New Quasi-cyclic and Quasi-twisted Codes and an Optimal Family of Polynomial Codes
Ray-Chaudhuri, Dijen K.
Algebraic Coding Theory uses mathematical constructs to improve reliability of communication through a noisy channel. Error-correcting codes have a wide spectrum of practical applications ranging from satellite communication to quality of sound in compact disks.
It was shown by Shannon in 1948 that efficient codes exist, but his argument was existential, not constructive. It is an important problem in coding theory to explicitly construct efficient codes which have sufficient structure to facilitate encoding and decoding. A class of codes known as quasi-cyclic (QC) codes have been shown to be promising toward solving this problem. On the one hand, they contain asymptotically good sequences of codes, while on the other hand, many best-known codes are of this type. In this dissertation we find new codes which have parameters better than the previously best-known ones, those of QC type. Moreover, we investigate the algebraic structure of a larger class of codes, called quasi-twisted (QT) codes. Making use of their structures, we develop search methods to look for new codes of this type. A large number of new codes which are QC or QT have been discovered.
Another important research area in algebraic coding theory is the study of codes over finite rings. We also study QC and QT codes over integer residue rings, especially over Z4. Using the usual Gray map we find a number of (nonlinear) binary codes with good parameters. One of these codes, obtained by lifting the generator polynomial of the famous binary Golay code, turns out to have the best parameters among all known binary codes. It is a (92,224,28)2-code.
In the final chapter, we introduce algebraic methods of constructing
families of polynomial codes over finite fields which make use of elements of
an extension field. We show that these classes of codes contain an infinite
family of optimal codes meeting the Griesmer bound as well as several families
of near optimal codes. We call a code near optimal if its minimum distance is
at most one unit less than that of an optimal one.