The dimension of subfield-subcodes of a subclass of generalized Reed-Solomon codes
Kyle D Marshall · California State University ScholarWorks (system-wide DSpace) · 2012
A fundamental problem in coding theory is to find codes with good parameters over a fixed field. Of particular interest is the case when the field is the prime subfield of a finite field, in which case many of the best known codes arise as subfield-subcodes of codes defined over larger fields. In general, the dimension of a subfield-subcode is non-trivial to compute. This thesis presents a particular class of subfield-subcodes of generalized Reed-Solomon codes and derives a lower bound for the true dimension of the subfield-subcodes of this class. The codes are the subfield-subcodes of shortened generalized Reed-Solomon codes with a particular twist. In order to understand these codes, an alternative but equivalent definition of generalized Reed-Solomon codes is introduced. Results from computational experiments are presented, showing that this family of codes contains many best known codes over F_, F_, and F_