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. The trace gives a method to go down from a code defined over an extension field to a code defined over the ground field. It is obvious that trace codes and subfield codes are linear codes over the ground field Fq, and it is equally obvious how to control the minimum distance of the subfield code. The minimum distance of the subfield code of is at least as large as the minimum distance of itself. The relationship between trace code and subfield code is clarified by a famous theorem due to Delsarte, which is the main result of the present section. There is a relation of duality.
