Duality for Several Families of Evaluation Codes

Abstract

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.