On the next-to-minimal weight of affine Cartesian codes
type: Item
instance of: scholarly article
title: On the next-to-minimal weight of affine Cartesian codes
zbMATH Open document ID: 1354.14040
author: Victor G. L. Neumann
author: Cicero Carvalho
published in: Finite Fields and their Applications
publication date: 2016-12-27T00:00:00Z
review text: This paper is interested in the value of the next-to-minimal weight of affine Cartesian codes. The next-to-minimal weight of an error-correcting code \(\mathcal{C}\)\, is the second lowest Hamming weight of the non-zero codewords of \(\mathcal{C}\). An affine Cartesian code \(\mathcal{C}_{\chi}(d)\)\, is the image of the evaluation map from polynomials of degree up to \(d\)\, of \(\mathbb{F}_q[X_1, \dots,X_n]\)\, on the points of a Cartesian set \(\chi=A_1\times \dots \times A_n \subseteq \mathbb{F}_q^n\). When \(A_i=\mathbb{F}_q,\,\, i=1,\dots,n\)\, we have the generalized Reed-Muller codes. The minimal weight of \(\mathcal{C}_{\chi}(d)\)\, was already known, see [\textit{H. H. López} et al., Des. Codes Cryptography 71, No. 1, 5--19 (2014; Zbl 1312.94118)], result quoted here as Theorem 1.2. The present paper deepens in the study of the next-to-minimal weight of those codes. The paper assumes that \(d_i=\sharp(A_i)\), \(3\leq d_1\leq \dots \leq d_n\), \(d\geq d_1 -1\), \(n\geq 3\). Section 2 finds, considering the restriction to some particular \(n\)-tuples of \(\chi\), an upper bound for the next-to-minimal weight of \(\mathcal{C}_{\chi}(d)\) (Theorem 2.4) and Section 3, using methods from Gröbner basis, determines a lower bound (Proposition 3.9). These two bounds allow then to know the exact value of the next-to-minimal weight of \(\mathcal{C}_{\chi}(d)\) codes in some particular cases (Theorem 3.10), generalizing similar results of \textit{R. Rolland} for generalized Reed-Muller codes [Cryptogr. Commun. 2, No. 1, 19--40 (2010; Zbl 1192.94107)].
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 94B60
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 6668667
zbMATH Keywords: next-to-minimal Hamming weight
zbMATH Keywords: affine Cartesian codes
zbMATH Keywords: Reed-Muller type codes
zbMATH Keywords: Gröbner bases
reviewed by: Juan Tena Ayuso
MaRDI profile type: Publication
full work available at URL: j.ffa.2016.11.005
OpenAlex ID: W2559713965
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DOI: 10.1016/J.FFA.2016.11.005
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description: scientific article; zbMATH DE number 6668667