Codes and Curves
This paper gives an example-driven overview of algebraic-geometry codes, with
attention confined to bivariate Goppa codes. J. Walker's _Codes and
Curves_ uses notation that is standard for graduate mathematics, but
unfortunately does not discuss decoding; O. Pretzel's _Codes and Algebraic
Curves_ gives a full discussion, but using non-standard notation. The current
work is a synthesis of the two, extending Walker's work by including a
discussion of the Skorobogatov-Vladut (SV) decoding algorithm.
First, notation for finite fields is given; then, the engineering problem is
defined. Selected concepts from algebraic geometry are introduced, illustrated
by examples. The construction and encoding of Goppa codes is described,
followed by an exposition of the SV decoding algorithm. Several worked
examples are shown. Software implementations of the encoder and decoder are
discussed, followed by performance analysis and comparison with other codes.
Finally, directions for further research are sketched.
John Kerl
Master's thesis defense
April 20, 2005