## Student: Jon Hamkins

## Advisor: Ken Zeger

## School: University of Illinois, Urbana-Champaign

Date of PhD: October 1996

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 analogous 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 superior 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