Self-dual codes, subcode structures, and applications.

摘要

The classification of self-dual codes has been an extremely active area in coding theory since 1972. A particularly interesting class of self-dual codes is those of Type II which have high minimum distance (called extremal or near-extremal). It is notable that this class of codes contains famous unique codes: the extended Hamming code, the extended Golay code, and the extended quadratic residue code. We examine the subcode structures of Type II codes for lengths up to 24, extremal Type II codes of length 32, and give partial results on the extended quadratic residue code. We also develop a generalization of self-dual codes to Network Coding Theory and give some results on existence of self-dual network codes with largest minimum distance for lengths up to 10. Complementary Information Set (CIS for short) codes, a class of classical codes recently developed in, have important applications to Cryptography. CIS codes contain self-dual codes as a subclass. We give a new classification result for CIS codes of length 14 and a partial result for length 16.