Defining Set

Let be a Cyclic Code of length (where is odd)
Let be a field extension of
A defining set of is a set with the property:

Note that necessarily for nonempty .
Also need not be unique

We say a Cyclic Code is defined by if:

Remark

We usually work with the Splitting Field of
(which can be shown to be where )

This wasn’t done in lectures like this

See Cyclic Code with Defining Set