论文标题
评估代码的双重
The dual of an evaluation code
论文作者
论文摘要
这项工作的目的是使用标准单元和指标函数研究评估代码的双重和代数双重。我们表明，评估代码的双重是代数双重的评估代码。我们开发了用于计算代数双重基础的算法。令$ c_1 $和$ c_2 $是标准单元跨越线性代码。我们给出了$ C_1 $和双$ C_2^\ perp $的单一等价性的组合条件。此外，我们对$ C_2^\ perp $的发电机矩阵的明确说明就$ C_1 $的量子和指示功能的系数表示。对于芦苇 - 毛刺型代码，我们就V-M-number和Hilbert函数的二元性标准提供了二元性标准。作为应用程序，我们为与Gorenstein理想相对应的芦苇型代码提供了明确的双重性。另外，当评估代码是单一的，评估点的集合是一个退化的仿射空间时，我们在二元为单一代码时进行分类。
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.
