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Research on the Problem of Sphere Packings and the Packing on the Surface of a Sphere
Author: XuLanJu
Tutor: FengKeQin
School: Tsinghua University
Course: Mathematics
Keywords: sphere packings errorcorrecting codes spherical codes algebraicgeometry codes superballs
CLC: O157.4
Type: PhD thesis
Year: 2006
Downloads: 83
Quote: 1
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Abstract
The problems of packing spheres in Euclidean space and of packing points on the surface of a sphere have been studied for several centuries. Up to now it is still unsolved. In this dissertation, we investigate the two problems. In more detail, we get the following results:With the help from good errorcorrecting codes, we make use of prime ideals over an imaginary quadratic number field to give the new constructions of sphere packings. In this way, we construct the dense sphere packings in dimensions n ≤ 62, some of which meet the bestknown densities. Particularly, we give the better densities than all previous ones for dimensions 59 and 61. In some higher dimensions, a number of other interesting results can also be given from our constructions.The problem of packing points on the surface of a sphere is equivalent to the study of spherical codes. N.J.A. Sloane has given a method for constructing spherical codes from binary codes. We introduce another method of converting ternary codes into spherical codes. By employing algebraicgeometry codes, we give an asymptotic lower bound of spherical code sequences, constructed in polynomial time. By making use of the idea involved in the proof of the GilbertVarshamov bound in coding theory, we construct a spherical code sequence in exponential time which achieves the bestknown asymptotic nonconstructive bound by E.A. Shamsiev and A.D. Wyner.We also discuss the generalization of sphere packings and kissing numbers in the case of superballs. First we give a propagation rule using a packing over superballs and codes. By good codes we improve asymptotic lower bounds given by J.A. Rush and N.J.A. Sloane. We also derive two GilbertVarshamov type bounds for classical sphere packings and by numerical computation we improve the previously bestknown densities for dimensions 512 — 1048584. Then we investigate the asymptotic quantity of the translative kissing number of a superball. We derive a GilbertVarshamov type lower bound and also give two lower bounds from binary codes in exponential time and polynomial time, one from binary code achieving the GilbertVarshamov bound and the other from algebraicgeometry codes. All the three bounds improve the bound givenby D.G. Larman and C. Zong.

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