A computational investigation of Goppa codes.
Philip McVey Yoder · 1976
In recent years, V. D. Goppa has defined a new class of linear codes commonly referred to in the literature as Goppa codes. This thesis briefly discusses this new class of linear error-correcting codes and describes a system of computer subroutines designed for the study of particular Goppa codes. In the development of these coding theory subroutines, much emphasis was placed on the design of the minimum distance calculation subroutine. A technique is presented which results in reduced computer execution time ov^er the amount of time that would be required if the straightforward approach to determining minimum distance were applied directly. Another minimum distance algorithm, not actually implemented but which could show much promise, is discussed in Appendix B. The coding theory subroutines were used to study Goppa codes with location sets over a variety of Galois fields. The location sets of the codes studied consist of all elements of the same order from a Galois field, while the generator polynomials consist of a single root which can be repeated a number of times. It was found that the order of the root seems to determine the attributes of the resulting code. It was shown that some Galois fields of the form 2 GF(2 ) may be partitioned by element order into sets which each have a number of elements equivalent to a power of 2. Some codes from these types of fields were found to be Reed-