Mathematics--Research

Properties of Goppa codes are studied. These are
"good" codes in the sense that they asymptotically
approach the Varshamov-Gilbert bound. E. N. Gilbert and
R. R. Varshamov have shown (independently) that it is
possible to construct an (n, k) linear code over GF(q)
with minimum distance d if [equation] and there are long Goppa codes which achieve this bound [10]. Subclasses of Goppa codes which remain invariant under symmetries are given special attention.

In this thesis we give a self-contained exposition
of the group-theoretic proofs of the Burnside p^a g^b theorem. The Burnside p^a g^b theorem states that all
groups of order p^a g^b are solvable, where p and q are primes. The proof was suggested by Thompson,
and published by Goldschmidt, Bender, and Matsuyama.