The dual of an evaluation code

• Published in: Designs, codes, and cryptography Vol. 89; no. 7; pp. 1367 - 1403
• Main Authors: López, Hiram H., Soprunov, Ivan, Villarreal, Rafael H.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2021; Springer Nature B.V
• Subjects: Algebra; Algorithms; Coding and Information Theory; Combinatorial analysis; Computer Science; Cryptology; Discrete Mathematics in Computer Science; Evaluation; Mathematical analysis; Finite field; Standard monomials; Affine torus; Secondary 14G50; 11T71; Evaluation codes; Degree; Indicator functions; Minimum distance; Toric codes; 94B27; Reed–Muller codes; Dual codes; Primary 13P25
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-021-00872-w

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.