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https://hdl.handle.net/2142/20964
Description
Title
Design and analysis of spherical codes
Author(s)
Hamkins, Jon
Issue Date
1996
Doctoral Committee Chair(s)
Vardy, Alexander
Department of Study
Electrical and Computer Engineering
Discipline
Electrical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Engineering, Electronics and Electrical
Language
eng
Abstract
"A spherical code is a finite set of points on the surface of a multidimensional unit radius sphere. This thesis gives two constructions for large spherical codes that may be used for channel coding and for source coding. The first construction ""wraps"" a finite subset of any sphere packing onto the unit sphere in one higher dimension. The second construction is similar to the recursive construction of laminated lattices. Both constructions result in codes that are asymptotically optimal with respect to minimum distance, and the first construction can be efficiently used as part of a vector quantizer for a memoryless Gaussian source. Both constructions are structured so that codepoints may be identified without having to store the entire codebook. For several different rates, the distortion performance of the proposed quantizer is better than previously published results of quantizers with equivalent complexities."
The construction techniques are motivated by the relationship between asymptotically large spherical codes and sphere packings in one lower dimension. It is shown that the asymptotically maximum density of a k-dimensional spherical code equals the maximum density of a sphere packing in $\IR\sp{k-1}$. Similar relationships hold for the quantization coefficient and covering thickness. Previously published upper and lower bounds on the size of spherical codes of given minimum distances are analyzed and shown to be loose for asymptotically small minimum distances.
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