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 $C_2^\perp$. Moreover, we give an explicit description of a generator matrix of $C_2^\perp$ 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.

翻译：本文旨在利用标准单项式和指示函数研究求值码的对偶与代数对偶。我们证明求值码的对偶是代数对偶的求值码，并开发了一种计算代数对偶基的算法。设$C_1$和$C_2$为由标准单项式张成的线性码，我们给出了$C_1$与对偶码$C_2^\perp$单项式等价的组合条件。此外，我们基于$C_1$的生成矩阵及指示函数系数，显式描述了$C_2^\perp$的生成矩阵。对于Reed-Muller型码，我们利用消逝理想的v-数与Hilbert函数给出了对偶性判据。作为应用，我们得到了对应于Gorenstein理想的Reed-Muller型码的显式对偶。另外，当求值码为单项式码且求值点集为退化仿射空间时，我们分类了其对偶码为单项式码的情形。