Title: An Introduction to Error-Correcting Codes
Abstract: Error-correcting codes are widely used to correct errors in
either the transmission or storage of information. An error-correcting code
provides the high fidelity on compact discs. Because of their demonstrated
practical usefulness, electrical engineers started studying these codes
about fifty years ago. Now they are studied by engineers, mathematicians
and computer scientists and a wide theory has been developed with many
connections to mathematical topics.
I will give all the basic definitions with examples and main problems in
error-correcting codes. We will discuss Hamming decoding and perfect codes.
We will use this to determine "what color is my hat".
Self-dual codes are an interesting class of codes which include
many very good codes. These codes are also related to interesting
designs, interesting groups and lattices. The extended Golay codes
over GF(2) and GF(3) are self-dual as is the [8,4,4] binary Hamming
code and the [6,3,4] Hexacode over GF(4). The weight enumerators
of these codes provide an upper bound on their minimum weights.
Self-dual codes over GF(2), GF(3),and GF(4) were classified from
the early seventies until the early eighties. We describe new
classifications of codes over Z4, formally self-dual codes and additive
GF(4) codes (for quantum computing).