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The dual of an evaluation code

Hiram H. López, Ivan Soprunov, Rafael H. Villarreal

TL;DR

The paper develops a comprehensive framework for understanding the duals of evaluation codes by linking the classical evaluation map to the algebraic dual inside the footprint basis. It proves that the dual of an evaluation code is the evaluation code of the algebraic dual, and it provides a practical Gaussian-elimination-based algorithm to compute bases for these dual spaces. Key contributions include a duality criterion for standard monomial codes, a Reed–Muller-type duality criterion tied to Hilbert function symmetry and v-numbers, and explicit descriptions of algebraic duals in monomial and degenerate torus/affine-space settings. The results unify dualities across toric, Cartesian, and degenerate geometric contexts, yield conditions under which duals remain monomial codes, and enable explicit self-dual constructions under suitable hypotheses, with Macaulay2 implementations to facilitate computation. These advances advance the design and analysis of evaluation codes by providing concrete duality formulas, algorithmic tools, and concrete duality instances in algebraic geometry–coding theory.

Abstract

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.

The dual of an evaluation code

TL;DR

The paper develops a comprehensive framework for understanding the duals of evaluation codes by linking the classical evaluation map to the algebraic dual inside the footprint basis. It proves that the dual of an evaluation code is the evaluation code of the algebraic dual, and it provides a practical Gaussian-elimination-based algorithm to compute bases for these dual spaces. Key contributions include a duality criterion for standard monomial codes, a Reed–Muller-type duality criterion tied to Hilbert function symmetry and v-numbers, and explicit descriptions of algebraic duals in monomial and degenerate torus/affine-space settings. The results unify dualities across toric, Cartesian, and degenerate geometric contexts, yield conditions under which duals remain monomial codes, and enable explicit self-dual constructions under suitable hypotheses, with Macaulay2 implementations to facilitate computation. These advances advance the design and analysis of evaluation codes by providing concrete duality formulas, algorithmic tools, and concrete duality instances in algebraic geometry–coding theory.

Abstract

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 and be linear codes spanned by standard monomials. We give a combinatorial condition for the monomial equivalence of and the dual . Moreover, we give an explicit description of a generator matrix of in terms of that of 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.

Paper Structure

This paper contains 12 sections, 48 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Hilbert functions and vanishing ideals
  3. The dual of evaluation codes
  4. Computing a basis
  5. Indicator functions and v-numbers of vanishing ideals
  6. Duality of standard monomial codes
  7. A duality criterion for Reed--Muller-type codes
  8. The algebraic dual of monomial evaluation codes
  9. Monomial evaluation codes on a degenerate torus
  10. Monomial evaluation codes on a degenerate affine space
  11. Examples
  12. Procedures for Macaulay$2$

Key Result

Theorem 2.1

monalg-rev If $I$ is an ideal of $S$ and $I=\mathfrak{q}_1\bigcap\cdots\bigcap\mathfrak{q}_m$ is an irredundant primary decomposition with ${\rm rad}(\mathfrak{q}_i)=\mathfrak{p}_i$, then the set of zero-divisors $\mathcal{Z}_S(S/I)$ of $S/I$ is equal to $\bigcup_{i=1}^m\mathfrak{p}_i$, and $\mathfr

Theorems & Definitions (111)

  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 101 more