Construction of new self-dual codes and quantum codes and their connections

Coding theory has been a fascinating topic since the 1940's. It was started by Claude Shannon's paper “A Mathematical Theory of Communication.” Coding theory deals with source encoding and error-correcting coding. Source encoding is concerned with how to encode messages with high efficiency, on the other hand, error-correcting coding is about codes which are used to correct or detect errors over a noisy channel. In the latter case fast decoding algorithms are required.; My thesis deals with error-correcting codes. In particular I am interested in classical self-dual codes over finite fields and additive quantum error-correcting codes over GF(4) as the latter codes can be constructed using techniques from classical coding theory.; In this thesis, we construct several new binary or quaternary extremal self-dual codes using a new construction method which I call the building-up method. We also construct new quantum error-correcting codes by generalizing the building-up construction. Besides the costruction of good codes, we demonstrate a hand decoding of the binary Reed-Muller code of length 32 and dimension 16 by projecting it onto the binary Hamming code of length 8 and dimension 4 over GF(4). We study combinatorial designs such as classical t-designs and generalized t-designs in self-dual quantum error-correcting codes. We propose some open problems on codes at the end.