Subfield Subcodes and Trace Codes

A very useful method of constructing codes over IF _q is to restrict codes which are defined over an extension field IF _qm . This means that, given a code C ⊆ (IF _qm )ⁿ, one considers the subfield subcode C∣_IFq = C∩IFⁿ _q . Many well-known codes can be defined in this way for instance BCH codes, Goppa codes and, more generally, alternant codes (cf. Section 2.3).

There is yet another method of defining a code over IF _q if a code over IF _qm is given. This construction uses the trace mapping Tr : IF _qm → IF _q . An important class of codes which can be represented as trace codes in a natural manner is the class of cyclic codes (see Example 9.2.4 below). The subfield subcode construction and the trace construction are closely related by Delsarte's Theorem 9.1.2.