tailieunhanh - Đề tài " New upper bounds on sphere packings I "

We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24. Contents 1. Introduction 2. Lattices, Fourier transforms, and Poisson summation 3. Principal theorems 4. Homogeneous spaces 5. Conditions for a sharp bound 6. Stationary points 7. Numerical results 8. Uniqueness Appendix A. . | Annals of Mathematics New upper bounds on sphere packings I By Henry Cohn and Noam Elkies Annals of Mathematics 157 2003 689 714 New upper bounds on sphere packings I By HENRy Cohn and Noam Elkies Abstract We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes and use it to prove upper bounds for the density of sphere packings which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24. Contents 1. Introduction 2. Lattices Fourier transforms and Poisson summation 3. Principal theorems 4. Homogeneous spaces 5. Conditions for a sharp bound 6. Stationary points 7. Numerical results 8. Uniqueness Appendix A. Technicalities about density Appendix B. Other convex bodies Appendix C. Numerical data Acknowledgements References 1. Introduction The sphere packing problem asks for the densest packing of spheres into Euclidean space. More precisely what fraction of Rra can be covered by congruent balls that do not intersect except along their boundaries This problem fits into a broad framework of packing problems including error-correcting codes Cohn was supported by an NSF Graduate Research Fellowship and by a summer internship at Lucent Technologies and currently holds an American Institute of Mathematics five-year fellowship. Elkies was supported in part by the Packard Foundation. 690 HENRY COHN AND NOAM ELKIES and spherical codes. Linear programming bounds D are the most powerful known technique for producing upper bounds in such problems. In particular KL uses this technique to prove the best bounds known for sphere packing density in high dimensions. However KL does not study sphere packing directly but rather passes through the intermediate problem of spherical codes. In this paper we develop linear programming bounds that apply directly to sphere packing and study these bounds numerically to prove the best bounds

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