The dual of an evaluation code

TL;DR

This paper investigates the duals of evaluation codes, providing algebraic characterizations, algorithms for basis computation, and explicit duality descriptions, especially for Reed--Muller-type codes and monomial evaluation codes.

Contribution

It establishes the dual of an evaluation code as the evaluation of the algebraic dual, introduces an algorithm for basis computation, and characterizes duality for Reed--Muller-type and monomial codes.

Findings

Dual of an evaluation code is the evaluation of the algebraic dual.

Provides an explicit basis computation algorithm for the algebraic dual.

Characterizes duality for Reed--Muller-type and monomial evaluation codes.

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.