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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 error-correcting codes spherical codes algebraic-geometry 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 error-correcting 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 best-known 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 algebraic-geometry 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 Gilbert-Varshamov bound in coding theory, we construct a spherical code sequence in exponential time which achieves the best-known 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 Gilbert-Varshamov type bounds for classical sphere packings and by numerical computation we improve the previously best-known densities for dimensions 512 — 1048584. Then we investigate the asymptotic quantity of the translative kissing number of a superball. We derive a Gilbert-Varshamov type lower bound and also give two lower bounds from binary codes in exponential time and polynomial time, one from binary code achieving the Gilbert-Varshamov bound and the other from algebraic-geometry codes. All the three bounds improve the bound givenby D.G. Larman and C. Zong.

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Combinatorics ( combinatorics ) > Coding theory ( on behalf of the digital theory )
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