Date on Master's Thesis/Doctoral Dissertation
5-2012
Document Type
Doctoral Dissertation
Degree Name
Ph. D.
Department
Mathematics
Committee Chair
Kim, Jon-Lark
Author's Keywords
Error-correcting codes; Coding theory; Self-dual codes; Optimum distance profile; Complementary information set; Network coding theory
Subject
Coding theory; Error-correcting codes (Information theory)
Abstract
The classification of self-dual codes has been an extremely active area in coding theory since 1972 [33]. 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 [8,4,4] code, the extended Golay [24,12,8] code, and the extended quadratic residue [48,24,12] 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 [48,24,12] 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 [7], 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.
Recommended Citation
Freibert, Finley James, "Self-dual codes, subcode structures, and applications." (2012). Electronic Theses and Dissertations. Paper 457.
https://doi.org/10.18297/etd/457
