  We consider generalizations of Reed-Muller codes, toric codes, and codes from
certain plane curves, such as those defined by norm and trace functions on
finite fields. In each case we are interested in codes defined by evaluating
arbitrary subsets of monomials, and in identifying when the dual codes are also
obtained by evaluating monomials. We then move to the context of order domain
theory, in which the subsets of monomials can be chosen to optimize decoding
performance using the Berlekamp-Massey-Sakata algorithm with majority voting.
We show that for the codes under consideration these subsets are well-behaved
and the dual codes are also defined by monomials.
