Some applications of mathematics to coding theory
This thesis deals with the transmission of data over a channel that is subject to noise, or interference. There are many different methods of trying to achieve reliable communication of data in the presence of noise. This thesis considers some of these methods, in particular, those aspects involving the use of error-correcting codes. A number of specific applications are considered, as well as some more general theory. One general class of codes is that of cyclic codes (where every cyclic shift of a codeword is also a codeword). Chapter 2 of this thesis reviews a decoding scheme for cyclic codes proposed by Professor P.M. Cohn. The scheme is a modification of standard array syndrome decoding. It is shown that Cohn's scheme does not perform as well as the original version of syndrome decoding. Chapter 3 considers Cyclotomically Shortened Reed Solomon codes (a class of codes introduced by J.L. Dornstetter) and their relationship with the Chinese Remainder Theorem codes of J.J. Stone. The blocklength and dimension of these codes is established, together with the best possible lower bound on the minimum distance. The notion of cyclotomic shortening is then extended to Alternant codes. Chapter 4 deals with the subject of interleaving for channels that are subject to bursts of errors. An optimal solution is given to a problem posed by Inmarsat when interleaving is used with a convolutional code. It is shown how to improve the method of interleaving which feeds data column-wise into an array and then transmits row-wise, by careful selection of the order in which the rows are transmitted. The final chapter discusses the concept of an error-correcting code with two different codeword lengths. Some general results about such codes are presented. A method of forming these codes is given for the case when one word length is twice the other. A specific example of this type of code is considered. Both theoretical and simulated performance results are presented for the example.