinds, deciphering of algebraic geometric codes, hint codes, Exponen- tial sums, speedy multiplication in finite fields, Asymptotic variety of issues on algebraic curves, Sphere packings.

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Extra info for Coding Theory and Algebraic Geometry: Proceedings of the International Workshop held in Luminy, France, June 17–21, 1991

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References [1] M. Eck and D. Lasser, B-Spline-Bezier Representation of Geometric Spline Curves. Quartzes and Quintics, Preprint 1254, Fachbereich Mathematik, Technische Hochschule Darmstadt, Darmstadt, 1989. [2] T. A. Foley, Local control of interval tension using weighted splines, Comput. Aided Geom. , 3 (1986), pp. 281-294. [3] , Interpolation with interval and point tension controls using cubic weighted v-sphnes, ACM Trans. Math. Software, 13 (1987), pp. 68-96. [4] H. Hagen, Geometric spline curves, Comput, Aided Geom.

2. 19) is the transition condition at Pt, i — 1 , - - - , 7 V — 1, which represents the fact that 9* 6 C ! ([0,/]). 21). 20) is replaced by Minimal-Energy Splines 31 or respectively. The following theorem describes one relation between minimal-energy splines and minimal-energy spline segments, the proof of which is easy and is omitted. 2. ,TV-1. 4. , the angles at both end points PI and PI are prescribed. For this case, the constraint set: Let TV be a positive integer. 1, we have the following statements concerning the approximate problem.

If the scale invariance property is used to normalize the given data, the parameters of the scaled problem can be obtained by scaling the parameters of the corresponding normalized problem. 1. 31) can be constructed using the relations: Proof. 33) for the curvature functions of y and x. 1). 8. The Shape of Minimal-Energy Splines The behaviour of minimal-energy curves is guided by the principle of avoiding regions with extreme curvature because large curvature results in a large energy value. This principle creates round-looking shapes (see Figs.

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