ABSTRACT
A general construction idea for good codes over some field Fq is to first construct a code over a larger field Fqr and then to go down in some way from the large field to the small field. The most fruitful methods of "going down" are the trace codes and the subfield codes. Some call this the freshman's dream. It means that F respects both the multiplicative and the additive field structure. It is clear that F is a one-to-one mapping. Our main objective is a study of subfield codes and trace codes of Reed-Solomon codes and codes obtained from puncturing Reed-Solomon codes. When will we consider two codes as essentially the same? Such codes will be called equivalent or also isomorphic. This question is closely related to the automorphism group of a code. Surprisingly there are several different notions of equivalence. The most popular one uses only permutations of coordinates.
