The dual of an evaluation code
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The aim of this work is to study the dual and the algebraic dual of an evaluation code using standard monomials and indicator functions. We show that the dual of an evaluation code is the evaluation code of the algebraic dual. We develop an algorithm for computing a basis for the algebraic dual. Let C_1 and C_2 be linear codes spanned by standard monomials. We give a combinatorial condition for the monomial equivalence of C_1 and the dual C_2^⊥. Moreover, we give an explicit description of a generator matrix of C_2^⊥ in terms of that of C_1 and coefficients of indicator functions. For Reed–Muller-type codes we give a duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide an explicit duality for Reed–Muller-type codes corresponding to Gorenstein ideals. In addition, when the evaluation code is monomial and the set of evaluation points is a degenerate affine space, we classify when the dual is a monomial code.
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